Superstrings, Geometry, Topology, and C*-algebras 9780821848876, 0821848879

Table of contents :
Contents
Preface
Conference Attendees
Conference Speakers
Introduction
Functoriality of Rieffel’s Generalised Fixed-Point Algebras for Proper Actions
Twists of K-theory and TMF
Division Algebras and Supersymmetry I
K-homology and D-branes
Riemann-Roch and Index Formulae in Twisted K-theory
Noncommutative Principal Torus Bundles via Parametrised Strict Deformation Quantization
A Survey of Noncommutative Yang-Mills Theory for Quantum Heisenberg Manifolds
From Rational Homotopy to K-Theory for Continuous Trace Algebras
Distances between Matrix Algebras that Converge to Coadjoint Orbits
Geometric and Topological Structures Related to M-branes
Landau-Ginzburg Models, Gerbes, and Kuznetsov’s Homological Projective Duality

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Superstrings, Geometry, Topology, and C*-algebras

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Proceedings of Symposia in

PURE MATHEMATICS Volume 81

Superstrings, Geometry, Topology, and C*-algebras

Robert S. Doran Greg Friedman Jonathan Rosenberg Editors

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American Mathematical Society Providence, Rhode Island

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NSF-CBMS REGIONAL CONFERENCE ON MATHEMATICS ON TOPOLOGY, C ∗ -ALGEBRAS, AND STRING DUALITY, HELD AT TEXAS CHRISTIAN UNIVERSITY, FORT WORTH, TEXAS, MAY 18–22, 2009 with support from the National Science Foundation, Grant DMS-0735233 2000 Mathematics Subject Classification. Primary 81–06, 55–06, 46–06, 46L87, 81T30.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Library of Congress Cataloging-in-Publication Data NSF-CBMS Conference on Topology, C ∗ -algebras, and String Duality (2009 : Texas Christian University) Superstrings, geometry, topology, and C ∗ -algebras : NSF-CBMS Conference on Topology, C ∗ algebras, and String Duality, May 18–22, 2009, Texas Christian University, Fort Worth, Texas / Robert S. Doran, Greg Friedman, Jonathan Rosenberg, editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 81) Includes bibliographical references and index. ISBN 978-0-8218-4887-6 (alk. paper) 1. Algebraic topology—Congresses. 2. Quantum theory—Congresses. 3. Functional analysis— Congresses. I. Doran, Robert S., 1937– II. Friedman, Greg, 1973– III. Rosenberg, J.(Jonathan), 1951– IV. Conference Board of the Mathematical Sciences. V. National Science Foundation (U.S.) VI. Title. QA612.N74 2009 530.12—dc22 2010027233

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Contents Preface

vii

Conference Attendees

xi

Conference Speakers

xiii

Introduction Jonathan Rosenberg

1

Functoriality of Rieffel’s Generalised Fixed-Point Algebras for Proper Actions Astrid an Huef, Iain Raeburn, and Dana P. Williams

9

Twists of K-theory and T M F Matthew Ando, Andrew J. Blumberg, and David Gepner

27

Division Algebras and Supersymmetry I John C. Baez and John Huerta

65

K-homology and D-branes Paul Baum

81

Riemann-Roch and Index Formulae in Twisted K-theory Alan L. Carey and Bai-Ling Wang

95

Noncommutative Principal Torus Bundles via Parametrised Strict Deformation Quantization Keith C. Hannabuss and Varghese Mathai 133 A Survey of Noncommutative Yang-Mills Theory for Quantum Heisenberg Manifolds Sooran Kang

149

From Rational Homotopy to K-Theory for Continuous Trace Algebras John R. Klein, Claude L. Schochet, and Samuel B. Smith

165

Distances between Matrix Algebras that Converge to Coadjoint Orbits Marc A. Rieffel

173

Geometric and Topological Structures Related to M-branes Hisham Sati

181

Landau-Ginzburg Models, Gerbes, and Kuznetsov’s Homological Projective Duality Eric Sharpe

237

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Preface The Conference Board of the Mathematical Sciences (CBMS) hosted a regional conference, funded by the National Science Foundation, during the week of May 18–22, 2009, entitled Topology, C*-algebras, and String Duality at Texas Christian University in Fort Worth, Texas. The principal lecturer was Jonathan Rosenberg of the University of Maryland, whose conference lectures have been published in Volume 111 of the CBMS’s Regional Conference Series in Mathematics. In addition to Professor Rosenberg’s lectures, the conference featured talks by fifteen other speakers on topics related to his lectures and the general theme of the conference. The purpose of this volume is to collect the contributions of these speakers and other participants. All papers have been carefully refereed and will not appear elsewhere. At first sight these papers, which are highly interdisciplinary, may appear unrelated. To provide direction and historical context for the reader, a technical introduction describing how the various papers fit together in a natural way has been written by Professor Rosenberg. It appears as the first article in the volume. The editors express their sincere gratitude and thanks to the speakers for their beautiful talks and their willingness to spend many hours writing them up so that the results would be available to the larger scientific community. In addition, we acknowledge the hard work and help of the referees. We thank the Conference Board of the Mathematical Sciences and the National Science Foundation for their support via NSF Grant DMS-0735233. We thank Sergei Gelfand, Christine Thivierge, and the dedicated staff at the American Mathematical Society for their efforts in publishing these proceedings. Finally, we thank Texas Christian University and all the participants who helped ensure a wonderfully successful conference. Robert S. Doran Greg Friedman Jonathan Rosenberg

vii

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1. Greg Friedman 2. Jonathan Rosenberg 3. Robert Doran 4. Shilin Yu 5. Matthew Ando 6. Huichi Huang 7. Varghese Mathai 8. Efton Park 9. Ruth Gornet 10. Rishni Ratnam 11. Stefan Mendez-Diez 12. Sooran Kang 13. Pedram Hekmati 14. Jacques Distler 15. Phu Chung 16. Seunghun Hong 17. Valentin Deaconu 18. Dorin Dumitrascu 19. Alan Carey 20. Rebecca Chen 21. Marc A. Rieffel 22. Letty Reza 23. Mart Abel 24. Hisham Sati 25. Jon Sjogren 26. Eric Sharpe 27. Ken Richardson 28. John Skukalek 29. Michael Tseng 30. Jody Trout 31. Peter Bouwknegt 32. Wang Qingyun 33. Nigel Higson 34. Daniel Freed 35. Mark Tomforde 36. Magnus Goffeng 37. Claude Schochet 38. Bruce Doran 39. John Huerta 40. Jacob Shotwell 41. Daniel Pape 42. James West 43. Jonathan Block 44. Loredana Ciurdariu 45. Loren Spice 46. Anna Spice 47. Dana Williams 48. Braxton Collier Not pictured: Paul Baum, Alexander A. Katz, Snigdhayan Mahanta, Scott Nollet

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Conference Attendees Mart Abel University of Tartu

Daniel Freed University of Texas at Austin

Matthew Ando University of Illinois

Greg Friedman Texas Christian University

Paul Baum Pennsylvania State University

Magnus Goffeng Chalmers University of Technology and University of Gothenburg

Jonathan Block Pennsylvania State University

Ruth Gornet University of Texas at Arlington

Peter Bouwknegt Australian National University

Pedram Hekmati Royal Institute of Technology

Alan Carey Australian National University

Nigel Higson Pennsylvania State University

Rebecca Chen University of Houston

Seunghun Hong Pennsylvania State University

Phu Chung University at Buffalo

Huichi Huang University at Buffalo

Loredana Ciurdariu University Politechnic of Timisoara

John Huerta University of California, Riverside

Braxton Collier University of Texas at Austin

Sooran Kang University of Colorado at Boulder

Valentin Deaconu University of Nevada, Reno

Alexander A. Katz St. John’s University

Jacques Distler University of Texas at Austin

Snigdhayan Mahanta Johns Hopkins University

Robert Doran Texas Christian University

Varghese Mathai University of Adelaide

Bruce Doran Accenture

Stefan Mendez-Diez University of Maryland

Dorin Dumitrascu Northern Arizona University

Scott Nollet Texas Christian University xi

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xii

PARTICIPANTS

Daniel Pape Mathematisches Intitut G¨ ottingen

James West University of Houston

Efton Park Texas Christian University

Dana Williams Dartmouth College

Wang Qingyun Washington University in St. Louis

Shilin Yu Pennsylvania State University

Rishni Ratnam Australian National University Letty Reza University of Houston Ken Richardson Texas Christian University Marc A. Rieffel University of California, Berkeley Jonathan Rosenberg University of Maryland Hisham Sati Yale University Claude Schochet Wayne State University Eric Sharpe Virginia Tech Jacob Shotwell Arizona State University Jon Sjogren Air Force Office of Scientific Research John Skukalek Pennsylvania State University Anna Spice University of Michigan Loren Spice University of Michigan Mark Tomforde University of Houston Jody Trout Dartmouth College Michael Tseng Pennsylvania State University

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Conference Speakers Jonathan Rosenberg Topology, C ∗ -algebras, and String Duality

Marc Rieffel Vector Bundles for “Matrix Algebras Converge to the Sphere”

Matthew Ando Twisted Generalized Cohomology and Twisted Elliptic Cohomology

Hisham Sati Fivebrane Structures in String Theory and M-theory

Paul Baum Equivariant K Homology

Claude Schochet An Update on the Unitary Group Eric Sharpe GLSMs, Gerbes, and Kuznetsov’s Homological Projective Duality

Jonathan Block Homological Mirror Symmetry and Noncommutative Geometry

Dana Williams Proper Actions on C*-algebras

Peter Bouwknegt The Geometry Behind Non-geometric Fluxes Alan Carey Twisted Geometric Cycles Jacques Distler Geometry and Topology of Orientifolds I Dan Freed Geometry and Topology of Orientifolds II Nigel Higson The Baum-Connes Conjecture and Parametrization of Representations Sooran Kang The Yang-Mills Functional and Laplace’s Equation on Quantum Heisenberg manifolds Varghese Mathai The Index of Projective Families of Elliptic Operators

xiii

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Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Introduction Jonathan Rosenberg Abstract. The papers in this volume are the outgrowth of an NSF-CBMS Regional Conference in the Mathematical Sciences, May 18–22, 2009, organized by Robert Doran and Greg Friedman at Texas Christian University. This introduction explains the scientific rationale for the conference and some of the common themes in the papers.

During the week of May 18–22, 2009, Robert Doran and Greg Friedman organized a wonderfully successful NSF-CBMS Regional Research Conference at Texas Christian University. I was the primary lecturer, and my lectures have now been published in [29]. However, Doran and Friedman also invited many other mathematicians and physicists to speak on topics related to my lectures. The papers in this volume are the outgrowth of their talks. The subject of my lectures, and the general theme of the conference, was highly interdisciplinary, and had to do with the confluence of superstring theory, algebraic topology, and C ∗ -algebras. While with “20/20 hindsight” it seems clear that these subjects fit together in a natural way, the connections between them developed almost by accident. Part of the history of these connections is explained in the introductions to [11] and [17]. The authors of [11] begin as follows: Until recently the interplay between physics and mathematics followed a familiar pattern: physics provides problems and mathematics provides solutions to these problems. Of course at times this relationship has led to the development of new mathematics. . . . But physicists did not traditionally attack problems of pure mathematics. This situation has drastically changed during the last 15 years. Physicists have formulated a number of striking conjectures (such as the existence of mirror symmetry) . . . . The basis of the physicists’ intuition is their belief that underlying quantum field theory 2010 Mathematics Subject Classification. Primary 81-06; Secondary 55-06, 46-06, 46L87, 81T30. Key words and phrases. string theory, supersymmetry, D-brane, C ∗ -algebra, crossed product, topological K-theory, twisted K-theory, classifying space, noncommutative geometry, LandauGinzburg theory, Yang-Mills theory. Partially supported by NSF grant DMS-0805003. c Mathematical 0000 (copyright Society holder) c 2010 American

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2

JONATHAN ROSENBERG

and string theory is a (as yet undiscovered) self-consistent mathematical framework. Of course this was written over 10 years ago. In the last 10 years, the same principle has been borne out time and again. As far as the subject matter of this volume is concerned, there are a few key developments from the last 35 years that one can point to, that played an essential role: (1) The Baum-Douglas [2, 3] and Kasparov [19, 20] approaches to (respectively) topological and analytic K-homology, and the realization that these theories are naturally isomorphic. (2) The “Second Superstring Revolution” around 1995. Geometric objects, known as D-branes, were shown to play a fundamental role in string theory, and as time went on, it was realized that they naturally carry vector bundles and topological charges (see for example [23, 31, 22, 32]), living in K-theory or K-homology (or still more complicated generalized homology theories). (3) The development of Connes’ theory of “noncommutative differential geometry,” epitomized by the book [9], and the gradual acceptance of noncommutative geometry as a natural tool in quantum field theories. (4) The invention of “twisted K-theory,” and the realization that it has a natural realization in terms of continuous trace C ∗ -algebras (see [28, 1, 18]). My own interest in combining string duality with topology and noncommutative geometry followed a rather circuitous route. A classical theorem of Grothendieck and Serre [15] computed the Brauer group Br C(X) for X a finite complex, and found that it is isomorphic to the torsion subgroup of H 3 (X, Z). In the 1970’s, Phil Green [14] worked out a more general theory of the Brauer group of C0 (X), for X a locally compact Hausdorff space. Green had the idea to drop all technical conditions on X and to allow continuous-trace algebras with infinite fiber dimension, not just classical Azumaya algebras, so as to get an isomorphism of the Brauer group Br C0 (X) with all of H 3 (X, Z), not just with its torsion subgroup. (When X is a finite complex, it doesn’t matter what cohomology theory one uses, but for ˇ general locally compact spaces, Cech cohomology is appropriate here.) Now it so happens that Donovan and Karoubi [12] had used classical Azumaya algebras to define twisted K-theory with torsion twistings, so Green’s idea of using more general continuous-trace algebras to replace Azumaya algebras made possible defining twisted K-theory of X with arbitrary twistings from H 3 (X, Z). In [27, §6] I pointed this out and explained how to generalize the Atiyah-Hirzebruch spectral sequence to make this twisted K-theory somewhat computable. But for the most part, the idea just sat around for a while since nobody had any immediate use for it. A number of years later, Raeburn and I [24] happened to study crossed products of continuous trace algebras by smooth actions, and we discovered the following interesting “reciprocity law” [24, Theorem 4.12]: Theorem 1. Let p : X → Z be a principal T-bundle, where T = R/Z is the circle group. Also assume X and Z are second-countable, locally compact Hausdorff, with finite homotopy type. Let H ∈ H 3 (X, Z) and let A = CT (X, H) be the corresponding stable continuous-trace algebra with Dixmier-Douady class H. Then the free action of T = R/Z on X lifts (in a unique way, up to exterior equivalence)

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INTRODUCTION

3

 = X. The crossed to an action of R on A inducing the given action of R/Z on A product A  R is again a stable continuous-trace algebra A# = CT (X # , H # ), with p# : X # → Z again a principal T-bundle. Furthermore, the characteristic classes of p and p# are related to the Dixmier-Douady classes H and H # by p! (H) = [p# ],

(p# )! (H # ) = [p],

where p! and (p# )! are the Gysin maps of the circle bundles. At the time, Raeburn and I regarded this entirely as a curiosity, and we certainly didn’t expect any physical applications. A bit later [28], I continued my studies of continuous-trace algebras and twisted K-theory, but I still didn’t expect any physical applications. Much to my surprise, I discovered many years later that my studies of continuous-trace algebras and twisted K-theory were starting to show up in the physics literature in papers such as [7] and [4]. In fact, twisted K-theory seemed to be exactly the mathematical framework needed to studying D-brane charges in string theory. Not only that, but the “reciprocity law” of [24] for continuous-trace algebras associated to circle bundles also showed up in physics, as the recipe for topology change and H-flux change in T-duality [5, 6]. Since that time, there has been a fruitful continuing interaction between the subjects of string theory, topology, and C ∗ -algebras, an interaction that led to the organization of the CBMS conference at TCU in 2009. With this as background, I can now explain how the various papers in this volume fit together. The papers of Baum and of Carey and Wang deal with Dbrane charges in K-homology and twisted K-homology, a natural continuation of the combination of items (1), (2), and (4) on the list of key developments above (page 2). Baum’s paper deals with the extension to the twisted case of the BaumDouglas approach to topological K-homology. While Baum does not go into the associated physics, D-branes in type II string theories come with precisely the structures he is discussing, and thus produce “topological charges” in the twisted K-homology of spacetime. The paper of Carey and Wang goes into more detail on the same subject, and also discusses a Riemann-Roch theorem in twisted K-theory. Carey and Wang explain how D-brane charges in twisted K-theory arise in both type II and type I string theories. The papers of Ando and Sati deal with roughly the same theme as those of Baum and Carey-Wang, but in a somewhat generalized context. Ando explains (from the point of view of a stable homotopy theorist) how to construct twisted generalized cohomology theories in general, and then specializes to the cases of twisted K, twisted elliptic cohomology, and twisted TMF. TMF [16], topological modular forms, is a version of elliptic cohomology that seems to play an important role in M-theory, the “master” theory that gives rise (on reduction from 11 dimensions to 10) to the five superstring theories. Sati’s paper concentrates on the physics side of the same topic, and explains how the physics of M-branes (which play the same role in M-theory that the D-branes play in string theory) leads to twisted String and Fivebrane structures. (These are higher-dimensional analogues of Spin and Spinc structures.) Sati also discusses the kinds of orientation conditions that arise for branes in F-theory [30], a 12-dimensional theory that is supposed to reduce to M-theory in certain circumstances.

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4

JONATHAN ROSENBERG

Two of the papers in this volume, by Kang and by Baez and Huerta, deal with Yang-Mills gauge theory and its connection with noncommutative geometry. The  basic Yang-Mills action on a spacetime manifold M is (up to a scalar factor) − M Tr(F ∧ F ), where F is the curvature of a connection on a principal G-bundle over M . Here G is some Lie group which depends on the details of the theory; for example, in the “standard model” of particle physics it is SU (3) × SU (2) × U (1). Physicists have known for some time [8] that in some circumstances one can make this action supersymmetric, by adding in a fermionic term of the form (again, up to a constant factor) ψ, ∂ / ψ, where ψ is a spinor field and /∂ is the Dirac operator. However, this only appears to work in three, four, six and ten dimensions. The paper of Baez and Huerta gives an explanation for this fact in terms of the fact that division algebras over R only occur in dimensions 1, 2, 4, and 8 (where one has the reals, complexes, quaternions, and octonions, respectively). Kang’s paper deals with noncommutative Yang-Mills in the sense of Connes and Rieffel [10], where the basic Yang-Mills action becomes − Tr({Θ, Θ}), where Θ is the curvature 2-form for a connection on a finitely generated projective module (the natural analogue of a vector bundle) over the smooth subalgebra of some C ∗ -algebra A. Connes and Rieffel took A to be Aθ , the irrational rotation algebra generated by two unitaries U and V with U V = e2πiθ V U . Kang considers the somewhat more complicated case of the “quantum Heisenberg manifold” in the sense of Rieffel [25]; this is a deformation quantization of the algebra of functions on a Heisenberg nilmanifold. Just to relate the papers of Baez-Huerta and Kang to the rest of the volume, it is perhaps worthwhile to explain how Yang-Mills and super-Yang-Mills are related to string theory. There are two interconnected ties between the two subjects. On the one hand, as we mentioned already, D-branes naturally carry certain Chan-Paton vector bundles; on these there is a natural Yang-Mills action. In addition, there is a duality, known as the AdS/CFT correspondence, between type IIB string theory on S 5 × AdS 5 (AdS 5 is anti-de Sitter space, a 5-dimensional Lorentz manifold with a metric of constant negative curvature) and 4-dimensional super-Yang-Mills on S 4 in the large-N limit [21]. The paper of Sharpe deals with Landau-Ginzburg models, a class of models which were originally constructed to model superconductivity, but which have turned out to be extremely useful for superstring theory as well. A LandauGinzburg model in string theory describes propagation of strings on a noncompact spacetime (always a complex manifold) with a holomorphic superpotential W , often having a degenerate critical point. One of the results explained in Sharpe’s paper is that A-twisted correlation functions in the Landau-Ginzburg model on π X = Tot(E ∨ − → B), E → B a holomorphic vector bundle, with W = p · π ∗ s, p a tautological section of π ∗ E ∨ and s a holomorphic section of E, should match correlation functions in the nonlinear sigma model on {s = 0}. Since the complex geometry of the Landau-Ginzburg spacetime is usually quite different from the one which the sigma model lives, sometimes one gets interesting relations in enumerative algebraic geometry which are hard to explain directly. The papers of Hannabuss-Mathai, Reiffel, Klein-Schochet-Smith, and an HuefRaeburn-Williams all deal with various aspects of C ∗ -algebraic noncommutative geometry. Several of them also have ties to quantum physics and to topology. Rieffel’s paper gives explicit examples of sequences of matrix algebras with dimensions going to ∞ whose “proximity” in a rather precise but technical sense goes to 0.

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INTRODUCTION

5

This sort of calculation is motivated by the use of “matrix models” to approximate quantum field theories on spaces with complicated geometry. The paper of Klein-Schochet-Smith computes the rational homotopy type of the group U (A) of unitary elements in the Azumaya algebra A of sections of a bundle of matrix algebras Mn over a compact space X. This turns out to be independent of what Azumaya algebra one chooses (so that one might as well take A = C(X) ⊗ Mn ), basically because the Brauer group of C(X) is torsion, and the authors are only interested in rational information. This paper also computes the map πj (U (A)) ⊗ Q → Kj (A) ⊗ Q; this gives explicit information on the stable range for rationalized topological K-theory of X. The paper of Hannabuss and Mathai deals with Rieffel’s theory of strict deformation quantization [26] and the theory of noncommutative principal bundles due to Echterhoff, Nest, and Oyono-Oyono [13]. The main theorem of this paper is that for every such bundle with a suitable smooth structure A∞ (X), there is a principal torus bundle T → X and a corresponding ∞ strict deformation quantization σ of Cfibre (Y ) (the continuous functions on Y that ∞ ∞ are fibrewise smooth), so that A (X) ∼ (Y )σ . = Cfibre Finally, the paper by an Huef, Raeburn, and Williams talks about functoriality issues in the theories of C ∗ -crossed products and fixed-point algebras for proper actions. Issues like this come up when one tries to use C ∗ -algebraic noncommutative geometry to study the geometry of spacetime in various physical theories. We hope the diversity of the papers in this volume will give the reader some idea of the breadth and vitality of the current interplay between superstring theory, geometry/topology, and noncommutative geometry. Acknowledgments I would like to thank Robert Doran and Greg Friedman again for their excellent work in organizing the conference. In addition, all three of us would like to thank the Conference Board of the Mathematical Sciences and the National Science Foundation for their financial support. NSF Grant DMS-0735233 supported the conference, and NSF Grant DMS-0602750 supported the entire Regional Conference program. Finally, we would like to thank the American Mathematical Society for encouraging the publication of this volume in the Proceedings of Symposia in Pure Mathematics series. References [1] Michael Atiyah and Graeme Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287– 330; Engl. translation, Ukr. Math. Bull. 1 (2004), no. 3, 291–334, arxiv.org: math/0407054. MR2172633 (2006m:55017) [2] Paul Baum and Ronald G. Douglas, Index theory, bordism, and K-homology, in Operator algebras and K-theory (San Francisco, Calif., 1981), Contemp. Math., vol. 10, Amer. Math. Soc., Providence, RI, 1982, pp. 1–31. MR658506 (83f:58070) , K-homology and index theory, in Operator algebras and applications, Part I [3] (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, RI, 1982, pp. 117–173. MR679698 (84d:58075) [4] Peter Bouwknegt, Alan L. Carey, Varghese Mathai, Michael K. Murray, and Danny Stevenson, Twisted K-theory and K-theory of bundle gerbes, Comm. Math. Phys. 228 (2002), no. 1, 17–45, arxiv.org: hep-th/0106194. MR1911247 (2003g:58049) [5] Peter Bouwknegt, Jarah Evslin, and Varghese Mathai, T -duality: topology change from H-flux, Comm. Math. Phys. 249 (2004), no. 2, 383–415, arxiv.org: hep-th/0306062. MR2080959 (2005m:81235)

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[6] [7] [8] [9] [10]

[11]

[12] [13]

[14] [15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23] [24] [25] [26]

JONATHAN ROSENBERG

, Topology and H-flux of T -dual manifolds, Phys. Rev. Lett. 92 (2004), no. 18, 181601, 3, arxiv.org: hep-th/0312052. MR2116165 (2006b:81215) Peter Bouwknegt and Varghese Mathai, B-fields and twisted K-theory, J. High Energy Phys. 2000, no. 3, Paper 7, arxiv.org: hep-th/0002023. MR1756434 (2001i:81198) Lars Brink, John H. Schwarz, and J. Scherk, Supersymmetric Yang-Mills theories, Nuclear Phys. B 121 (1977), no. 1, 77–92. MR0446217 (56 #4545) Alain Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994, now available at http://www.alainconnes.org/en/downloads.php. MR1303779 (95j:46063) Alain Connes and Marc A. Rieffel, Yang-Mills for noncommutative two-tori, in Operator algebras and mathematical physics (Iowa City, Iowa, 1985), 237–266, Contemp. Math., 62, Amer. Math. Soc., Providence, RI, 1987. MR0878383 (88b:58033) Pierre Deligne, Pavel Etingof, Daniel S. Freed, David Kazhdan, John W. Morgan, and David R. Morrison (eds.), Quantum fields and strings: a course for mathematicians. Vol. 1, 2, American Mathematical Society, Providence, RI, 1999, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. MR1701618 (2000e:81010) P. Donovan and M. Karoubi, Graded Brauer groups and K-theory with local coefficients, ´ Inst. Hautes Etudes Sci. Publ. Math. No. 38 (1970), 5–25. MR0282363 (43 #8075) Siegfried Echterhoff, Ryszard Nest, and Herv´e Oyono-Oyono, Principal non-commutative torus bundles, Proc. Lond. Math. Soc. (3) 99 (2009), no. 1, 1–31, arxiv.org: 0810.0111. MR2520349 Philip Green, The Brauer group of a commutative C ∗ -algebra, handwritten manuscript, 1978. Alexander Grothendieck, Le groupe de Brauer. I. Alg` ebres d’Azumaya et interpr´ etations diverses, in S´ eminaire Bourbaki, Vol. 9, Exp. No. 290, Soc. Math. France, Paris, 1995, pp. 199– 219. (Also printed in Dix expos´ es sur la cohomologie des sch´ emas, Advanced Studies in Pure Mathematics, Vol. 3, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris 1968.) MR0244269 (39 #5586a), MR1608798 M. J. Hopkins, Algebraic topology and modular forms, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291–317, Higher Ed. Press, Beijing, 2002, arxiv.org: math.AT/0212397. MR1989190 (2004g:11032). Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry, Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI, 2003, With a preface by C. Vafa. MR2003030 (2004g:14042) Max Karoubi, Twisted K-theory, old and new, in K-Theory and Noncommutative Geometry (Valladolid, Spain, 2006), EMS Ser. Congr. Rep., European Math. Soc., Z¨ urich, 2008, pp. 117–149, arxiv.org: math.KT/0701789. MR2513335 G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 796–838; Engl. translation, Math. USSR-Izv. 9 (1975), no. 4, 751–792 (1976). MR0488027 (58 #7603) , The operator K-functor and extensions of C ∗ -algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719; Engl. translation, Math. USSR Izv. 16, 513–572 (1981). MR0582160 (81m:58075) Juan Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), no. 2, 231–252, arxiv.org: hep-th/9711200. MR1633016 (99e:81204a) Ruben Minasian and Gregory Moore, K-theory and Ramond-Ramond charge, J. High Energy Phys. 1997, no. 11, Paper 2, see updated version at arxiv.org: hep-th/9710230. MR1606278 (2000a:81190) Joseph Polchinski, Dirichlet branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995), no. 26, 4724–4727, arxiv.org: hep-th/9510017. MR1366179 (96m:81185) Iain Raeburn and Jonathan Rosenberg, Crossed products of continuous-trace C ∗ -algebras by smooth actions, Trans. Amer. Math. Soc. 305 (1988), no. 1, 1–45. MR920145 (89e:46077) Marc A. Rieffel, Deformation quantization of Heisenberg manifolds. Comm. Math. Phys. 122 (1989), no. 4, 531–562. MR1002830 (90e:46060) , Deformation quantization and operator algebras, in Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411–423, Proc. Sympos. Pure Math., 51, Part 1, Amer. Math. Soc., Providence, RI, 1990. MR1077400 (91h:46120)

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INTRODUCTION

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[27] Jonathan Rosenberg, Homological invariants of extensions of C ∗ -algebras, in Operator algebras and applications, Part 1 (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, RI, 1982, pp. 35–75. MR679694 (85h:46099) , Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math. [28] Soc. Ser. A 47 (1989), no. 3, 368–381. MR1018964 (91d:46090) , Topology, C ∗ -algebras, and string duality, CBMS Regional Conference Series in [29] Mathematics, vol. 111, American Mathematical Society, Providence, RI, 2009. MR2560910 [30] Cumrun Vafa, Evidence for F -theory, Nuclear Phys. B 469 (1996), no. 3, 403–415, arxiv.org: hep-th/960202. MR1403744 (97g:81059) [31] Edward Witten, Bound states of strings and p-branes, Nuclear Phys. B 460 (1996), no. 2, 335–350, arxiv.org: hep-th/9510135. MR1377168 (97c:81162) , D-branes and K-theory, J. High Energy Phys. 1998, no. 12, Paper 19, arxiv.org: [32] hep-th/9810188. MR1674715 (2000e:81151) Department of Mathematics, University of Maryland, College Park, MD 207424015 E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681756

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Functoriality of Rieffel’s Generalised Fixed-Point Algebras for Proper Actions Astrid an Huef, Iain Raeburn, and Dana P. Williams Abstract. We consider two categories of C ∗ -algebras; in the first, the isomorphisms are ordinary isomorphisms, and in the second, the isomorphisms are Morita equivalences. We show how these two categories, and categories of dynamical systems based on them, crop up in a variety of C ∗ -algebraic contexts. We show that Rieffel’s construction of a fixed-point algebra for a proper action can be made into functors defined on these categories, and that his Morita equivalence then gives a natural isomorphism between these functors and crossed-product functors. There are interesting applications to nonabelian duality for crossed products.

Introduction Let α be an action of a locally compact group G on a C ∗ -algebra A. In [38], Rieffel studied a class of proper actions for which there is a Morita equivalence between the reduced crossed product Aα,r G and a generalised fixed-point algebra Aα sitting inside the multiplier algebra M (A). Rieffel subsequently proved that α is proper whenever there is a free and proper G-space T and an equivariant embedding ϕ : C0 (T ) → M (A) [39, Theorem 5.7]. In [15], inspired by previous work of Kaliszewski and Quigg [13], it was observed that Rieffel’s hypothesis says precisely that ((A, α), ϕ) is an object in a comma category of dynamical systems. It therefore becomes possible to ask questions about the functoriality of Rieffel’s construction, and about the naturality of his Morita equivalence. These questions have been tackled in several recent papers [15, 7, 8], which we believe contain some very interesting results. In particular, they have substantial applications to nonabelian duality for C ∗ -algebraic dynamical systems. However, these papers also contain a confusing array of categories and functors. So our goal here is to discuss the main categories and explain why people are interested in them. We will then review some of the main results of the papers [13, 15, 7, 8], and try to explain why we find them interesting. In all the categories of interest to us, the objects are either C ∗ -algebras or dynamical systems involving actions or coactions of a fixed group on C ∗ -algebras. 2000 Mathematics Subject Classification. 46L55. This research was supported by the Australian Research Council and the Edward Shapiro fund at Dartmouth College. c Mathematical 0000 (copyright Society holder) c 2010 American

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AN HUEF, RAEBURN, AND WILLIAMS

But when we decide what morphisms to use, we have to make a choice, and what we choose depends on what sort of theorems we are interested in. Loosely speaking, we have to decide whether we want the isomorphisms in our category to be the usual isomorphisms of C ∗ -algebras, or to be Morita equivalences. We think that, once we have made that decision, there is a “correct” way to go forward. We begin in §1 with a discussion of commutative C ∗ -algebras; since Morita equivalence does not preserve commutativity, it is clear that in this case we want isomorphisms to be the usual isomorphisms. However, even then we have to do something a little odd: we want the morphisms from A to B to be homomorphisms ϕ : A → M (B). Once we have the right category, we can see that operator algebraists have been implicitly working in this category for years. The motivating example for Kaliszewski and Quigg was a duality theory for dynamical systems due to Landstad [19], and our main motivation is, as we said above, to understand Rieffel’s proper actions. We discuss Landstad duality in §2. In §3, we discuss its analogue for crossed products by coactions, which is due to Quigg [30], and how this makes contact with Rieffel’s theory of proper actions. We begin §4 by showing how the search for naturality results leads us to a different category C* of C ∗ -algebras in which the morphisms are based on right-Hilbert bimodules. Categories of this kind have been around much longer, and [2, 3], for example, contain a detailed discussion of how imprimitivity theorems provide natural isomorphisms between functors with values in C*. In §5, we discuss a theorem from [7] which says that Rieffel’s Morita equivalences give a natural isomorphism between a crossed-product functor and a fixed-point-algebra functor. This powerful result implies, for example, that the version in [9] of Mansfield imprimitivity for arbitrary subgroups is natural. We finish with a brief survey of one of the main results of [8] which uses an approach based on Rieffel’s theory to establish induction-in-stages for crossed products by coactions. ∗ 1. The Category C* nd and Commutative C -algebras

In our first course in C ∗ -algebras, we learned that commutative unital C ∗ algebras are basically the same things as compact topological spaces. To make this formal, we note that the assignment X → C(X) is the object map in a contravariant functor C from the category Cpct of compact Hausdorff spaces and continuous functions to the category CommC*1 of unital commutative C ∗ -algebras and unital homomorphisms (which for us are always ∗-preserving); the morphism C(f ) associated to a continuous map f : X → Y sends a ∈ C(Y ) to a ◦ f ∈ C(X). Then the Gelfand-Naimark theorem implies that the functor C is an equivalence of categories. (This result goes back to [25], and we will go into the details of what it means in the proof of Theorem 2 below.) The Gelfand-Naimark theorem for non-unital algebras says that commutative C ∗ -algebras are basically the same things as locally compact topological spaces. However, it is not so easy to put this version in a categorical context, and in doing so we run into some important issues which are very relevant to problems involving crossed products and nonabelian duality. So we will discuss these issues now as motivation for our later choices. There is no doubt what the analogue of the functor C does to objects: it takes a locally compact Hausdorff space X to the C ∗ -algebra C0 (X) of continuous functions a : X → C which vanish at infinity. However, there is a problem with

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PROPER ACTIONS

11 3

morphisms: composing with a continuous function f : X → Y does not necessarily map C0 (Y ) into C0 (X). For example, consider the function f : R → R defined by f (x) = (1 + x2 )−1 : any function a ∈ C0 (R) which is identically 1 on [0, 1] satisfies a ◦ f = 1, and hence a ◦ f does not vanish at infinity. One way out is to restrict attention to the category in which the morphisms from X to Y are the proper functions f : X → Y for which inverse images of compact sets are compact, and then on the C ∗ -algebra side one has to restrict attention to the homomorphisms ϕ : A → B such that the products ϕ(a)b span a dense subspace of B. In [28], Pedersen does exactly this, and calls these proper homomorphisms. It turns out, though, that there is a very satisfactory way to handle arbitrary continuous functions between locally compact spaces, in which we allow morphisms which take values in Cb (X). A homomorphism ϕ of one C ∗ -algebra A into the multiplier algebra M (B) of another C ∗ -algebra B is called nondegenerate if ϕ(A)B := span{ϕ(a)b : a ∈ A, b ∈ B} is all of B. (This notation is suggestive: the Cohen factorisation theorem says that everything in the closed span factors as ϕ(a)b.) We want to think of the nondegenerate homomorphisms ϕ : A → M (B) as morphisms from A to B. Every nondegenerate homomorphism ϕ extends to a unital homomorphism ϕ¯ : M (A) → M (B) (see [35, Corollary 2.51], for example); the extension has to satisfy ϕ(m)(ϕ(a)b) ¯ = ϕ(ma)b, and hence the nondegeneracy implies that there is exactly one such extension, and that it is strictly continuous. The following fundamental proposition is implicit in a number of earlier works, including [43, 44], [41] and [13, §1]. ∗ Proposition 1. There is a category C* nd in which the objects are C -algebras, the morphisms from A to B are the nondegenerate homomorphisms from A to M (B), and the composition of ϕ : A → M (B) and ψ : B → M (C) is ψ ◦ ϕ := ψ¯ ◦ ϕ. The isomorphisms in this category are the usual isomorphisms of C ∗ -algebras.

Proof. It is easy to check that the composition ψ¯ ◦ ϕ : A → M (C) is non¯ ¯ is a homomorphism degenerate, and hence defines a morphism in C* nd . Since ψ ◦ ϕ ¯ from M (A) to M (C) which extends ψ ◦ ϕ, it must be the unique extension ψ¯ ◦ ϕ. Thus if θ : C → M (D) is another nondegenerate homomorphism, we have ¯ ◦ϕ θ ◦ (ψ ◦ ϕ) = θ¯ ◦ (ψ ◦ ϕ) = θ¯ ◦ (ψ¯ ◦ ϕ) = (θ¯ ◦ ψ) = (θ¯ ◦ ψ) ◦ ϕ = (θ ◦ ψ) ◦ ϕ = (θ ◦ ψ) ◦ ϕ, and composition in C* nd is associative. The identity maps idA : A → A, viewed as ¯ A = idM (A) , and hence have the properties homomorphisms into M (A), satisfy id one requires of the identity morphisms in C* nd . Thus C* nd is a category, as claimed. For the last assertion, notice first that every isomorphism is trivially nondegenerate, and hence defines a morphism in C* nd , which is an isomorphism because it has an inverse. Conversely, suppose that ϕ : A → M (B) and ψ : B → M (A) are ¯ inverses of each other in C* ¯ ◦ ψ = idB . Using first the nd , so that ψ ◦ ϕ = idA and ϕ nondegeneracy of ψ and then the nondegeneracy of ϕ, we obtain ϕ(A) = ϕ(ψ(B)A) = ϕ(ψ(B))ϕ(A) ¯ = Bϕ(A) = B. ¯ B = ψ, we have ψ ◦ ϕ = idA . The same arguments Thus ϕ has range B, and since ψ| show that ϕ ◦ ψ = idB , so ϕ is an isomorphism in the usual sense. 

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12 4

AN HUEF, RAEBURN, AND WILLIAMS

If f : X → Y is a continuous map between locally compact spaces and a ∈ C0 (Y ), then a ◦ f is a continuous bounded function which defines a multiplier of C0 (X). For every b in the dense subalgebra Cc (X), we can choose a ∈ Cc (Y ) such that a = 1 on f (supp b), and then b = (a ◦ f )b, so C0 (f ) : a → a ◦ f is a nondegenerate homomorphism from C0 (Y ) to M (C0 (X)); the extension C0 (f ) to Cb (X) = M (C0 (X)) is again given by composition with f . We now have a functor C0 from the category LCpct of locally compact spaces and continuous maps to the ∗ full subcategory CommC* nd of C* nd whose objects are commutative C -algebras. This functor has the properties we expect: Theorem 2. The contravariant functor C0 : LCpct → CommC* nd is an equivalence of categories. Proof. To say that C0 is an equivalence means that there is a functor G : CommC* nd → LCpct such that C0 ◦ G and G ◦ C0 are naturally isomorphic to the identity functors. To verify that it is an equivalence, though, it suffices to show that every object in CommC* nd is isomorphic to one of the form C0 (X), which is exactly what the Gelfand-Naimark theorem says, and that C0 is a bijection on each set Mor(X, Y ) of morphisms (see [21, page 91]). Injectivity is easy: since C0 (Y ) separates points of Y , a ◦ f = a ◦ g for all a ∈ C0 (Y ) implies that f (x) = g(x) for all x ∈ X. For surjectivity, we suppose that ϕ : C0 (Y ) → C0 (X) is a nondegenerate homomorphism. Then for each x ∈ X, the composition x ◦ ϕ with the evaluation map is a homomorphism from C0 (Y ) to C, and the nondegeneracy of ϕ implies that x ◦ ϕ is non-zero. Since  : y → y is a homeomorphism of Y onto the maximal ideal space of C0 (Y ), there is a unique f (x) ∈ Y such that x ◦ ϕ = f (x) , and f = −1 ◦ ϕ∗ ◦  is continuous. The equation x ◦ ϕ = f (x) then says precisely that ϕ = C0 (f ).  The result in [21, page 91] which we have just used in the proof of Theorem 2 is a little unnerving to analysts. (Well, to us, anyway.) Its proof, for example, makes carefree use of the axiom of choice. So it is perhaps reassuring that in the situation of Theorem 2, there is a relatively concrete inverse functor Δ which takes a commutative C ∗ -algebra A to its maximal ideal space Δ(A). (We say “relatively concrete” here because the axiom of choice is also used in the proof that the Gelfand transform is an isomorphism.) The argument on page 92 of [21] shows that, once we have chosen isomorphisms ηA : A → C(Δ(A)) for every commutative C ∗ -algebra A, there is exactly one way to extend Δ to a functor in such a way that η := {ηA : A ∈ Obj(CommC* nd )} is a natural isomorphism. If we choose ηA : A → C0 (Δ(A)) to be the Gelfand transform, then the functor Δ takes a morphism ϕ : A → M (B) to the map Δ(ϕ) : ω → ω ◦ ϕ. So we have the following naturality result: Corollary 3. The Gelfand transforms {ηA : A ∈ Obj(CommC* nd )} form a natural isomorphism between the identity functor on CommC* and the composition C0 ◦Δ. nd Of course, modulo the existence of the isomorphisms ηA , which is the content of the (highly non-trivial) Gelfand-Naimark theorem, this result can be easily proved directly: we just need to check that for every morphism ϕ : A → M (B) the following

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diagram commutes in CommC* nd : A

ηA

ϕ

 B

/ C0 (Δ(A)) C0 (Δ(ϕ))

ηB

 / C0 (Δ(B)).

2. Crossed Products and Landstad Duality Although the category C* nd has only been studied recently, nondegenerate homomorphisms have been around for years. For example, the unitary representations U : G → U (H) of a locally compact group G on a Hilbert space H are in one-to-one correspondence with the nondegenerate representations πU of the group algebras L1 (G) or C ∗ (G) on H. In this context, “nondegenerate” usually means that the elements πU (a)h span a dense subspace of H, but this is equivalent to the nondegeneracy of πU as a homomorphism into B(H) = M (K(H)). More generally, if u : G → U M (B) is a strictly continuous homomorphism into the unitary group of a multiplier algebra, then there is a unique nondegenerate homomorphism πu : C ∗ (G) → M (B), called the integrated form of u, from which we can recover u by composing with a canonical unitary representation kG : G → U M (C ∗ (G)). The composition here is taken in the spirit of the category C* nd : it is the composition in the usual sense of the extension of πu to M (C ∗ (G)) with kG . We say that kG is universal for unitary representations of G. One application of this circle of ideas which will be particularly relevant here is the existence of the comultiplication δG on C ∗ (G), which is the integrated form of the unitary representation kG ⊗ kG : G → U M (C ∗ (G) ⊗ C ∗ (G)). Thus δG is by definition a nondegenerate homomorphism of C ∗ (G) into M (C ∗ (G) ⊗ C ∗ (G)). Its other crucial property is coassociativity: (δG ⊗ id) ◦ δG = (id ⊗ δG ) ◦ δG , where the compositions are interpreted as being those in the category C* nd . Now suppose that α : G → Aut A is an action of a locally compact group G on a C ∗ -algebra. Nondegeneracy is then built into the notion of covariant representation of the system: a covariant representation (π, u) of a dynamical system (A, G, α) in a multiplier algebra M (B) consists of a nondegenerate homomorphism π : A → M (B) and a strictly continuous homomorphism u : G → U M (B) such that π(αt (a)) = ut π(a)u∗t . The crossed product is then, either by definition [33] or by theorem [42, 2.34–36], a C ∗ -algebra A α G which is generated (in a sense made precise in those references) by a universal covariant representation (iA , iG ) of (A, G, α) in M (A α G). Each covariant representation (π, u) in M (B) has an integrated form π  u which is a nondegenerate homomorphism of A α G into M (B) such that π = (π  u) ◦ iA and u = (π  u) ◦ iG . The crossed product A α G carries a dual coaction α, ˆ which is the integrated form of iG ⊗ kG : G → U M ((A α G) ⊗ C ∗ (G)). This is another nondegenerate homomorphism, and the crucial coaction identity (α ˆ ⊗ id) ◦ α ˆ = (id ⊗ δG ) ◦ α ˆ again has to be interpreted in the category C* . (Makes you wonder how we ever managed nd without C* nd .) There is another version of the crossed-product construction which can be more suitable for spatial arguments, and which is particularly important for the issues we discuss in this paper. For any representation π : A → B(Hπ ), there is a regular representation (˜ π , U ) of (A, G, α) on L2 (G, Hπ ) such that (˜ π (a)h)(r) =

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AN HUEF, RAEBURN, AND WILLIAMS

π(αr−1 (a))(h(r)) and λs h(r) = h(s−1 r) for h ∈ L2 (G, Hπ ). The reduced crossed product A α,r G is the quotient of A α G which has the property that every π ˜ λ factors through a representation of A α,r G, and then π ˜  λ is faithful whenever π is [42, §7.2]. The reduced crossed product is also generated by a canonical covariant representation (irA , irG ), and the dual coaction α ˆ factors through a coaction α ˆ n : A α,r G → M ((A α,r G) ⊗ C ∗ (G)) characterised by (1)

α ˆ n ◦ irA (a) = irA (a) ⊗ 1 and α ˆ n ◦ irG (s) = irG (s) ⊗ kG (s).

This coaction is called the normalisation of α ˆ , and is in particular normal in the sense that the canonical map jAG of A α,r G into M ((A α,r G) αˆ G) is injective (see Proposition A.61 of [3]). Kaliszewski and Quigg’s motivation for working in the category C* nd came from the following characterisation of the C ∗ -algebras which arise as reduced crossed products. Theorem 4 (Landstad, Kaliszewski-Quigg). Suppose that B is a C ∗ -algebra and G is a locally compact group. Then there is a dynamical system (A, G, α) such that B is isomorphic to A α,r G if and only if there is a morphism π : C ∗ (G) → M (B) in C* nd and a nondegenerate (see Remark 6 below) normal coaction δ : B → M (B ⊗ C ∗ (G)) such that (2)

(π ⊗ id) ◦ δG = δ ◦ π.

In [13] the authors say that this result follows from a theorem of Landstad [19], and it is certainly true that most of the hard work is done by Landstad’s result. But we think it is worth looking at the proof; those who are not interested in the subtleties of coactions should probably skip to the end of the proof below. We begin by stating Landstad’s theorem in modern terminology. Theorem 5 (Landstad, 1979). Suppose that B is a C ∗ -algebra and G is a locally compact group. Then there is a dynamical system (A, G, α) such that B is isomorphic to A α,r G if and only if there are a strictly continuous homomorphism u : G → U M (B) and a reduced coaction δ : B → M (B ⊗ Cr∗ (G)) such that ¯ s ) = us ⊗ λs for s ∈ G, and (a) δ(u (b) δ(B)(1 ⊗ Cr∗ (G)) = B ⊗ Cr∗ (G). The “reduced coaction” appearing in Landstad’s theorem is required to have slightly different properties from the full coactions which we use elsewhere in this paper, and which are used in [3] and [13], for example. A reduced coaction on B is an injective nondegenerate homomorphism of B into M (B ⊗ Cr∗ (G)) rather than M (B ⊗ C ∗ (G)), and it is required to be coassociative with respect to the r comultiplication δG on Cr∗ (G). Remark 6. Nowadays, the second condition (b) in Theorem 5 is usually absorbed into the assertion that δ is a coaction. Everyone agrees that for δ to be a coaction δ(B)(1⊗Cr∗ (G)) must be contained in B ⊗Cr∗ (G), and Landstad described the requirement of equality as “nondegeneracy”, which in view of our emphasis on C* nd has turned out to be unfortunate terminology. Coactions of amenable or discrete groups are automatically nondegenerate in Landstad’s sense, and dual coactions are always nondegenerate. We therefore follow modern usage and assume that all coactions satisfy (b), or its analogue in the case of full coactions. (So (b) can now be deleted from Theorem 5 and the word “nondegenerate” from Theorem 4.)

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15 7

PROPER ACTIONS

Proof of Theorem 4. For B = A α,r G, we take δ = α ˆ n and π = πirG . The second equation in (1) implies that (π ⊗ id) ◦ δG (kG (s)) = π ⊗ id(kG (s) ⊗ kG (s)) = irG (s) ⊗ kG (s) =α ˆ n ◦ π(kG (s)) for all s ∈ G, which implies (2). r Now suppose that there exist π and δ as described. Then we define u := π ¯ ◦ kG , and consider the reduction δ r of δ, which since δ is normal is just δ r := (id ⊗ πλ ) ◦ δ. Now we compute: δ r (us ) = (id ⊗ πλ ) ◦ δ(¯ π (kr (s))) = id ⊗ πλ ◦ δ¯ ◦ π ¯ (kr (s)) G

= id ⊗ πλ ◦ π ⊗ id ◦ =π ¯◦

r (s) kG

r δG (kG (s))

=π⊗

G r πλ (kG (s)

⊗ kG (s))

⊗ λ s = us ⊗ λ s .

r

Thus u and δ satisfy the hypotheses of Landstad’s theorem (Theorem 5), and we can deduce from it that B is isomorphic to a reduced crossed product.  Kaliszewski and Quigg then made two further crucial observations. First, they recognised that there is a category of coactions associated to C*: the objects in C*coactnd (G) consist of full coactions δ on C ∗ -algebras B, and the morphisms from (B, δ) to (C, ) are nondegenerate homomorphisms ϕ : B → M (C) such that (ϕ ⊗ id) ◦ δ =  ◦ ϕ. Then (2) says that the homomorphism π in Corollary 4 is a morphism in C*coactnd (G) from (C ∗ (G), δG ) to (B, δ). Second, they knew that for every object a and every subcategory D in a category C there is a comma category a ↓ D in which objects are morphisms f : a → x in C from a to objects in D, and the morphisms from (x, f ) to (y, g) are morphisms h : x → y in D such that h ◦ f = g. Thus Landstad’s theorem identifies the reduced crossed products as the C ∗ -algebras which can be augmented with a coaction δ and a homomorphism π to form an object in the comma category (C ∗ (G), δG ) ↓ C*coactnnd (G), where n C*coactnd (G) is the full subcategory of normal coactions. The main results in [13] concern crossed-product functors defined on the category C*actnd (G) whose objects are dynamical systems (A, G, α) and whose morphisms ϕ : (A, α) → (B, β) are nondegenerate homomorphisms ϕ : A → M (B) such that ϕ ◦ αs = βs ◦ ϕ for s ∈ G (where yet again the composition on the right is taken in C* nd ). The following theorem is Theorem 4.1 of [13]. Theorem 7 (Kaliszewski-Quigg, 2009). There is a functor CPr from C*actnd (G) to the comma category (C ∗ (G), δG ) ↓ C*coactnnd (G) which takes the object (A, α) to (A α,r G, α ˆ r , irG ), and this functor is an equivalence of categories. Landstad’s theorem, in the form of Theorem 4, says that CPr is essentially surjective: every object in the comma category is isomorphic to one of the form CPr (A, α) = A α,r G. Thus Theorem 7 can be viewed as an extension of Landstad’s theorem, and Kaliszewski and Quigg call it “categorical Landstad duality for actions”. They also obtain an analogous result for full crossed products. 3. Proper Actions and Landstad Duality for Coactions Quigg’s version of Landstad duality for crossed products by coactions [30] is also easy to formulate in categories based on C* nd . Suppose that δ is a coaction of G on a C ∗ -algebra C, and let wG denote the function s → kG (s), viewed as a multiplier

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16 8

AN HUEF, RAEBURN, AND WILLIAMS

of C0 (G, C ∗ (G)). A covariant representation of (C, δ) in a multiplier algebra M (B) consists of nondegenerate homomorphisms π : C → M (B) and μ : C0 (G) → M (B) such that (π ⊗ id) ◦ δ(c) = μ ⊗ id(wG )(π(c) ⊗ 1)μ ⊗ id(wG )∗ for c ∈ C, where, as should seem usual by now, the composition is interpreted in C* nd . The crossed product C δ G is generated by a universal covariant representation (jC , jG ) in M (C δ G), in the sense that products jC (c)jG (f ) span a dense subspace of C δ G. The crossed product carries a dual action δˆ such that δˆs (jC (c)jG (f )) = jC (c)jG (rts (f )), where rt is defined by rts (f )(t) = f (ts). Quigg’s theorem identifies the C ∗ -algebras which are isomorphic to crossed products by coactions. Theorem 8 (Quigg, 1992). Suppose that G is a locally compact group and A is a C ∗ -algebra. There is a system (C, δ) such that A is isomorphic to C δ G if and only if there are a nondegenerate homomorphism ϕ : C0 (G) → M (A) and an action α of G on A such that (A, α, ϕ) is an object in the comma category (C0 (G), rt) ↓ C*actnd (G). ˆ and the hard bit is to When A = C δ G, we can take ϕ := jG and α := δ, prove the converse. This is done in [30, Theorem 3.3]. It is then natural to look for a “categorical Landstad duality for coactions” which parallels the results of [13]. However, triples (A, α, ϕ) of the sort appearing in Theorem 8 had earlier (that is, before [13]) appeared in important work of Rieffel on proper actions, and it has proved very worthwhile to follow up this circle of ideas in Rieffel’s context. To explain this, we need to digress a little. If α : G → Aut A is an action of a compact abelian group, then information about the crossed product can be recovered from the fixed point algebra Aα , and, more generally, from the spectral subspaces  Aα (ω) := { a ∈ A : αs (a) = ω(s)a } for ω ∈ G. A fundamental result of Kishimoto and Takai [16, Theorem 2] says that if the spectral subspaces are large in the sense that Aα (ω)∗ Aα (ω) is dense in Aα for every  then A α G is Morita equivalent to Aα . There is as yet no completely ω ∈ G, satisfactory notion of a free action of a group on a C ∗ -algebra (see [29], for example), but having large spectral subspaces is one example of such a notion. When G is locally compact, the fixed-point algebra is often trivial. For example, if rt is the action of G = Z on R by right translation, then f ∈ C0 (R)rt if and only if f is periodic with period 1, which since f vanishes at ∞ forces f to be identically zero. However, if the orbit space for an action is nice enough, then the algebra of continuous functions on the orbit space can be used as a substitute for the fixedpoint algebra. A right action of a locally compact group G on a locally compact space T is called proper if the map (x, s) → (x, x·s) : T ×G → T ×T is proper. The orbit space T /G for a proper action is always Hausdorff [42, Corollary 3.43], and a classical result of Green [5] says that if the action of G on T is free and proper, then C0 (T ) rt G is Morita equivalent to C0 (T /G) (for this formulation of Green’s result see [42, Remark 4.12]). We want to think of C0 (T /G) as a subalgebra of the multiplier algebra M (C0 (G)) = Cb (T ) which is invariant under the extensions rts of the automorphisms rts . In the past twenty-five years, many researchers have investigated analogues of free and proper actions for noncommutative C ∗ -algebras [34, 38, 4, 24, 10,

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PROPER ACTIONS

17 9

39, 11]. Here we are interested in the notion of proper action α : G → Aut A introduced by Rieffel [38]. He assumes that there is an α-invariant subalgebra A0 of A with properties like those of the subalgebra Cc (T ) of C0 (T ), and that there is an M (A)α -valued inner product on A0 . The completion Z(A, α) of A0 in this inner product is a full Hilbert module over a subalgebra Aα of M (A)α , which Rieffel calls the generalised fixed-point algebra for α. The algebra K(Z(A, α)) of generalised compact operators on Z(A, G, α) sits naturally as an ideal E(α) in the reduced crossed product A α,r G [38, Theorem 1.5]. The action α is saturated when E(α) is all of the reduced crossed product. Thus when α is proper and saturated, A α,r G is Morita equivalent to Aα . Saturation is a freeness condition: if G acts properly on T , then rt : G → Aut(C0 (T )) is proper with respect to Cc (G), and the action is saturated if and only if G acts freely [23, §3]. On the face of it, though, Rieffel’s bimodule Z(A, α) and the fixed-point algebra Aα depend on the choice of subalgebra A0 , and it seems unlikely that Rieffel’s process is functorial. The connection with our categories lies in a more recent theorem of Rieffel which identifies a large family of proper actions for which there is a canonical choice of the dense subalgebra A0 [39, Theorem 5.7]. Theorem 9 (Rieffel, 2004). Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T , and (A, G, α) is a dynamical system such that there is a nondegenerate homomorphism ϕ : C0 (T ) → M (A) satisfying ϕ ◦ rt = α ◦ ϕ (with composition in the sense of C* nd ). Then α is proper and saturated with respect to the subalgebra A0 = ϕ(Cc (T ))Aϕ(Cc (T )). Example 10. A closed subgroup H of a locally compact group G acts freely and properly on G, and hence we can apply Theorem 9 to the pair (T, G) = (G, H) and to the canonical map jG : C0 (G) → M (C δ G). In this case, highly nontrivial ˆ results of Mansfield [22] can be used to identify the fixed-point algebra (C δ G)δ with the crossed product C δ,r (G/H) by the homogeneous space [9, Remark 3.4]. (These crossed products were introduced in [1]; the relationship with the crossed product C δ| (G/H) by the restricted coaction, which makes sense when H is normal, is discussed in [1, Remark 2.2].) Then Theorem 3.1 of [9] shows that Rieffel’s δˆ Morita equivalence between (C δ G) δ,r ˆ H and (C δ G) extends Mansfield’s imprimitivity theorem for coactions to arbitary closed subgroups (as opposed to the amenable normal subgroups in Mansfield’s original theorem [22, Theorem 27] and the normal ones in [12]). From our categorical point of view, the hypotheses on ϕ in Theorem 9 say precisely that (A, α, ϕ) := ((A, α), ϕ) is an object in the comma category (C0 (T ), rt) ↓ C*actnd (G). Then Rieffel’s theorem implies that (A, α, ϕ) → Aα is a construction which takes objects in the comma category to objects in the category C* nd . One naturally asks: is this construction functorial? More precisely, is there an analogous construction on morphisms which makes (A, α, ϕ) → Aα into a functor from (C0 (T ), rt) ↓ C*actnd (G) to C* nd ? This question was answered in [15, §2] using a new construction of Rieffel’s generalised fixed-point algebra. The crucial ingredient is an averaging process E of Olesen and Pedersen [26, 27], which was subsequently developed by Quigg in [31, 32] and used extensively in his proof of Theorem 8. This averaging process E

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AN HUEF, RAEBURN, AND WILLIAMS

makes sense on the dense subalgebra A0 = ϕ(Cc (T ))Aϕ(Cc (T )), and satisfies  ϕ(f )E(ϕ(g)aϕ(h)) = ϕ(f )αs (ϕ(g)aϕ(h)) ds for f, g, h ∈ Cc (T ) and a ∈ A; G

the integral on the right has an unambiguous meaning because properness implies that s → f rts (g) has compact support. It is shown in [15, Proposition 2.4] that the closure of E(A0 ) is a C ∗ -subalgebra of M (A), which we denote by Fix(A, α, ϕ) to emphasise all the data involved in the construction. It is shown in [15, Proposition 3.1] that Fix(A, α, ϕ) and Rieffel’s Aα are exactly the same subalgebra of M (A). If σ : (A, α, ϕ) → (B, β, ψ) is a morphism in the comma category, so that in particular σ is a nondegenerate homomorphism from A to M (B), then the extension σ ¯ maps Fix(A, α, ϕ) into M (Fix(B, β, ψ)), and is nondegenerate. (This is Proposition 2.6 of [15]; a gap in the proof of nondegeneracy is filled in Corollary 2.3 of [8].) Theorem 11 (Kaliszewski-Quigg-Raeburn, 2008). Suppose that a locally compact group G acts properly on the right of a locally compact space T . Then the assignments (A, α, ϕ) → Fix(A, α, ϕ) and σ → σ ¯ |Fix(A,α,ϕ) form a functor from (C0 (T ), rt) ↓ C*actnd (G) to C* . nd To return to the setting of Quigg-Landstad duality, we take (T, G) = (G, G) in this theorem. This gives us a functor Fix from (C0 (G), rt) ↓ C*actnd (G) to C* nd . Because the fixed-point algebra Fix(A, α, ϕ) is defined using the same averaging process E as Quigg used in [30, §3], Fix(A, α, ϕ) is the same as the algebra C constructed by Quigg (unfortunately for us, he called it B). So Quigg proves in [30] that δA (c) = ϕ ⊗ πλ (wG )(c ⊗ 1)ϕ ⊗ πλ (wG )∗ defines a reduced coaction of G on C = Fix(A, α, ϕ), and that A is isomorphic to the crossed product C δA G. An examination of the proof of [31, Theorem 4.7] shows that the similar formula f δA (c) = ϕ ⊗ id(wG )(c ⊗ 1)ϕ ⊗ id(wG )∗

defines the unique full coaction with reduction δA . The argument on page 2960 of [15] shows that this construction respects morphisms, so that Fix extends to a n functor FixG from (C0 (G), rt) ↓ C*actnd (G) to C*coactnd (G). The following very satisfactory “categorical Landstad duality for coactions” is Corollary 4.3 of [15]. Theorem 12 (Kaliszewski-Quigg-Raeburn, 2008). Let G be a locally comˆ jG ) and π → π  id form a functor from pact group. Then (C, δ) → (C δ G, δ, n C*coactnd (G) to (C0 (G), rt) ↓ C*actnd (G). This functor is an equivalence of categories with quasi-inverse FixG . In fact, this is a much more satisfying theorem than its analogue for actions because we have a specific construction of a quasi-inverse. We would be interested to see an analogous process for Fixing over coactions. 4. Naturality Now that we have a functorial version Fix of Rieffel’s generalised fixed-point algebra, we remember that the main point of Rieffel’s paper [38] was to construct a Morita equivalence between Aα = Fix(A, α, ϕ) and the reduced crossed product

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19 11

PROPER ACTIONS

RCP(A, α, ϕ) := A α,r G. This equivalence is implemented by an (A α,r G) – Fix(A, α, ϕ) imprimitivity bimodule Z(A, α, ϕ). There is another category C* of C ∗ -algebras in which the isomorphisms are given by imprimitivity bimodules, so it makes sense to ask whether these isomorphisms are natural. Of course, before discussing this problem, we need to be clear about what the category C* is. If A and B are C ∗ -algebras, then a right-Hilbert A – B bimodule is a right Hilbert B-module X which is also a left A-module via a nondegenerate homomorphism of A into the algebra L(X) of bounded adjointable operators on X. (These are sometimes called A – B correspondences.) The objects in C* are C ∗ -algebras, and the morphisms from A to B are the isomorphism classes [X] of full rightHilbert A – B bimodules. Every nondegenerate homomorphism ϕ : A → M (B) gives a right-Hilbert bimodule: view B as a right Hilbert B-module over itself with

b1 , b2 B := b∗1 b2 , and define the action of A by a · b := ϕ(a)b. We denote the isomorphism class of this bimodule by [ϕ]. In [2], it is shown that [ϕ] = [ψ] if and only if there exists u ∈ U M (B) such that ψ = (Ad u) ◦ ϕ, so we are not just adding more morphisms to C* nd , we are also slightly changing the morphisms we already have. If A XB and B YC are right Hilbert bimodules, then we define the composition using the internal tensor product: [Y ][X] := [X ⊗B Y ]. The identity morphism 1A on A is [A AA ] = [idA ]. Now we can see why we have had to take isomorphism classes of bimodules as our morphisms: the bimodule A ⊗A X representing [X]1A = [X][idA ] is only isomorphic to X. A similar subtlety arises when checking that composition of morphisms is associative. The details are in [2, Proposition 2.4]. In [2, Proposition 2.6], it is shown that the isomorphisms from A to B in C* are the classes [X] in which X is an imprimitivity bimodule, so that X also carries a left inner product A x , y such that A x , y · z = x · y , zB . Similar results were obtained independently by Landsman [17, 18] and by Schweizer [40], and a slightly more general category in which the bimodules are not required to be full as right Hilbert modules was considered in [3]. Theorem 3.2 of [15] says that, for every nondegenerate homomorphism σ : A → M (B), the diagram A α,r G

(3)

[Z(A,α,ϕ)]

[σid]

 B β,r G

/ Fix(A, α, ϕ) [σ|]

[Z(B,β,ψ)]

 / Fix(B, β, ψ)

commutes in C*, which means that Z(A, α, ϕ) ⊗Fix(A,α,ϕ) Fix(B, β, ψ) and (B β,r G) ⊗Bβ,r G Z(B, β, ψ) are isomorphic as right-Hilbert (A α,r G) – Fix(B, β, ψ) bimodules. Thus Rieffel’s bimodules (or rather, the morphisms in C* which they determine) implement a natural isomorphism between the functors RCP and Fix from (C0 (T ), rt) ↓ C*actnd (G) to C*. This naturality theorem certainly has interesting applications to nonabelian duality, where it gives naturality for the extension in [9] of Mansfield’s imprimitity theorem to closed subgroups (see [15, Theorem 6.2]). However, it is slightly

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AN HUEF, RAEBURN, AND WILLIAMS

unsatisfactory: we were forced to change the target category from C* nd to C* because the bimodules Z do not define morphisms in C* , but in the diagram (3) we nd have not fully committed to the change. Our goal in [7] was to find versions of the same functors defined on a category built from C* — that is, ones in which the morphisms are implemented by bimodules — to establish that Rieffel’s Morita equivalence gives a natural isomorphism between these functors, and to apply the results to nonabelian duality. We will describe our progress in the next section. 5. Upgrading to C* Proposition 3.3 of [2] says that for every locally compact group G, there is a category C*act(G) whose objects are dynamical systems (A, α) = (A, G, α) and whose morphisms are obtained by adding actions to the morphisms of C*. Formally, if (A, α) and (B, β) are objects in C*(G) and A XB is a right-Hilbert bimodule, then an action of G on a right-Hilbert bimodule X is a strongly continuous homomorphism u of G into the linear isomorphisms of X such that   us (a · x · b) = αs (a) · us (x) · βs (b) and us (x) , us (y)B = βs x , yB , and the morphisms in C*act(G) are isomorphism classes of pairs (X, u). Next we consider a free and proper action of G on a locally compact space T and look for an analogue of the comma category for the system (C0 (T ), rt). The objects are easy: to ensure that Fix is defined on objects, we need to insist that every system (A, α) is equipped with a nondegenerate homomorphism ϕ : C0 (T ) → M (A) which is rt – α equivariant. We choose to use the semi-comma category C*act(G, (C0 (T ), rt)) in which the objects are triples (A, α, ϕ), and the morphisms from (A, α, ϕ) to (B, β, ψ) are just the morphisms from (A, α) to (B, β) in C*act(G). In [7, Remark 2.4] we have discussed our reasons for adding the maps ϕ to our objects and then ignoring them in our morphisms, and the discussion below of how we Fix morphisms should help convince sceptics that this is appropriate. We know how to Fix objects in the semi-comma category C*act(G, (C0 (T ), rt)), and we need to describe how to Fix a morphism [X, u] from (A, α, ϕ) to (B, β, ψ). We begin by factoring the morphism [X] in C* as the composition [K(X) XB ][κA ] of the isomorphism associated to the imprimitivity bimodule K(X) XB with the morphism coming from the nondegenerate homomorphism κA : A → M (K(X)) = L(X) describing the left action of A on X (see Proposition 2.27 of [3]). The action u of G on X gives an action μ of G on K(X) such that μs (Θx,y ) = Θus (x),us (y) , and then κA satisfies κA ◦ αs = μs ◦ κA . So the morphism [(A,α) (X, u)(B,β) ] in C*act(G) factors as [(K(X),μ) (X, u)(B,β) ][κA ]. Now κA is a morphism in (C0 (G), rt) ↓ C*actnd (G) from (A, α, ϕ) to (K(X), μ, κA ◦ ϕ), and hence by Theorem 11 restricts to a morphism κA | from Fix(A, α, ϕ) to Fix(K(X), μ, κA ◦ ϕ). We want to define Fix so that it is a functor, so our definition must satisfy (4)

Fix([X, u]) = Fix([(K(X),μ) (X, u)(B,β) ]) Fix([κA ]).

Since we don’t want to change the meaning of Fix on morphisms in C* nd , our strategy is to define Fix([κA ]) := [κA |], figure out how to Fix imprimitivity bimodules, and then use (4) to define Fix([X, u]). So we suppose that (A, α, ϕ) and (B, β, ψ) are objects in the semi-comma category C*act(G, (C0 (T ), rt)), and that [X, u] is an equivariant (A, α) – (B, β) imprimitivity bimodule. We emphasise that, because of our choice of morphisms in C*act(G, (C0 (T ), rt)), we do not make any assumption relating the actions of ϕ

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PROPER ACTIONS

21 13

 := {(x) : x ∈ X} be the dual bimodule, and form the and ψ on X. We let X linking algebra   A X L(X) :=  , X B as in the discussion following [35, Theorem 3.19]. Then     α u ϕ 0 and ϕL := L(u) := (u) β 0 ψ define an action L(u) of G on L(X) and a nondegenerate homomorphism ϕL of C0 (T ) into M (L(X)) which intertwines rt and L(u). Then (L(X), L(u), ϕL ) is an object in C*act(G, (C0 (T ), rt)), and (reverting to Rieffel’s notation to simplify the formulas) we can form L(X)L(u) := Fix(L(X), L(u), ϕL ). It follows quite easily from the construction of Fix in [15, §2] that the diagonal corners in L(X)L(u) are Aα and B β , and we define X u to be the upper right-hand corner, so that   α Xu A ; L(X)L(u) = ∗ Bβ with the actions and inner products coming from the operations in L(X)L(u) , X u becomes an Aα – B β -imprimitivity bimodule (see [35, Proposition 3.1]). We now define Fix([X, u]) := [X u ], and use (4) to define Fix in general, as described above. With this definition, Theorem 3.3 of [7] says: Theorem 13. Suppose that T is a free and proper right G-space. Then the assignments (A, α, ϕ) → Fix(A, α, ϕ)

and

[X, u] → Fix([X, u])

form a functor Fix from the semi-comma category C*act(G, (C0 (T ), rt)) to C*. Proving that Fix preserves the composition of morphisms is surprisingly complicated, and involves several non-trivial steps. For example, we needed to show that if (A,α) (X, u)(B,β) and (B,β) Y(C,γ) are imprimitivity bimodules implementing isomorphisms in C*act(G, (C0 (T ), rt)), then (X ⊗B Y )u⊗v is isomorphic to X u ⊗B β Y v as Aα – C γ imprimitivity bimodules. It follows from [3, Theorem 3.7] that RCP is a functor from C*act(G, (C0 (T ), rt)) to C* which takes a morphism [X, u] to the class of the Combes bimodule [X u,r G]. We can now state the main naturality result, which is Theorem 3.5 of [7]. Theorem 14. Suppose that a locally compact group G acts freely and properly on a locally compact space T . Then the Morita equivalences Z(A, α, ϕ) implement a natural isomorphism between the functors RCP and Fix from C*act(G, (C0 (T ), rt)) to C*. The proof of Theorem 14 relies on factoring morphisms; then Theorem 3.2 of [15] gives the result for the nondegenerate homomorphism, and standard linking algebra techniques give the other half. We saw in Example 10 that Rieffel’s Morita equivalence can be used to generalise Mansfield’s imprimitivity theorem to crossed products by homogeneous spaces, and we want to deduce from Theorem 14 that this imprimitivity theorem gives a natural isomorphism. To get the imprimitivity theorem in Example 10, we applied Rieffel’s Theorem 9 to a crossed product C δ G. So the naturality result we seek relates the compositions of RCP and Fix with a crossed-product functor.

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AN HUEF, RAEBURN, AND WILLIAMS

Suppose as in Example 10 that H is a closed subgroup of a locally compact group G. We know from Theorem 2.15 of [3] that there is a category C*coactn (G) whose objects are normal coactions (B, δ), and whose morphisms are isomorphism classes of suitably equivariant right-Hilbert bimodules. We also know from Theorem 3.13 of [3] that there is a functor CP : C*coactn (G) → C*act(H), and adding the canonical map jG makes CP into a functor with values in the comma category (C0 (G), rt |H ) ↓ C*act(H). We show in [7, Proposition 5.5] that there is a functor RCPG/H which sends (B, δ) to the crossed product B δ,r (G/H) by the homogeneous space G/H, and that this functor coincides with Fix ◦ CP. We saw in ˆ H , jG ) implement a Morita equivExample 10 that Rieffel’s bimodules Z(B δ G, δ| alence between (B δ G) δ|,r ˆ H and B δ,r G/H. Write RCPH for the functor from C*act(G) to C* sending (C, γ) → C γ|,r H. Then the general naturality result above gives the following theorem, which is Theorem 5.6 of [7]. Corollary 15. Let H be a closed subgroup of G. Then Rieffel’s Morita equivˆ H , jG ) implement a natural isomorphism between the functors alences Z(G δ G, δ| RCPH ◦ CP and RCPG/H from C*coactn (G) to C*. Corollary 15 extends Theorem 4.3 of [3] to non-normal subgroups, and extends Theorem 6.2 of [15] to categories based on C* rather than ones based on C* nd . 6. Induction-in-stages and Fixing-in-stages Rieffel’s theory of proper actions seems to be a powerful tool for studying systems in the comma or semi-comma category associated to a pair (T, G). Corollary 15 is, we think, an impressive first example. As another example, we discuss an approach to induction-in-stages which works through the same general machinery, and which we carried out in [8]. The original purpose of an imprimitivity theorem was to provide a way of recognising induced representations (as in, for example, [20]), and Rieffel’s theory of Morita equivalence for C ∗ -algebras was developed to put imprimitivity theorems in a C ∗ -algebraic context [36, 37]. One can reverse the process: a Morita equivalence X between a crossed product C α G and another C ∗ -algebra B gives an induction process X-Ind which takes a representation of B on H to a representation of C on X ⊗B H, and for which there is a ready-made imprimitivity theorem (see, for example, [6, Proposition 2.1]). The situation is slightly less satisfactory when one has a reduced crossed product, but one can still construct induced representations and prove an imprimitivity theorem. Mansfield’s imprimitivity theorem, as extended to homogeneous spaces in [9], gives an induction process IndG G/H from Bδ,r (G/H) to Bδ G which comes with an imprimitivity theorem. One then asks whether this induction process has the other properties which one would expect. For example, we ask whether we can induct-inG/H G/K stages: if we have subgroups H, K and L with H ⊂ K ⊂ L, is IndG/K (IndG/L π) G/H

unitarily equivalent to IndG/L π? If the subgroups are normal and amenable, then the induction processes are those defined by Mansfield [22], and induction-in-stages was established in [14, Theorem 3.1]. For non-normal subgroups, not much seems to be known. There are clearly issues: for example, the subgroups H and K have to be normal in L for the three induction processes to be defined. We tackled this problem in [8] using our semi-comma category. Suppose that (T, G) is as usual, N is a closed normal subgroup of G, and (A, α, ϕ) is an object

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PROPER ACTIONS

23 15

in C*act(G, (C0 (T ), rt)). Then N also acts freely and properly on T , so we can form the fixed-point algebra FixN (A, α|N , ϕ). The quotient G/N has a natural action αG/N on Aα|N := FixN (A, α|N , ϕ), and the map ϕ induces a homomorphism ϕN : C0 (T /N ) → M (Aα|N ) such that (Aα|N , αG/N , ϕN ) is an object in the semicomma category C*act(G/N, (C0 (T /N ), rt)). We prove in [8] that FixN extends to a functor G/N

FixN

: C*act(G, C0 (T ), rt) → C*act(G/N, (C0 (T /N ), rt)), G/N

and that the functors FixG/N ◦ FixN and FixG are naturally isomorphic (see [8, Theorem 4.5]). The first difficulty in the proof is showing that the functor FixN has an equivariant version: because the functor Fix is defined using the factorisation of morphisms, we have to track carefully through the constructions in [7] to make sure that they all respect the actions of G/N . Applying this result on “fixing-in-stages” with (T, G) = (L/H, K/H), gives the following version of induction-in-stages, which is Theorem 7.3 of [8]. Theorem 16. Suppose that δ is a normal coaction of G on B, and that H, K and L are closed subgroups of G such that H ⊂ K ⊂ L and both H and K are normal in L. Then for every representation π of B δ,r (G/L), the representation G/H G/K G/H IndG/K (IndG/L π) is unitarily equivalent to IndG/L π. Obviously this is not the last word on the subject, and the normality hypotheses on subgroups are irritating. However, Mansfield’s induction process is notoriously hard to work with, and it seems remarkable that one can prove very much at all about an induction process which is substantially more general than his. We think that Rieffel’s theory of proper actions is proving to be a remarkably malleable and powerful tool. References [1] S. Echterhoff, S. Kaliszewski and I. Raeburn, Crossed products by dual coactions of groups and homogeneous spaces, J. Operator Theory 39 (1998), 151–176. [2] S. Echterhoff, S. Kaliszewski, J. Quigg and I. Raeburn, Naturality and induced representations, Bull. Austral. Math. Soc. 61 (2000), 415–438. , A categorical approach to imprimitivity theorems for C ∗ -dynamical systems, Mem. [3] Amer. Math. Soc. 180 (2006), no. 850, viii+169. [4] R. Exel, Morita-Rieffel equivalence and spectral theory for integrable automorphism groups of C ∗ -algebras, J. Funct. Anal. 172 (2000), 404–465. [5] P. Green, C ∗ -algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), 71–97. [6] A. an Huef, S. Kaliszewski, I. Raeburn and D. P. Williams, Extension problems for representations of crossed-product C ∗ -algebras, J. Operator Theory 62 (2009), 171–198. , Naturality of Rieffel’s Morita equivalence for proper actions, Algebr. Represent. [7] Theory, to appear. , Fixed-point algebras for proper actions and crossed products by homogeneous spaces, [8] preprint; arXiv:math/0907.0681. [9] A. an Huef and I. Raeburn, Mansfield’s imprimitivity theorem for arbitrary closed subgroups, Proc. Amer. Math. Soc. 132 (2004), 1153–1162. [10] A. an Huef, I. Raeburn and D. P. Williams, Proper actions on imprimitivity bimodules and decompositions of Morita equivalences, J. Funct. Anal. 200 (2003), 401–428. , A symmetric imprimitivity theorem for commuting proper actions, Canad. J. Math. [11] 57 (2005), 983–1011. [12] S. Kaliszewski and J. Quigg, Imprimitivity for C ∗ -coactions of non-amenable groups, Math. Proc. Camb. Phil. Soc. 123 (1998), 101–118.

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[13] , Categorical Landstad duality for actions, Indiana Univ. Math. J. 58 (2009), 415–441. [14] S. Kaliszewski, J. Quigg and I. Raeburn, Duality of restriction and induction for C ∗ coactions, Trans. Amer. Math. Soc. 349 (1997), 2085–2113. , Proper actions, fixed-point algebras and naturality in nonabelian duality, J. Funct. [15] Anal. 254 (2008), 2949–2968. [16] A. Kishimoto and H. Takai, Some remarks on C ∗ -dynamical systems with a compact abelian group, Publ. Res. Inst. Math. Sci. 14 (1978), 383–397. [17] N. P. Landsman, Bicategories of operator algebras and Poisson manifolds, Mathematical Physics in Mathematics and Physics, Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, 2001, pages 271–286. , Quantized reduction as a tensor product, Quantization of Singular Symplectic Quo[18] tients, Progress in Math., vol. 198, Birkhuser, Basel, 2001, pages 137–180. [19] M. B. Landstad, Duality theory for covariant systems, Trans. Amer. Math. Soc. 248 (1979), 223–267. [20] G. W. Mackey, Imprimitivity for representations of locally compact groups, Proc. Nat. Acad. Sci. USA 35 (1949), 537–545. [21] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Math, vol. 5, Springer, Berlin, 1971. [22] K. Mansfield, Induced representations of crossed products by coactions, J. Funct. Anal. 97 (1991), 112–161. [23] D. Marelli and I. Raeburn, Proper actions which are not saturated, Proc. Amer. Math. Soc. 137 (2009), 2273–2283. [24] R. Meyer, Generalised fixed-point algebras and square-integrable group actions, J. Funct. Anal. 186 (2001), 167–195. [25] J. W. Negrepontis, Duality in analysis from the point of view of triples, J. Algebra 15 (1971), 228–253. [26] D. Olesen and G. K. Pedersen, Applications of the Connes spectrum to C ∗ -dynamical systems, J. Funct. Anal. 30 (1978), 179–197. , Applications of the Connes spectrum to C ∗ -dynamical systems II, J. Funct. Anal. [27] 36 (1980), 18–32. [28] G. K. Pedersen, Pullback and pushout constructions in C ∗ -algebra theory, J. Funct. Anal. 167 (1999), 243–344. [29] N. C. Phillips, Equivariant K-Theory and Freeness of Group Actions on C ∗ -Algebras, Lecture Notes in Math., vol. 1274, Springer, Berlin, 1987. [30] J. C. Quigg, Landstad duality for C ∗ -coactions, Math. Scand. 71 (1992), 277–294. , Full and reduced C ∗ -coactions, Math. Proc. Camb. Phil. Soc. 116 (1994), 435–450. [31] [32] J. C. Quigg and I. Raeburn, Induced C ∗ -algebras and Landstad duality for twisted coactions, Trans. Amer. Math. Soc. 347 (1995), 2885–2915. [33] I. Raeburn, On crossed products and Takai duality, Proc. Edinburgh Math. Soc. 31 (1988), 321–330. [34] I. Raeburn and D. P. Williams, Pull-backs of C ∗ -algebras and crossed products by certain diagonal actions, Trans. Amer. Math. Soc. 287 (1985), 755–777. , Morita Equivalence and Continuous-Trace C ∗ -Algebras, Math. Surveys and Mono[35] graphs, vol. 60, Amer. Math. Soc., Providence, 1998. [36] M. A. Rieffel, Induced representations of C ∗ -algebras, Adv. in Math. 13 (1974), 176–257. , Unitary representations of group extensions; an algebraic approach to the theory [37] of Mackey and Blattner, Studies in Analysis, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, 1979, pages 43–82. , Proper actions of groups on C ∗ -algebras, Mappings of Operator Algebras, Progress [38] in Math., vol. 84, Birkh¨ auser, Boston, 1990, pages 141–182. , Integrable and proper actions on C ∗ -algebras, and square-integrable representations [39] of groups, Expositiones Math. 22 (2004), 1–53. [40] J. Schweizer, Crossed products by C ∗ -correspondences and Cuntz-Pimsner algebras, C ∗ Algebras (M¨ unster, 1999), Springer, Berlin, 2000, pages 203–226. ebres de Hopf et C ∗ -alg` ebres de Kac, Proc. London Math. Soc. 50 (1985), [41] J.-L. Vallin, C ∗ -alg` 131–174. [42] D. P. Williams, Crossed Products of C ∗ -Algebras, Math. Surveys and Monographs, vol. 134, Amer. Math. Soc., Providence, 2007.

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[43] S. L. Woronowicz, Pseudospaces, pseudogroups and Pontriagin duality, Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., vol. 116, Springer, Berlin, 1980, pages 407–412. , Unbounded elements affiliated with C ∗ -algebras and noncompact quantum groups, [44] Comm. Math. Phys. 136 (1991), 399–432. School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia Current address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand E-mail address: [email protected] School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia E-mail address: [email protected] Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681757

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Twists of K-theory and TMF Matthew Ando, Andrew J. Blumberg, and David Gepner Abstract. We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and ∞-category theory provided by [ABGHR]. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how this approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO0 . . . 4. We also discuss Poincar´e duality and umkehr maps in this setting.

Contents 1. Introduction 2. Classical Examples of Twisted Generalized Cohomology 3. Bundles of Module Spectra 4. The Generalized Thom Spectrum 5. Twisted Generalized Cohomology 6. Multiplicative Orientations and Comparison of Thom Spectra 7. Application: K(Z, 3), Twisted K-theory, and the Spinc Orientation 8. Application: Degree-four Cohomology and Twisted Elliptic Cohomology 9. Application: Poincare Duality and Twisted Umkehr Maps 10. Motivation: D-brane Charges in K-theory 11. An Elliptic Cohomology Analogue 12. Twists of Equivariant Elliptic Cohomology References

27 30 32 40 42 45 52 54 56 58 59 59 61

1. Introduction In [ABGHR], we and our co-authors generalize the classical notion of Thom spectrum. Let R be an A∞ ring spectrum: it has a space of units GL1 R which deloops to give a classifying space BGL1 R. To a space X and a map ξ : X → BGL1 R M. Ando was supported in part by NSF grant DMS-0705233. A. J. Blumberg was supported in part by NSF grant DMS-0906105. c Mathematical 0000 (copyright Society holder) c 2010 American

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ANDO, BLUMBERG, AND GEPNER

we associate an R-module Thom spectrum X ξ . Letting S denote the sphere spectrum, one finds that BGL1 S is the classifying space for stable spherical fibrations of virtual rank 0, and X ξ is equivalent to the classical Thom spectrum of the spherical fibration classified by ξ (as in, for example, [LMSM86]). We remark in the introduction to [ABGHR] that BGL1 R classifies the twists of R-theory. More precisely, we define the ξ-twisted R-homology of X to be def Rk (X)ξ = π0 R-mod(Σk R, X ξ ) ∼ = πk X ξ

and the ξ-twisted cohomology to be def

Rk (X)ξ = π0 R-mod(X ξ , Σk R), where here Σk denotes the k-fold suspension, or equivalently smashing with S k . In this paper, we expand on that remark, explaining how this definition generalizes both singular cohomology with local coefficients and the twists of K-theory studied by [DK70, Ros89, AS04]. The key maneuver is to focus on the ∞categorical approach to Thom spectra developed in [ABGHR] (where by ∞categories we mean the quasicategories of [Joy02, HTT]). We show that the ∞-category LineR of R-modules L which admit a weak equivalence R  L is a model for BGL1 R: we have a weak equivalence of spaces |LineR |  BGL1 R. One appeal of LineR is that, by construction, it classifies what one might call “homotopy local systems” of free rank-one R-modules. This flexible notion generalizes both classical local coefficient systems and bundles of spaces (such as bundles of Fredholm operators). As one might expect, our work is closely related to the parametrized spectra of May and Sigurdsson [MS06]; we discuss the relationship further in Section 3.4. As applications of our approach to twisted generalized cohomology, we explain how the twisting of K-theory by degree three cohomology is related to the Spinc orientation of Atiyah-Bott-Shapiro. Similarly, recall that there is a map (unique up to homotopy) λ

BSpin − → K(Z, 4) whose restriction to BSU is the second Chern class. The fiber of λ is called BString, and if V is a Spin vector bundle on X, then a String structure on V is a trivialization of λ(V ); that is, a map g in the diagram g

X

v

v

V

v

BString v; v π  / BSpin,

together with a homotopy πg ⇒ V. The work of Ando, Hopkins, and Rezk [AHR] constructs an E∞ String orientation of tmf , the spectrum of topological modular forms. (The discussion in this paper applies equally to the connective spectrum tmf and to the periodic spectrum T M F .) Associating to a vector bundle its underlying spherical fibration gives a map BSpin → BGL1 S, and associated to the unit S → tmf is a map BGL1 S → BGL1 tmf. Composing these, we have a map k : BSpin → BGL1 tmf.

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We show that the E∞ String orientation of tmf of [AHR] implies the following. Theorem 1.1. (1) tmf admits twists by degree-four integral cohomology. More precisely, there is a map h : K(Z, 4) → BGL1 tmf making the diagram k

BSpin −−−−→ BGL1 tmf ⏐ ⏐ ⏐ ⏐= λ  h

K(Z, 4) −−−−→ BGL1 tmf commute up to homotopy. Thus given a map z : X → K(Z, 4) (representing a class in H 4 (X, Z)) we can define def

tmf ∗ (X)z = tmf ∗ (X)hz . (2) If V is a Spin-bundle over X, classified by V

X −→ BSpin, then a homotopy hλ(V ) ⇒ k(V ) determines an isomorphism tmf ∗ (X V ) ∼ = tmf ∗ (X)λ(V ) . of modules over tmf ∗ (X). Theorem 1.1 is well-known to the experts (e.g. Hopkins, Lurie, Rezk, and Strickland). As the reader will see, with the approach to twisted cohomology presented here, it is an immediate consequence of the String orientation. We also explain how twisted generalized cohomology is related to Poincar´e duality. We briefly describe some work in preparation, concerning twisted umkehr maps in generalized cohomology. As special cases, we recover the twisted K-theory umkehr map constructed by Carey and Wang, and we construct an umkehr map in twisted elliptic cohomology. As we explain, our interest in the twisted elliptic cohomology umkehr map arose from conversations with Hisham Sati. Finally, we report two applications of twisted equivariant elliptic cohomology: we recall a result (due independently to the first author [And00] and Jacob Lurie), relating twisted equivariant elliptic cohomology to representations of loop groups, and we explain work of the first author and John Greenlees, relating twisted equivariant elliptic cohomology to the equivariant sigma orientation. Remark 1.2. If E is a cohomology theory and X is a space, then E ∗ (X) will refer to the unreduced cohomology. If Z is a spectrum, then we write E ∗ (Z) for the spectrum cohomology. We write Σ∞ + for the functor disjoint basepoint

Σ∞

−−−−−−−−−−→ (pointed spaces) −−→ (spectra), Σ∞ + : (spaces) − so we have by definition

E ∗ (X) ∼ = E ∗ (Σ∞ + X),

while the reduced cohomology is  ∗ (X) ∼ E = E ∗ (Σ∞ X). If V is a vector bundle over X of rank r, then we write X V for its Thom spectrum: this is equivalent to the suspension spectrum of the Thom space, so E ∗ (X V ) is the

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reduced cohomology of the Thom space. Thus the Thom isomorphism, if it exists, takes the form E ∗ (X) ∼ = E ∗+r (X V ). Remark 1.3. The ∞-category LineR is not the largest category we could use to construct twists of R-theory. If R is an E∞ ring spectrum, i.e. a commutative ring spectrum, then we could consider the ∞-category Pic(R), consisting of invertible Rmodules: R-modules L for which there exists an R-module M such that L ∧R M  R. Acknowledgments This paper is the basis for a talk given by the first author at the CBMS conference on C ∗ -algebras, topology, and physics at Texas Christian University in May 2009. We thank the organizers, Bob Doran and Greg Friedman, for the opportunity. It is a pleasure to acknowledge stimulating conversations with Alan Carey, Dan Freed, and Hisham Sati which directly influenced this write-up. 2. Classical Examples of Twisted Generalized Cohomology 2.1. Geometric models for twisted K-theory. Let H be a complex Hilbert space, and let F be its space of Fredholm operators. Then F is a representing space for K-theory: Atiyah showed [Ati69] that K(X) ∼ = π0 map(X, F) = π0 Γ(X × F → X). Atiyah and Segal [AS04] develop the following approach to twisted K-theory. The unitary group U = U (H) of H acts on the space F of Fredholm operators by conjugation. Associated to a principal P U -bundle P → X, then, we can form the bundle ξ = P ×P U F → X with fiber F. They define the P -twisted K-theory of X to be K(X)P = π0 Γ(ξ → X). Thus one twists K(X) by P U -bundles over X; isomorphism classes of these are classified by π0 map(X, BP U ); as BP U is a model for K(Z, 3), we have have π0 map(X, BP U ) ∼ = H 3 (X; Z). We warn the reader that this summary neglects important and delicate issues which Atiyah and Segal address with care, for example concerning the choice of topology on U and P U . Another approach to twisted K-theory passes through algebraic K-theory; again we neglect important operator-theoretic matters, referring the reader to [Ros89, BM00, BCMMS] for details. Conjugation induces an action of P U on the algebra K of compact operators on H. Thus from the P U -bundle P → X we can form the bundle P ×P U K. Let A = Γ(P ×P U K → X). This is a (non-unital) C ∗ -algebra, and K(X)P ∼ = K(A). If P is the trivial bundle, then A ∼ = map(X, K), and we have isomorphisms ∼ K(map(X, C)) = ∼ K(X). K(A) = Both of these approaches to twisted K-theory are based on the idea that from a P U -bundle we can build a bundle of copies of the representing space for K-theory,

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and both have had a number of successes. They demand a good deal of information about K-theory, and they exploit features of models of K-theory which may not be available in other cohomology theories. May and Sigurdsson show how to implement the construction of Atiyah and Segal in the setting of their theory of parametrized stable homotopy theory [MS06, §22]. Specifically, they give a construction of certain twisted cohomology theories associated to parametrized spectra, and explain how the Atiyah-Segal definition fits into their framework. However, the approach of May and Sigurdsson also takes advantage of good features of known models for K-theory which may not be available in other cohomology theories. In this paper we explain another way to locate twisted K-theory in stable homotopy theory. Our constructions continue to demand a good deal of K-theory, for example that it be an A∞ or E∞ ring spectrum, but many generalized cohomology theories E satisfy our demands, and so our approach works in those cases as well. Our approach incorporates and generalizes the construction of Thom spectra of vector bundles, and so clarifies standard results concerning twisted E-theory, such as the relationship to the Thom isomorphism and Poincar´e duality. It also generalizes the classical notion of (co)homology with local coefficients, as we now explain. 2.2. Cohomology with local coefficients. Let X be a space, and let Π≤1 (X) be its fundamental groupoid. We recall that a local coefficient system on X is a functor1 {A} : Π≤1 (X) → (Abelian groups). Given a local system {A} on X, we can form the twisted singular homology H∗ (X; {A}) and cohomology H ∗ (X; {A}). Example 2.1. For example, if π : E → X is a Serre fibration, then associating to a point p ∈ X the fiber Fp = π −1 (p) gives rise to a representation F• : Π≤1 (X) − → Ho(spaces). (Here Ho denotes the homotopy category obtained by inverting the weak equivalences.) Applying singular cohomology in degree r produces the local coefficient system {H r (F• )}. Example 2.2. If V is a vector bundle over X of rank r, then taking the fiberwise one-point compactification V + provides a Serre fibration, and so we have the local coefficient system (2.3)

 r (V•+ ); Z} {H

 r (Vp+ ; Z). whose value at p ∈ X is the cohomology group H From the Serre spectral sequence it follows that there is an isomorphism  r (V•+ )}) ∼ H ∗ (X; {H = H ∗+r (X V ; Z)  r (V•+ )} between the twisted cohomology of X with coefficients in the system {H and the cohomology of the Thom spectrum of V . 1Since the fundamental groupoid is a groupoid, it is equivalent to consider covariant or contravariant functors.

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An orientation of V is a trivialization of the local system (2.3), that is, an isomorphism of functors  r (V•+ )} ∼ {H =Z (where Z denotes the evident constant functor). It follows immediately that an orientation of V determines a Thom isomorphism H ∗ (X; Z) ∼ = H ∗+r (X V ; Z). 3. Bundles of Module Spectra 3.1. The problem. Let X be a space. We seek a notion of local system of spectra ξ on X, generalizing the bundles of Fredholm operators in §2.1 and the local systems of §2.2. In particular, if E → X is a Serre fibration as in Example 2.1, then the classical local system {H r (F• )} should arise from a bundle of spectra F• ∧ HZ by passing to homotopy groups. From this example, we quickly see that while it is reasonable to ask ξ to associate to each point p ∈ X a spectrum ξp , it is too much to expect to associate to a path γ : I → X from p to q an isomorphism of fibers; instead, we expect a homotopy equivalence ξ γ : ξp → ξq . Moreover, an (endpoint-preserving) homotopy of paths H : γ → γ  should give rise to a path ξH : ξ γ → ξγ  in the space of homotopy equivalences from ξp to ξq . These homotopy coherence issues quickly lead one to consider representations of not merely the fundamental groupoid Π≤1 (X), but the whole fundamental ∞groupoid Π≤∞ (X), that is, the singular complex Sing X. Quasicategories make it both natural and inevitable to consider such representations. 3.2. ∞-categories from spaces and from simplicial model categories. Recall that a quasicategory is a simplicial set which has fillings for all inner horns. Thus one source of quasicategories is spaces. If X is a space, then its singular complex Sing X is a Kan complex: it has fillings for all horns. From the point of view of quasicategories, where 1-simplices correspond to morphisms, this means that all morphisms are invertible up to (coherent higher) homotopy. Thus Kan complexes may be identified with “∞-groupoids”. We also recall (from [HTT, Appendix A and 1.1.5.9]) how simplicial model categories give rise to quasicategories. This procedure is an important source of quasicategories, and it provides intuition about how quasicategories encode homotopy theory. If M is a simplicial model category, then we can define M ◦ to be the full subcategory consisting of cofibrant and fibrant objects. The simplicial nerve of M ◦, C = N M ◦, is the quasicategory associated to M . By construction, C is the simplicial set in which (1) the vertices C0 are cofibrant-fibrant objects of M ;

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(2) C1 consists of maps L→M between cofibrant-fibrant objects; (3) C2 consists of diagrams (not necessarily commutative) /M L@ @@ @@ g h @@  N, f

together with a homotopy from gf to h in the mapping space (simplicial set) M (L, M ); and so forth. In particular, in C the equivalences correspond to weak equivalences in M ◦ , that is, homotopy equivalences. Thus we may sometimes refer to the equivalences in a quasicategory as weak equivalences or homotopy equivalences. A simplicial model category M has an associated homotopy category ho M , and an ∞-category C has a homotopy category ho C . As one would expect, there is an equivalence of categories (enriched over the homotopy category of spaces) ho M  ho N M ◦ . By analogy to the model category situation, if C is a quasicategory and ho C  D, then we shall say that C is a “model for D”. 3.3. The ∞-category of A-modules. Let S be a symmetric monoidal ∞category of spectra. Lurie constructs such an ∞-category from scratch ([DAGI] introduces an ∞-category of spectra, which is shown to be monoidal in [DAGII], and symmetric monoidal in [DAGIII]). Lurie shows that his ∞-category is equivalent to the symmetric monoidal ∞-category arising from the symmetric spectra of [HSS00], and so by [MMSS01] it is equivalent to the symmetric monoidal ∞categories of spectra arising from various classical symmetric monoidal simplicial model categories of spectra. Let S be the sphere spectrum. Definition 3.1. An S-algebra is a monoid (strictly speaking, an algebra, since the relevant monoidal structure is not given by the cartesian product) in S . We write Alg(S) for the ∞-category of S-algebras, and CommAlg(S) for the ∞-category of commutative S-algebras. Using [DAGII, DAGIII] and [MMSS01] as above, one learns that the symmetric monoidal structure on S is such that Alg(S) is a model for A∞ ring spectra, and CommAlg(S) is a model for E∞ ring spectra, so the reader is free to use his or her favorite method to produce ∞-categories equivalent to Alg(S) and CommAlg(S). Definition 3.2. If A is an S-algebra, we let ModA be the ∞-category of Amodules. Example 3.3. An S-module is just a spectrum, and so ModS is the ∞-category of spectra.

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3.4. Bundles of spaces and spectra. The purpose of this section is to introduce the ∞-categorical model of parametrized spectra we work with in this paper and compare it to the May-Sigurdsson notion of parametrized spectra [MS06]. We begin by reviewing some models for the ∞-category of spaces over a cofibrant topological space X. On the one hand, we have the topological model category T /X of spaces over X, obtained from the topological model category of spaces by forming the slice category; i.e., the weak equivalences and fibrations are determined by the forgetful functor to spaces. We will refer to this model structure as the “standard” model structure on T /X. This is Quillen equivalent to the corresponding simplicial model category structure on simplicial sets over Sing X, which in turn is Quillen equivalent to the simplicial model category of simplicial presheaves on the simplicial category C[Sing X] (with, say, the projective model structure) [HTT, §2.2.1.2]. Here C denotes the left adjoint to the simplicial nerve; it associates a simplicial category to a simplicial set [HTT, §1.1.5]. Remark 3.4. The Quillen equivalence between simplicial presheaves and parametrized spaces depends on the fact that the base is an ∞-groupoid (Kan complex) as opposed to an ∞-category; there is a more general theory of “right fibrations” (and, dually, “left fibrations”), but over a Kan complex a right fibration is a left fibration (and conversely) and therefore a Kan fibration. On the level of ∞-categories, this yields an equivalence St : NSet◦Δ/ Sing X −→ Fun(Sing X op , NSet◦Δ ); the map, called the straightening functor, rigidifies a fibration over Sing X into a presheaf of ∞-groupoids on Sing X whose value at the point x is equivalent to the fiber over x [HTT, §3.2.1]. A distinct benefit of the presheaf approach is a particularly straightforward treatment of the base-change adjunctions. Given a map of spaces f : Y → X, we may restrict a presheaf of ∞-groupoids F on Sing X to a presheaf of ∞-groupoids f ∗ F on Sing Y . This gives a functor, on the level of ∞-categories, from spaces over X to spaces over Y , such that the fiber of f ∗ F over the point y of Y is equivalent to the fiber of F over f (y). Moreover, f ∗ admits both a left adjoint f! and a right adjoint f∗ , given by left and right Kan extension along the map Sing Y op → Sing X op , respectively. Note that this is left and right Kan extension in the ∞-categorical sense, which amounts to homotopy left and right Kan extension on the level of simplicial categories or model categories. On the level of model categories of presheaves, there is an additional subtlety: f ∗ : Fun(C[Sing X op ], SetΔ ) −→ Fun(C[Sing Y op ], SetΔ ) is a right Quillen functor for the projective model structure, with (derived) left adjoint f! , and a left Quillen functor for the injective model structure, with (derived) right adjoint f∗ , on the above categories of (simplicial) presheaves. Of course the identity adjunction gives a Quillen equivalence between these two model structures, but nevertheless one is forced to switch back and forth between projective and injective model structures if one wishes to simultaneously consider both base-change adjunctions. Now we may stabilize either of the equivalent ∞-categories NSet◦Δ/ Sing X  Fun(Sing X op , NSet◦Δ )

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TWISTS OF K-THEORY AND TMF

35 9

by forming the ∞-category of spectrum objects in C∗ ; here C∗ denotes the ∞category of pointed objects in C . If C is an ∞-category with finite limits, then so is C∗ , and Stab(C ) is defined as the inverse limit of the tower Ω

Ω

Stab(C ) = lim{· · · −→ C∗ −→ C∗ } associated to the loops endomorphism Ω : C∗ → C∗ of C∗ . In other words, a spectrum object in an ∞-category C (with finite limits) is a sequence of pointed objects A = {A0 , A1 , . . .} together with equivalences An  ΩAn+1 for each natural number n. Thus, our category of parametrized spectra is the stabilization Stab(N(SetΔ / Sing X)◦ ). For our purposes, it turns out to be much more convenient to use the presheaf model; there is an equivalence of ∞-categories Stab(N(SetΔ / Sing X)◦ )  Fun(Sing X op , S ). Note that a functor F : Sing X → S associates to each point x of X a spectrum Fx , to each path x0 → x1 in X a map of spectra (necessarily a homotopy equivalence) Fx0 → Fx1 , and so on for higher-dimensional simplices of X. Given a presentable ∞-category C , the stabilization Stab(C ) is itself presentable, and the functor Ω∞ : Stab(C ) → C admits a left adjoint Σ∞ : C → Stab(C ) [DAGI, Proposition 15.4]. Just as Ω∞ is natural in presentable ∞categories and right adjoint functors, dually, Σ∞ is natural in presentable ∞categories and left adjoint functors [DAGI, Corollary 15.5]. In particular, given a map of spaces f : Y → X, the adjoint pairs (f! , f ∗ ) and (f ∗ , f∗ ) defined above extend to the stabilizations, yielding a restriction functor f ∗ : Fun(Sing X op , S ) −→ Fun(Sing Y op , S ) which admits a left adjoint f! and a right adjoint f ∗ , again given by left and right Kan extension, respectively. We can also formally stabilize suitable model categories, using Hovey’s work on spectra in general model categories [Hov01]. Specifically, given a left proper cellular model category C and an endofunctor of C , Hovey constructs a cellular model category SpN C of spectra. When C is additionally a simplicial symmetric monoidal model category, the endofunctor given by the tensor with S 1 yields a simplicial symmetric monoidal model category of symmetric spectra SpΣ C (as well as a simplicial model category SpN C of prespectra). These models of the stabilization are functorial in left Quillen functors which are suitably compatible with the respective endofunctors (see [Hov01, 5.2]). In order to compare our model of parametrized spectra over X to the MaySigurdsson model, we use the following consistency result. Proposition 3.5. Let C be a left proper cellular simplicial model category and write SpN C for the cellular simplicial model category of spectra generated by the tensor with S 1 . Then there is an equivalence of ∞-categories N(SpN C )◦  Stab(NC ◦ ). Proof. The functors Evn : SpN C → C , which associate to a spectrum its n -space An , induce a functor (of ∞-categories) th

f : N(SpN C )◦ → lim{· · · −→ NC∗◦ −→ NC∗◦ }  Stab(NC ◦ ) Ω

Ω

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36 10

ANDO, BLUMBERG, AND GEPNER

which is evidently essentially surjective. To see that it is fully faithful, it suffices to check that for cofibrant-fibrant spectrum objects A and B in SpN C , there is an equivalence of mapping spaces Ω

Ω

map(A, B)  holim{· · · −→ map(A1 , B1 ) −→ map(A0 , B0 )}, where Ω : map(An+1 , Bn+1 ) → map(An , Bn ) sends An+1 → Bn+1 to An  ΩAn+1 → ΩBn+1  Bn . Since any cofibrant A is a retract of a cellular object, inductively we can reduce to the case in which A = Fm X, i.e., the shifted suspension spectrum on a cofibrant object X of C∗ . Then map(A, B)  map(X, Bm ) by adjunction. The latter is in turn equivalent to map(Σn−m X, Bn ), where we interpret Σn−m X = ∗ for m > n, in which case the homotopy limit is equivalent to that of the homotopically constant (above degree n) tower whose nth term is map(Σn−m X, Bn ).  We now recall the May-Sigurdsson setup. Given a space X, let (T /X)∗ denote the category of spaces over and under X (ex-spaces). Although this category has a model structure induced by the standard model structure on T /X, one of the key insights of May and Sigurdsson is that for the purposes of parametrized homotopy theory it is essential to work with a variant they call the qf -model structure [MS06, 6.2.6]. This model structure is Quillen equivalent to the standard model structure on ex-spaces [MS06, 6.2.7]; however, its cofiber and fiber sequences are compatible with classical notions of cofibration and fibration (described in terms of extension and lifting properties). May and Sigurdsson then construct a stable model structure on the categories SX of orthogonal spectra in (T /X)∗ [MS06, 12.3.10]. This model structure is based on the qf -model structure on ex-spaces, leveraging the diagrammatic viewpoint of [MMSS01, MM02]. Similarly, they construct a stable model structure on the category PX of prespectra in (T /X)∗ ; the forgetful functor SX → PX is a Quillen equivalence [MS06, 12.3.10]. Using [MS06, 12.3.14], we see that after passing to ∞-categories the category PX is in turn equivalent to the category SpN (T /X)∗ ; the formal stabilization of the qf -model structure on (T /X)∗ with respect to the fiberwise smash with S 1 . Using Proposition 3.5 and the fact that the qf -model structure is Quillen equivalent to the standard model structure, we obtain equivalences of ∞-categories N(SX )◦ → N(PX )◦ → N(SpN (T /X)∗ )◦ → Stab(N(T /X)◦ ) → Stab(N(SetΔ / Sing X)◦ ) → Fun(Sing X op , S ). Thus we obtain the following comparison theorem. Theorem 3.6. There is an equivalence of ∞-categories between the simplicial nerve of the May-Sigurdsson category of parametrized orthogonal spectra N(SX )◦ and the ∞-category F un(Sing X op , S ) of presheaves on X with values in spectra. Furthermore, the derived base-change functors we construct via the stabilization of the presheaves agree with the derived base-change functors constructed by May and Sigurdsson. To see this, observe that it suffices to check this for f ∗ ; compatibility then follows formally for the adjoints f∗ and f! . Moreover, since f ∗ on the categories of spectra is obtained as the suspension of f ∗ on spaces, we can reduce to

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checking that the right derived functor of f ∗ : (T /X)∗ → (T /Y )∗ in the qf -model structure is compatible with the right derived functor of f ∗ : Fun(C(Sing X op ), SetΔ ) → Fun(C(Sing Y op ), SetΔ ) in the projective model structure. By the work of [MS06, §9.3], it suffices to check the compatibility for f ∗ in the q-model structure. Since both versions of f ∗ that arise here are Quillen right adjoints, this amounts to the verification that the diagram Fun(C(Sing X op ), SetΔ ) 

f∗

/ Fun(C(Sing Y op ), SetΔ )

Un f∗

SetΔ /X



Un

/ SetΔ /Y

commutes when applied to fibrant objects, where here Un denotes the unstraightening functor (which is the right adjoint of the Quillen equivalence). Finally, this follows from [HTT, 2.2.1.1]. 3.5. Bundles of A-modules and A-lines. If X is a space, let Sing X be its singular complex. The work of the previous section justifies the following definition. Definition 3.7. A bundle or homotopy local system of A-modules over X is a map of simplicial sets f : Sing X → ModA . Similarly if Y is any ∞-groupoid, then a bundle of A-modules over Y is just a map of simplicial sets f : Y → ModA . Thus f assigns (0) to each point p ∈ X an A-module f (p); (1) to each path γ from p to q a map of A-modules f (γ) : f (p) → f (q);

(3.8)

(2) to each 2-simplex σ : Δ2 → X, say p? ?? ??σ02 σ01 ??   / r, q σ12

a path f (σ) in ModA (f (p), f (r)) from f (σ12 )f (σ01 ) to f (σ02 ); and so forth. Recall [HTT, 1.2.7.3] that if Y is a simplicial set and C is an ∞-category, then the simplicial mapping space C Y is the ∞-category Fun(Y, C) of functors from Y to C. Definition 3.9. The ∞-category of bundles of A-modules over X is the simplicial mapping space def

ModX A = Fun(Sing X, ModA ).

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ANDO, BLUMBERG, AND GEPNER

Remark 3.10. We have not set up the framework necessary to work directly with the bundle of A-modules associated to f : Sing X → ModA (although see Theorem 3.6). Nonetheless, the notation of bundle and pullback is compelling, and so we write M for the identity map ModA → ModA , and if f : Sing X → ModA is a map of ∞-categories, then we may write f ∗ M as a synonym for f , when we want to emphasize its bundle aspect. Recall that Sing X is a Kan complex or ∞-groupoid: it satisfies the extension condition for all horns. Viewing an ∞-category as a model for a homotopy theory, an ∞-groupoid models a homotopy theory in which all the morphisms are homotopy equivalences. In particular, the map f (γ) in (3.8) is necessarily an equivalence: the A-modules f (p) will vary through weak equivalences as p varies over a path component of X. We shall be particularly interested in the case that these fibers are free rank-one A-modules. Definition 3.11. An A-line is an A-module L which admits a weak equivalence 

L− → A. The ∞-category LineA is the maximal ∞-groupoid in ModA generated by the Alines. We write j for the inclusion j : LineA → ModA def

and L = j ∗ M for the tautological bundle of A-lines over LineA . By construction LineA is a Kan complex, and we regard it as the classifying space for bundles of A-lines. If X is a space, then a map f : Sing X → LineA . assigns (0) to each point p ∈ X an A-line f (p); (1) to each path γ from p to q an equivalence map of A-lines f (γ) : f (p)  f (q); (2) to each 2-simplex σ : Δ2 → X, say p? ?? ??σ02 σ01 ??   / r, q σ12

a path f (σ) in LineA (f (p), f (r)) from f (σ12 )f (σ01 ) to f (σ02 ); and so forth. Definition 3.12. The simplicial mapping space LineX A = Fun(Sing X, LineA ) is an ∞-category (in fact, a Kan complex); we call it the the ∞-category or space of A-lines over X. We develop twisted A-theory starting from LineA in §5. Before doing so, we briefly discuss other aspects of the ∞-category LineA .

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3.6. LineA and GL1 A. By construction, LineA is connected, and so equivalent to the maximal ∞-groupoid B Aut(A) on the single A-module A. As we discuss in [ABGHR, §6], it is an important point that the space of morphisms Aut(A) = LineA (A, A) is not a group, or even a monoid, but instead merely a group-like A∞ space. Nevertheless, LineA is not only the classifying space for bundles of A-lines, but it is a delooping of Aut(A). To see this, let Triv(A) be the ∞-category of A-lines  L, equipped with an equivalence L − → A. Then [ABGHR, Prop. 7.38] Triv(A) is contractible, and the map Triv(A) → LineA is a Kan fibration, with fiber Aut(A). Classical infinite loop space theory provides another model for homotopy type Aut(A). Namely, let A be an A∞ ring spectrum in the sense of [LMSM86]: so π0 Ω∞ A is a ring. Let GL1 A be the pull-back in the diagram GL1 A ⏐ ⏐ 

−−−−→

Ω∞ A ⏐ ⏐ 

(π0 Ω∞ A)× −−−−→ π0 Ω∞ A. Then GL1 A is a group-like A∞ space: π0 GL1 A is a group. We show that GL1 A  | Aut(A)|. Since the geometric realization of a Kan fibration is a Serre fibration [Qui68], the fibration Aut(A) → Triv(A) → LineA gives rise to a fibration (3.13)

GL1 A  | Aut(A)| → |Triv(A)|  ∗ → |LineA |.

Thus |LineA | provides a model for the delooping BGL1 A. It has the virtue that we have already given a precise description of the vertices of the simplicial mapping space LineX A = map(Sing X, LineA )  map(X, |LineA |). Example 3.14. If S is the sphere spectrum, then Ω∞ S is the space QS 0 = Ω Σ∞ S 0 , and GL1 S = Q±1 S 0 , ∞

i.e., the unit components. The space BGL1 S  |LineS | is the classifying space for stable spherical fibrations of virtual rank 0. It follows the space of S-lines over X is homotopy equivalent to the space of spherical fibration of virtual rank 0. Example 3.15. The classical J-homomorphism is a map J : O → GL1 S, which deloops to give a map BJ : BO → BGL1 S. One sees that this is the map which takes a virtual vector bundle of rank 0 to its associated stable spherical fibration; we may regard this as associating to a vector bundle its bundle of S-lines.

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40 14

ANDO, BLUMBERG, AND GEPNER

Example 3.16. If A is an S-algebra, then the unit of A induces a map BGL1 S → BGL1 A. In our setting, this map arises from the map of ∞-categories ModS → ModA given by M → M ⊗S A = M ∧S A, which restricts to give a map of ∞-categories LineS → LineA . Example 3.17 ([MQRT77]). Let HZ be the integral Eilenberg-MacLane spectrum. Then Ω∞ HZ  K(Z, 0)  Z, and so GL1 HZ  {±1}  Z/2, and BGL1 HZ  BZ/2  K(Z/2, 1). Remark 3.18. If A = K, the spectrum representing complex K-theory, then Aut(K) has the homotopy type of the space of K-module equivalences K → K. Atiyah and Segal [AS04] build twists of K-theory from P U -bundles. They remark that one can more generally build twists of K-theory from G-bundles, where G is the group of strict K-module automorphisms of K-theory. Our space Aut(K) generalizes this idea. Remark 3.19. As we explain in [ABGHR, §6], for many algebras A (including the sphere S), the group Autstrict (A) of strict A-module automorphisms of A cannot provide a sufficiently rich theory of bundles of A-modules. For example, Lewis’s Theorem [Lew91] implies that there is no model for the sphere spectrum S such that the classifying space B Autstrict (S) classifies stable spherical fibrations. (See also [MS06, §22.2] for discussion of this issue.) 4. The Generalized Thom Spectrum Let A be an S-algebra, let X be a space, and let f be a bundle of A-lines over X, that is, a map of simplicial sets f : Sing X → LineA . Although the ∞-category LineA is not cocomplete (it doesn’t even have sums), the ∞-category ModA of A-modules is complete and cocomplete. This allows us in [ABGHR] to make the following definition. Definition 4.1. The Thom spectrum of f is the colimit   j f def X f = colim Sing X − → LineA − → ModA . Equivalently, X f is the left Kan extension Lπ jf in the diagram / ModA . j5 j j j π jLj(f j) j p  j ∗j X

f

/ LineA

j

The colimit and left Kan extension here are ∞-categorical colimits: they are generalizations of the notion of homotopy colimit and homotopy left Kan extension. It is an important achievement of ∞-category theory to give a sensible definition of these colimits.

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41 15

Let AX : X → ∗ → LineA be the map which picks out A, considered as the constant A-line over X. The colimit means that we have an equivalence of mapping spaces ∗ ∗ ModA (X f , A)  ModX A (f j M , AX ). Notice also that we have a natural inclusion X ∗ ∗ ∗ LineX A (f L , AX ) → ModA (f j M , AX ) :

(4.2)

a map of bundles of A-modules f ∗ L → AX is a map of bundles of A-lines if it is an equivalence over every point of X, and one checks that the inclusion (4.2) is the inclusion of a set of path components. Definition 4.3. The space of orientations of X f is the pull-back in the diagram orient(X f , A) ⏐ ⏐ 

(4.4)

−−−−→

ModA (X f , A) ⏐ ⏐ 

X ∗ ∗ ∗ LineX A (f L , AX ) −−−−→ ModA (f j M , AX ).

That is, the space of orientations orient(X f , A) is the subspace of A-module maps X f → A which correspond, under the equivalence ∗ ∗ ModA (X f , A)  ModX A (f j M , AX ),

to fiberwise equivalences f ∗ L → AX . This appealing notion of orientation expresses orientations as fiberwise equivalences of bundles of spectra. The following results from [ABGHR] explain how our Thom spectra and orientations generalize the classical notions. Theorem 4.5. Suppose that ξ : Sing X → LineS corresponds to a map g : X → BGL1 S. Then X ξ is equivalent to the classical Thom spectrum X g of the spherical fibration classified by g. It follows (see Example 3.16) that if f is the composition ξ

f : Sing X − → LineS → LineA , then X f  X ξ ∧S A  X g ∧S A is equivalent to the classical Thom spectrum tensored with A. We can then study the space of orientations of X f via the equivalences (4.6)

ModA (X f , A)  ModA (X g ∧S A, A)  ModS (X g , A),

and we find that Proposition 4.7. A map α : X f → A ∈ ModA (X f , A) is in orient(X f , A) if and only if it corresponds to an orientation β : X g → A of the classical Thom spectrum, that is, if and only if z

Δ

z∧β

(X+ − → A) → (X g −→ X+ ∧ X g −−→ A ∧ A → A) induces an isomorphism

A∗ (X+ ) ∼ = A∗ (X g ).

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ANDO, BLUMBERG, AND GEPNER

Our theory leads to an obstruction theory for orientations. Let mapf (Sing X, Triv(A)) be the simplicial set which is the pull-back in the diagram mapf (Sing X, Triv(A)) −−−−→ map(Sing X, Triv(A)) ⏐ ⏐ ⏐ ⏐   {f }

−−−−→ map(Sing X, LineA ).

That is, mapf (Sing X, Triv(A)) is the mapping simplicial set of lifts in the diagram (4.8) s Sing X

s

s f

s

Triv(A) s9  / LineA .

The obstruction theory for orientations of the bundle of A-modules is given by the following. Theorem 4.9. Let f : Sing X → LineA be a bundle of A-lines over X, and let X f be the associated A-module Thom spectrum. Then there is an equivalence f mapf (Sing X, Triv(A))  LineX A (f, ι)  orient(X , A).

In particular, the bundle f ∗ L admits an orientation if and only if f is nullhomotopic. Example 4.10. This theorem recovers and slightly generalizes the obstruction theory of [MQRT77] (which treats the case that A is a E∞ ring spectrum, that is, a commutative S-algebra). Let g : X → BGL1 S be a stable spherical fibration. Then g admits a Thom isomorphism in A-theory if and only if the composition g

X− → BGL1 S  |LineS | → |LineA |  BGL1 A is null. Example 4.11. This example appears in [MQRT77]. Let HZ be the integral Eilenberg-MacLane spectrum. From Example 3.17 we have BGL1 HZ  K(Z/2, 1). The obstruction to orienting a vector bundle V /X in singular cohomology is the map V

BJ

X −→ BO −−→ BGL1 S − → BGL1 HZ  K(Z/2, 1); this is just the first Stiefel-Whitney class. 5. Twisted Generalized Cohomology Now we consider twisted generalized cohomology in the language of sections 3 and 4. Let A be an S-algebra, and let f : Sing X → LineA or, equivalently, f : X → BGL1 A (see §3.6) classify a bundle of A-lines over X. As in Definition 4.1, let   j f X f = colim Sing X − → LineA − → ModA

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43 17

be the indicated A-module. We think of X f as the f -twisted cohomology object associated to the bundle f , and we make the following Definition 5.1. The f -twisted A homology and cohomology groups of X are def

An (X)f = π0 ModA (X f , Σn A) def

An (X)f = π0 ModA (Σn A, X f ). Equivalently, we have An (X)f = π−n FA (X f , A) An (X)f = πn FA (A, X f ) ∼ = πn X f . Here if V and W are A-modules, then FA (V, W ) is the function spectrum of A-module maps from V to W : it is a spectrum such that Ω∞ FA (V, W )  ModA (V, W ). Thus for n ≥ 0, πn FA (V, W ) ∼ = πn ModA (V, W ) ∼ = ModA (Σn V, W ) ∼ = ModA (V, Σ−n W ). Example 5.2. Suppose that V is a vector bundle over X. Then we can form the map V

BJ

j(V ) : X −→ BO −−→ BGL1 S − → BGL1 A. and also the twisted cohomology A∗ (X)j(V ) = π0 ModA (X j(V ) , Σ∗ A). Since by Theorem 4.5 X j(V )  X V ∧ A, we have (5.3)

A∗ (X)j(V ) ∼ = π0 ModS (X V , Σ∗ A) = A∗ (X V ),

so in this case the twisted cohomology is just the cohomology of the Thom spectrum. The definition is not quite a direct generalization of that Atiyah and Segal. Let SX : Sing X → ModS be the constant functor which attaches to each point of X the sphere spectrum S. Theorem 3.6 and the work of [MS06, §22] allow us to describe their construction as attaching to the bundle of A-lines f ∗ L over X the group ∗ A(X)f,AS = π0 Γ(f /X) ∼ = π0 ModX S (SX , f L ). To compare this definition to ours, we must assume (as is the case for K-theory, for example) that A is a commutative S-algebra. In that case, if L is an A-module, then the dual spectrum L∨ = FA (L, A) is again an A-module. As in the case of classical commutative rings, the operation L → L∨ defines an involution def

( − )∨

LineA −−−−→ LineA on the ∞-category of A-lines, such that L∨ ∧A M  FA (L, M );

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44 18

ANDO, BLUMBERG, AND GEPNER

in particular ModS (S, L∨ )  ModA (A, L∨ )  ModA (L, A). Remark 5.4. As we have written it, we have an evident functor LineA → (LineA )op . As LineA is an ∞-groupoid, it admits an equivalence LineA  Lineop A . This is an ∞-categorical approach to the following: if A is a commutative S-algebra then GL1 A is a sort of commutative group. More precisely, it is a commutative group-like monoid in the ∞-category of spaces, or equivalently it is a group-like E∞ -space. As such it has an involution −1 : GL1 A → GL1 A which deloops to a map B(−1) : BGL1 A → BGL1 A. In any case, given a map f : Sing X → LineA , classifying the bundle f ∗ L , we may form the map f

( − )∨

→ LineA −−−−→ LineA −f = f ∨ : Sing X − so that (−f )∗ L is the fiberwise dual of f ∗ L , and then one has ∗ Γ((−f )∗ L ) = ModX S (SX , (−f ) L ) ∗  ModX A (AX , (−f ) L ) ∗  ModX A (f L , A)

 ModA (X f , A). That is, the cohomology object we associate to f : X → BGL1 A is the one which Atiyah and Segal associate to −f : X → BGL1 A, A∗ (X)f ∼ = A∗ (X)−f,AS . Of course Atiyah and Segal also explain how to construct a K-line from a P U -bundle over X: in our language, they construct a map BP U  K(Z, 3) → BGL1 K. From our point of view the existence of this map can be phrased as a question about the Spinc orientation of complex K-theory. To see this, recall that Atiyah, Bott, and Shapiro [ABS64] produce a Thom isomorphism in complex K-theory for Spinc -bundles. According to Theorem 4.9,

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this corresponds to the arrow labeled ABS in the diagram / GL1 K

K(Z, 2)  BSpinc  BSO

ABS

BJ

 / Triv(K)  ∗

 / LineS ( − )∧K/ LineK  BGL1 K

βw2

 BABS K(Z, 3) _ _ _ _ _ _ _ _ _ _/ BGL1 K

The map ABS induces a map (5.5)

ABS : K(Z, 2) → GL1 K,

and the diagram suggests that we ask whether this map deloops to give a map B(ABS) : K(Z, 3) → BGL1 K, as indicated. Not surprisingly, this question is related to the multiplicative properties of the orientation. The Thom spectrum associated M Spinc associated to BSpinc is a commutative S-algebra, as is K-theory. The construction of Atiyah-Bott-Shapiro produces a map of spectra t : M Spinc → K, and Michael Joachim [Joa04] shows that t can be refined to a map of commutative S-algebras. As we shall see in §6, it follows that K(Z, 2) → GL1 K is a map of infinite loop spaces. We use these ideas to twist K-theory by maps X → K(Z, 3) in §7. 6. Multiplicative Orientations and Comparison of Thom Spectra The most familiar orientations are exponential: the Thom class of a Whitney sum is the product of the Thom classes. For example, consider the case of Spin bundles and real K-theory, KO. Atiyah, Bott, and Shapiro show that the Dirac operator associates to a spin vector bundle V → X a Thom class t(V ) ∈ KO(X V ). If M Spin is the Thom spectrum of the universal Spin bundle over BSpin, then we can view their construction as corresponding to a map of spectra t : M Spin → KO. If W → Y is another spin vector bundle, then (X × Y )V ⊕W  X V ∧ Y W , and it turns out that with respect to the resulting isomorphism KO((X × Y )V ⊕W ) ∼ = KO(X V ∧ Y W ), one has (6.1)

t(V ⊕ W ) = t(V ) ∧ t(W ).

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ANDO, BLUMBERG, AND GEPNER

Now the sum of vector bundles gives M Spin the structure of a ring spectrum, and so the multiplicative property (6.1) (together with a unit condition, which says that t(0) = 1) corresponds to the fact that t : M Spin → KO is a map of monoids in the homotopy category of spectra. It is important that t is in fact a map of commutative monoids in the ∞-category of spectra. More precisely, M Spin and KO are both commutative S-algebras, and it turns out [Joa04, AHR] that t is a map of commutative S-algebras. The construction of classical Thom spectra such as M Spin, M SO, M U as commutative S-algebras (equivalently, E∞ ring spectra) is due to [MQRT77, LMSM86]. In this section, we discuss the theory from the ∞-categorical point of view. We’ll see (Remark 6.23) that this gives a way to think about the comparison of our Thom spectrum to classical constructions. It also provides some tools we use to build twists of K-theory from P U -bundles. We begin with a question. Suppose that A is a commutative S-algebra. Under what conditions on a map f should we expect that the Thom j

f

X f = colim(Sing X − → LineA − → ModA ) is a commutative A-algebra? And in that situation, how do we understand Aalgebra maps out of X f ? In the context of ∞-categories, spaces play the role which sets play in the context of classical categories, and so we begin by studying the situation of a discrete commutative ring R and a set X. In that case, an R-line is just a free rank-one R-module, and a bundle of R-lines ξ over X is just a collection of R-lines, indexed by the points x ∈ X. We can think of this as a functor ξ : X → LineR from X, considered as a discrete category, to the category of R-lines: free rank-one R-modules and isomorphisms. The “Thom spectrum”   ξ X ξ = colim X − → LineR − → ModR is easily seen to be the sum Xξ ∼ =



ξx .

x∈X

Now suppose that R is a commutative ring, so that ModR is a symmetric monoidal category, and LineR is the maximal sub-groupoid of ModR generated by R. If X is a discrete abelian group, then we may consider X as a symmetric monoidal category with objects the elements of X. It is then not difficult to check the following. Proposition 6.2. If ξ : X → LineR is a map of symmetric monoidal categories, then X ξ has structure of a commutative R-algebra. The analogue of this result holds in the ∞-categorical setting; see Theorem 6.21 below. It is possible to give a direct proof; instead we sketch the circuitous proof given in [ABGHR], as some of the results which arise along the way will be useful in sections 7 and 8.

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We begin with another construction of the R-module X ξ in the discrete associative case. Let GL1 R be the group of units of R. Note that the free abelian group functor Z : (sets) → (abelian goups)

(6.3) induces a functor

Z : (groups) o

(6.4)

/

(rings) : GL1

whose right adjoint is GL1 . In particular, we have a natural map of rings Z[GL1 R] → R and so the colimit-preserving functor Z[ − ]⊗Z[GL

R] R

(GL1 R-sets) −−−−−−−−1−−→ ModR . This functor restricts to an equivalence of categories /

Z[ − ] ⊗Z[GL1 R] R : Tors(GL1 R) o

(6.5)

LineR : T .

Here Tors(GL1 R) is the category of GL1 R-torsors, and the inverse equivalence is the functor T which associates to an R-line L the GL1 R-torsor def T (L) = LineR (R, L) ∼ = {u ∈ L|Ru ∼ = L}.

That is, we have the following diagram of categories which commutes up to natural isomorphism Line O R Z[ − ]⊗Z[GL1 R] R ∼ = T

 Tors(GL1 R)

/ ModR O Z[ − ]⊗Z[GL1 R] R

/ (GL1 R-sets).

Moreover, the vertical arrows preserve colimits, and the left vertical arrows comprise an equivalence. If ξ is a bundle of R-lines over X, we write P (ξ) for the GL1 R-set   ξ T P (ξ) = colim X − → LineR − → Tors(GL1 R) − → (GL1 R-sets) . That is, P (ξ) is the GL1 R-torsor over X whose fiber at x ∈ X is P (ξ)x = T (ξx ). Proposition 6.6. There is a natural isomorphism of R-modules Xξ ∼ = Z[P (ξ)] ⊗Z[GL1 R] R. Proof. We have   ξ X ξ = colim X − → LineR → ModR  Z[ − ]⊗Z[GL1 R] R Tξ ∼ → (GL1 R-sets) −−−−−−−−−−→ ModR = colim X −−→ Tors(GL1 R) − ∼ = Z[P (ξ)] ⊗Z[GL1 R] R. 

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ANDO, BLUMBERG, AND GEPNER

Figure 1. Selected instances of the analogy Sets:Spaces::Categories:∞-categories Alternate name/description Categorical notion ∞-categorical notion ∞-category Category ∞-groupoid Set Space/Kan complex monoidal ∞-groupoid Monoid A∞ space; ∗-algebra group-like monoidal group-like A∞ space Group ∞-groupoid group-like E∞ space; Abelian group group-like symmetric (−1)-connected spectrum monoidal ∞-groupoid spectrum Abelian group The sphere spectrum S The ring Z Ring Monoid in spectra S-algebra or A∞ ring spectrum Commutative ring Commutative monoid in spectra Commutative S-algebra or E∞ ring spectrum The functor Σ∞ The functor Z + ModA ModR BGL1 A LineA LineR GL1 A LineA (A, A) = AutA (A) GL1 R GL1 A-space A∞ GL1 A-space GL1 R-set Tors(GL1 A) GL1 R-torsors ⊗ ∧

Now suppose that R is a commutative ring, and X is an abelian group. If ξ : X → LineR is a symmetric monoidal functor, then GL1 R → P (ξ) → X is an extension of abelian groups. The adjunction (6.4) restricts further to an adjunction / (6.7) Z : (abelian groups) o (commutative rings) : GL1 , and so we have maps of commutative rings Z[P (ξ)] ← Z[GL1 R] → R. The isomorphism of Proposition 6.6 has the following consequence. Proposition 6.8. If R is a commutative ring and ξ : X → LineR is a symmetric monoidal functor, then X ξ is a commutative R-algebra; indeed, it is the pushout in the category of commutative rings Z[GL1 R] −−−−→ ⏐ ⏐ 

R ⏐ ⏐ 

Z[P (ξ)] −−−−→ X ξ . The preceding discussion generalizes elegantly and directly to spaces and spectra. The generalization illustrates how spaces play the role in ∞-categories that sets play in categories. The reader may find it useful to consult the table in Figure 6.

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Let T be an ∞-category of spaces. The analogue of the adjunction (6.3) is / Σ∞ (6.9) S : Ω∞ . + : T o Definition 6.10. Let Alg(∗) be the ∞-category of monoids in the ∞-category of T . We introduce the awkward name ∗-algebra to emphasize that these are more general than monoids with respect to the classical product of topological spaces. The symmetric monoidal structure on T is such that Alg(∗) is a model for the ∞-category of A∞ spaces. A ∗-algebra X is group-like if π0 X is a group. Definition 6.11. We write Alg(∗)× for the ∞-category of group-like monoids. The adjunction (6.9) restricts to an adjunction / × Σ∞ Alg(S) : GL1 . (6.12) + : Alg(∗) o Remark 6.13. If X is a group-like monoid in T , then Sing X is a group-like monoidal ∞-groupoid. One way to see that a group-like A∞ space is the appropriate generalization of a group is to observe that if Z is an object in a category, then End(Z) is a monoid, and Aut(Z) is a group. If Z is an object in an ∞-category, then End(Z) is a monoidal ∞-groupoid, and Aut(Z) is group-like. In particular, as we have already discussed in §3.6, one construction of the right adjoint GL1 is the following. If A is an S-algebra, then it has an ∞-category of modules ModA . In ModA we have the maximal sub-∞-groupoid LineA whose objects are weakly equivalent to A. We can define GL1 A = Aut(A) = LineA (A, A) to be the subspace of (the geometric realization of) ModA (A, A) consisting of homotopy equivalences: it is a group-like ∗-algebra. We also write BGL1 A for the full ∞-subcategory of LineA on the single object A: this is the ∞-groupoid with Aut A as its simplicial set of of morphisms. Since GL1 A is a (group-like) monoid in the symmetric monoidal ∞-category of spaces, we can form the ∞-category of GL1 A-spaces, and then define Tors(GL1 A) to be the maximal subgroupoid whose objects are GL1 A-spaces P which admit an equivalence of GL1 A-spaces GL1 A  P. The adjunction (6.12) provides a map of S-algebras Σ∞ + GL1 A → A, and so a (∞-category) colimit-preserving functor Σ∞ A : (GL1 A-spaces) − → ModA , + ( − ) ∧ Σ∞ + GL1 A which restricts to an equivalence of ∞-categories Σ∞ A : Tors(GL1 A) → LineA . + ( − ) ∧ Σ∞ + GL1 A The inverse equivalence is the functor T : LineA → Tors(GL1 A) which to an A-line L associates the GL1 A-torsor def

T (L) = LineA (A, L).

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ANDO, BLUMBERG, AND GEPNER

Putting all these together, we have the homotopy commutative diagram of ∞-categories / ModA Line O A O Σ∞ GL1 A A  T + ( − )∧Σ∞ +

Σ∞ + ( − )∧Σ∞ GL1 A A +

 Tors(GL1 A)

/ (GL1 A-spaces),

in which the vertical arrows preserve ∞-categorical colimits, and the left vertical arrows comprise an equivalence. Now let X be a space, and let ξ : X → LineA be a bundle of A-lines over X. Recall that   ξ → LineA − → ModA . X ξ = colim Sing X − On the other hand, let   ξ T P (ξ) = colim Sing X − → LineA − → Tors(GL1 A) − → (GL1 A-spaces) . We have the following analogue of Proposition 6.6. Proposition 6.14. There is a natural equivalence of A-modules (6.15)

X ξ  Σ∞ A. + P (ξ) ∧Σ∞ + GL1 A

Now we turn to the commutative case. Definition 6.16. We write CommAlg(∗) for the ∞-category of commutative monoids in T . It is equivalent to the nerve of the simplicial category of (cofibrant and fibrant) E∞ spaces. We write CommAlg(∗)× for the ∞-category of group-like commutative monoids, which models group-like E∞ spaces. The adjunction (6.12) restricts to the analogue of the adjunction (6.7), namely / × Σ∞ (6.17) CommAlg(S) : GL1 + : CommAlg(∗) o The reader will notice that in the table in Figure 6, we mention two models for “abelian groups”, namely, group-like E∞ spaces and spectra. It is a classical theorem of May [May72, May74], reviewed for example in [ABGHR, §3], that the functor Ω∞ induces an equivalence of ∞-categories Ω∞ : ((−1)-connected spectra)  CommAlg(∗)× , and so we may rewrite the adjunction (6.17) as (6.18) ∞ Σ∞ + Ω : ((−1)-connected spectra)

o

 Ω

∞

×

/ CommAlg(∗) o

Σ∞ +

/

CommAlg(S) : gl1

GL1

(The left adjoints are written on top, but the pair of adjoints on the left is an equivalence of ∞-categories). Note that we have introduced the functor gl1 , with the property that if A is a commutative S-algebra, then GL1 A  Ω∞ gl1 A. We also define bgl1 A = Σgl1 A: then Ω∞ bgl1 A  BGL1 A.

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51 25

Thus a map of spectra f : b → bgl1 A may be viewed equivalently as a map of group-like commutative ∗-algebras Ω∞ f : B = Ω∞ b → BGL1 A or as a map of symmetric monoidal ∞-groupoids ξ : Sing B − → LineA .

(6.19)

Form the pull-back diagram gl1 A ⏐ ⏐  (6.20)

gl1 A ⏐ ⏐ 

p ⏐ ⏐ 

−−−−→ egl1 A  ∗ ⏐ ⏐ 

b

−−−−→

f

bgl1 A.

One checks [ABGHR, Lemma 8.23] that P (ξ)  Ω∞ p, and so we have the following. Proposition 6.21. The Thom spectrum of the map ξ of symmetric monoidal ∞-groupoids (6.19) is a commutative A-algebra; indeed, we have ∞ ∞ X ξ  Σ∞ A  Σ∞ A. + P (ξ) ∧Σ∞ + Ω p ∧ Σ∞ + GL1 A + Ω gl1 A

Example 6.22. Taking A to be the sphere spectrum in Proposition 6.21, we recover the result of [LMSM86] that the Thom spectrum of an ∞-loop map B → BGL1 S is a commutative S-algebra. Remark 6.23. The formula (6.15) provides one way to see that our Thom spectrum coincides with the classical Thom spectrum of [MQRT77, LMSM86]. One way to compute the smash product in (6.15) is to realize it as a two-sided bar construction [EKMM96, Proposition 7.5]. In particular if ξ : X → BGL1 S classifies a spherical fibration, then one expects (6.24)

∞ X ξ  Σ∞ S  B(Σ∞ + P (ξ) ∧Σ∞ + P (ξ), Σ+ GL1 S, S). + GL1 S

It is not difficult to see that the constructions of [MQRT77, LMSM86] provide careful models for the two-sided bar construction on the right-hand side of (6.24). Note that some care is required to make this proposal precise; see §8.6 of [ABGHR] for details.

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ANDO, BLUMBERG, AND GEPNER

7. Application: K(Z, 3), Twisted K-theory, and the Spinc Orientation 7.1. Recall that BSpinc participates in a fibration of infinite loop spaces (7.1)

bw

2 K(Z, 2) → BSpinc → BSO −−→ K(Z, 3),

where bw2 is the composite of the usual w2 with the Z-Bockstein w

b

2 BSO −−→ K(Z/2, 2) − → K(Z, 3).

Passing to Thom spectra in (7.1) we have a map of commutative S-algebras c Σ∞ + K(Z, 2) → M Spin .

It’s a theorem of Joachim [Joa04] that the orientation M Spinc → K of Atiyah-Bott-Shapiro is map of commutative S-algebras, and so we have a sequence of maps of commutative S-algebras (7.2)

c Σ∞ + K(Z, 2) → M Spin → K.

∞ The (Σ∞ + Ω , gl1 ) adjunction (6.18) produces from (7.2) a map of infinite loop spaces K(Z, 2) → GL1 K which we deloop once to view as a map

T : K(Z, 3) → BGL1 K. That is, the fact that the Atiyah-Bott-Shapiro orientation is a map of (commutative) S-algebras implies that we have a homotopy-commutative diagram

(7.3)

K(Z, 2) −−−−→ ⏐ ⏐ 

GL1 K ⏐ ⏐ 

BSpinc −−−−→ ⏐ ⏐ 

∗ ⏐ ⏐ 

BSO ⏐ ⏐ βw2 

j

−−−−→ BGL1 K ⏐ ⏐=  T

K(Z, 3) −−−−→ BGL1 K. Now suppose given a map α : X → K(Z, 3). Then we may form the Thom spectrum X T α , and define def

K n (X)α = π0 ModK (X T α , Σn K). We then have the following. Proposition 7.4. A map α : X → K(Z, 3) gives rise to a twist K ∗ (X)α of the K-theory of X. A choice of homotopy T βw2 ⇒ j in the diagram (7.3) above determines, for every oriented vector bundle V over X, an isomorphism (7.5) K ∗ (X)βw (V ) ∼ = K ∗ (X V ). 2

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TWISTS OF K-THEORY AND TMF

53 27

In this way, the characteristic class βw2 (V ) determines the K-theory of the Thom spectrum of V . Proof. We prove the isomorphism (7.5) to show how simple it is from this point of view. Consider the homotopy commutative diagram V

X −−−−→

BSO ⏐ ⏐ βw2 

j

−−−−→ BGL1 K ⏐ ⏐=  T

K(Z, 3) −−−−→ BGL1 K. Omitting the gradings, we have K 0 (X)βw2 (V ) = π0 ModK (X T βw2 (V ) , K) ∼ = π0 ModK (X j(V ) , K) ∼ = π0 ModK (X V ∧ K, K) ∼ = π0 S (X V , K) = K 0 (X V ). The first isomorphism uses the construction of X ξ together with the fact that T βw2 and j are homotopic as maps BSO → BGL1 K. The second isomorphism is Theorem 4.5.  Remark 7.6. We needn’t have started with an oriented bundle. For example, let F be the fiber in the sequence F → BSpinc → BO. This is a fibration of infinite loop spaces, and so it deloops to give γ

→ BF. F → BSpinc → BO − The same argument produces an E∞ map c Σ∞ + F → M Spin → K

whose adjoint F → GL1 K deloops to ζ : BF → BGL1 K, and if V is any vector bundle then K(X V ) ∼ = K(X)ζγ(V ) . 7.2. Khorami’s theorem. At this point we are in a position to state a remarkable result of M. Khorami [Kho10]. K(Z, 2) is a group-like commutative ∗-algebra, and so we have a bundle K(Z, 2) → EK(Z, 2)  ∗ → BK(Z, 2)  K(Z, 3). (One can build this bundle a number of ways: by modeling K(Z, 2)  P U as mentioned in §2.1, or using infinite loop space theory, or using the ∞-category technology described above.)

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ANDO, BLUMBERG, AND GEPNER

Given a map ζ : X → K(Z, 3), we can form the pull-back K(Z, 2)-bundle Q −−−−→ EK(Z, 2) ⏐ ⏐ ⏐ ⏐   ζ

X −−−−→ K(Z, 3). On the other hand, let P be the pull-back P −−−−→ EGL1 K ⏐ ⏐ ⏐ ⏐   Tζ

X −−−−→ BGL1 K as in Proposition 6.21. One can check that ∞ Σ∞ Σ∞ + P  Σ+ Q ∧ Σ ∞ + GL1 K, + K(Z,2)

and so the Thom spectrum whose homotopy calculates the ζ-twisted K-theory is X T ζ  Σ∞ K + P ∧ Σ∞ + GL1 K Σ∞ K  Σ∞ + Q ∧ Σ∞ + GL1 K ∧Σ∞ + GL1 K + K(Z,2)

(7.7)

 Σ∞ K, + Q ∧ Σ∞ + K(Z,2) where the map of commutative S-algebras Σ∞ + K(Z, 2) → K is (7.2). The formula (7.7) implies that there is a spectral sequence (see for example [EKMM96, Theorem 4.1]) (7.8) TorK∗ K(Z,2) (K∗ Q, K∗ ) ⇒ π∗ X T ζ ∼ = K∗ (X)ζ . ∗

Note that K∗ K(Z, 2) ∼ = K∗ {β1 , β2 , . . .}, so K∗ is not a flat K∗ K(Z, 2)-module. Nevertheless Khorami proves the following. Theorem 7.9. In (7.8) one has Torq = 0 for q > 0, and so K∗ (X)ζ ∼ = K∗ Q ⊗K K(Z,2) K∗ . ∗

8. Application: Degree-four Cohomology and Twisted Elliptic Cohomology The arguments of §7.1 apply equally well to String structures and the spectrum of topological modular forms. Recall that spin vector bundles admit a degree four characteristic class λ, which we may view as a map of infinite loop spaces λ

BSpin − → K(Z, 4). Indeed this map detects the generator of H 4 BSpin ∼ = Z. The fiber of λ is called BString, and so we have maps of infinite loop spaces λ

K(Z, 3) − → BString → BSpin − → K(Z, 4). Passing to Thom spectra, we get maps of commutative S-algebras Σ∞ + K(Z, 3) → M String → M Spin. Let tmf be the spectrum of topological modular forms [Hop02]. Ando, Hopkins, and Rezk [AHR] have produced a map of commutative S-algebras σ

M String − → tmf,

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55 29

and so we have a map of commutative S-algebras Σ∞ + K(Z, 3) → tmf, whose adjoint (see (6.12,6.17,6.18)) K(Z, 3) → GL1 tmf deloops to T : K(Z, 4) → BGL1 tmf. By construction, the map T makes the diagram K(Z, 3) −−−−→ GL1 tmf ⏐ ⏐ ⏐ ⏐  

(8.1)

BString −−−−→ ⏐ ⏐ 

∗ ⏐ ⏐ 

j

BSpin −−−−→ BGL1 tmf ⏐ ⏐ ⏐ ⏐= λ  T

K(Z, 4) −−−−→ BGL1 tmf commute up to homotopy. If ζ : X → K(Z, 4) is a map, then we may define def

tmf (X)kζ = π0 Modtmf (X T ζ , Σk tmf ), and so we have the following. Proposition 8.2. A map ζ : X → K(Z, 4) gives rise to a twist tmf ∗ (X)ζ of the tmf -theory of X. A choice of homotopy Tλ ⇒ j in the diagram (8.1) above determines, for every map V:X− → BSpin, an isomorphism of tmf ∗ (X)-modules (8.3)

tmf ∗ (X)λ(V ) ∼ = tmf ∗ (X V ).

In this way, the characteristic class λ(V ) determines the tmf -cohomology of the Thom spectrum of V . Remark 8.4. As in Remark 7.6, we could have started with the fiber F in the fibration sequence of infinite loop spaces γ

F → BString → BO − → BF. We have a map of commutative S-algebras Σ∞ + F → M String → tmf whose adjoint F → GL1 tmf deloops to ζ : BF → BGL1 tmf,

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and if V : X → BO classifies a virtual vector bundle, then tmf ∗ (X V ) ∼ = tmf ∗ (X)ζγ(V ) 9. Application: Poincare Duality and Twisted Umkehr Maps Let M be a compact smooth manifold with tangent bundle T of rank d. Embed M in RN , and then perform the Pontrjagin-Thom construction: collapse to a point the complement of a tubular neighborhood of M . If ν is the normal bundle of the embedding, this gives a map SN → M ν . Desuspending N times then yields a map μ : S 0 → M −T .

(9.1)

As usual, the Thom spectrum admits a relative diagonal map −T M −T −→ Σ∞ +M ∧M Δ

(this is the map which gives the cohomology of the Thom spectrum the structure of a module over the cohomology of the base). If E is a spectrum, then to a map f : M −T → E we can associate the composition μ

1∧f

−T → M −T −→ Σ∞ −−→− → Σ∞ S0 − +M ∧M + M ∧ E. Δ

Milnor-Spanier-Atiyah duality says that this procedure yields an isomorphism E∗ (M ) ∼ = E −∗ (M −T ). In the presence of a Thom isomorphism ∼ E d−∗ (M ) E −∗ (M −T ) = we have Poincar´e duality

E∗ (M ) ∼ = E d−∗ (M ). Without a Thom isomorphism, we choose a map α α

→ BO M− classifying d − T , and then we define τ (−T ) to be the composition α

j

→ BO − → BGL1 S − → BGL1 E. τ (−T ) : M − Then, following Example 5.2, we have E d−∗ (M )τ (−T ) ∼ = E d−∗ (M d−T ) ∼ = E −∗ (M −T ) ∼ = E∗ (M ). Combining this with the results of sections 7.1 and 8, we have the following. Proposition 9.2. Suppose that M is an oriented compact manifold of dimension d. Then K∗ (M ) ∼ = K d−∗ (M )−βw2 (M ) . If M is a spin manifold, then tmf∗ (M ) ∼ = tmf d−∗ (M )−λ(M ) .

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9.1. Twisted umkehr maps. In this section we sketch the construction of some umkehr maps in twisted generalized cohomology. Note that similar constructions are studied in [CW, MS06, Wal06]. Also, since this paper was written, we have learned that Bunke, Schneider, and Spitzweck have independently developed a similar approach to twisted umkehr maps. Suppose that we have a family of compact spaces over X, that is, a map of ∞-categories ζ : Sing X → (compact spaces). We also have the trivial map ∗

∗X : Sing X − → (compact spaces). If M is a compact space, let 0 DM = F (Σ∞ + M, S )

be the Spanier-Whitehead dual of M+ : this is a functor of ∞-categories D : (compact spaces)op → S . The projection M →∗ gives rise to a map of spectra (9.3)

S0 ∼ = D∗ → DM.

Indeed if M is a compact manifold with tangent bundle T , then Milnor-SpanierAtiyah duality says that DM  M −T , in such a way that the Pontrjagin-Thom map (9.1) identifies with (9.3). 0 In any case, let SX = D∗X . We have a natural map 0 u : SX → Dζ

of bundles of spectra over X. Essentially, we are applying the map (9.3) fiberwise. It follows from Proposition 7.7 of [ABGHR] that X S X  Σ∞ + X. 0

As for X Dζ , in a forthcoming paper we prove the following. Proposition 9.4. Suppose that ζ arises from a bundle f

Y − →X of compact manifolds, and let T f be its bundle of tangents along the fiber. Then X Dζ  Y −T f . In particular, passing to Thom spectra on u gives a map of spectra (9.5)

−T f t : Σ∞ . +X →Y

This map is equivalent to the classical stable transfer map associated to f . The map t, and indeed the idea that it arises from applying the map (9.1) fiberwise, is classical; see for example [BG75]. Casting it in our setting enables us to construct twisted versions. More precisely, suppose that R is a commutative S-algebra, and suppose given a bundle of R-lines over X ξ : Sing X → − LineR .

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We then have a map of bundles of R-lines over X 0 u ∧ ξ : ξ  SX ∧ ξ → Dζ ∧ ξ.

Thus we have constructed a twisted umkehr map R∗ (X Dζ )ξ → R∗ (X)ξ . In the situation of Proposition 9.4, we have a twisted transfer map R∗ (Y −T f )ξ → R∗ (X)ξ .

(9.6)

About this we show the following. Proposition 9.7. Suppose that Tf

Y −−→ BO − → BGL1 S − → BGL1 R, regarded as a map Sing Y → LineR , is homotopic to ξf : Sing Y − → Sing X → LineR . A choice of homotopy determines an isomorphism R∗ (Y ) ∼ = R∗ (Y −T f )ξ , and composing with the twisted transfer (9.6) we have a twisted umkehr map R∗ (Y ) → R∗ (X)ξ . 10. Motivation: D-brane Charges in K-theory 10.1. The Freed-Witten anomaly. Let j : D → X be an embedded submanifold, let ν be the normal bundle of j, and suppose that D carries a complex vector bundle ξ. Suppose moreover that ν carries a Spinc -structure. Then we can form the K-theory push-forward j! : K(D) → K(X). In that situation Minasian and Moore and Witten discovered that it is sensible to think of the K-theory class j! (ξ) ∈ K(X) as the “charge” of the D-brane D with Chan-Paton bundle ξ. If ν does not carry a Spinc -structure, then we still have the Pontrjagin-Thom construction X → Dν . Suppose we have a map H : X → K(Z, 3) making the diagram ν

D −−−−→ ⏐ ⏐ j

BSO ⏐ ⏐bw  2

H

X −−−−→ K(Z, 3) commute up to homotopy. According to Proposition 9.7, a homotopy c : bw2  Hj determines an isomorphism K ∗ (D) ∼ = K ∗ (Dν )−H (since ν = −T j), and then we have a twisted umkehr map (10.1)

j! : K ∗ (D) − → K ∗ (X)−H .

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∗ The class j! (ξ) ∈ K−H (X) is evidently an analogue of the charge in this situation. The discovery of the condition that there exists a class H on X such that H|D = W3 (ν) is due to Freed and Witten [FW99]. Although we discovered this push-forward in an attempt to understand Freed and Witten’s condition, we were not the first: it appeared, formulated this way, in a paper of Carey and Wang [CW]. An important contribution of their work is the construction, using the twisted K-theory of [AS04], of the umkehr map (10.1).

11. An Elliptic Cohomology Analogue Now suppose that we are given an embedding of manifolds j : M → Y, and that ν = ν(j) is equipped with a Spin structure. Suppose we have a map H making the diagram ν

(11.1)

M −−−−→ BSpin ⏐ ⏐ ⏐ ⏐ j λ H

Y −−−−→ K(Z, 4) commute up to homotopy. By Proposition 9.7, a homotopy c : λν  Hj determines an isomorphism tmf ∗ (M )  tmf ∗ (Dν )−H , and then we have a homomorphism of tmf ∗ (Y )-modules (11.2)

tmf ∗ (M )  tmf ∗ (Dν )−H → tmf ∗ (Y )−H .

Remark 11.3. The data of a configuration like (12.3) together with the homotopy c was studied by Wang [Wan08], who calls it a twisted String structure. In fact, it was predicted by Kriz and Sati [KS04, Sat] that tmf should be the natural receptacle for M -brane charges. Remark 11.4. The authors are grateful to Hisham Sati for suggesting that we think about diagrams like (11.1). The on-going investigation of the resulting twisted umkehr maps is joint work with him. 12. Twists of Equivariant Elliptic Cohomology At present we do not know how to twist equivariant cohomology theories in general; for that matter, equivariant Thom spectra are poorly understood. However, twists by degree four Borel cohomology play an important role in equivariant elliptic cohomology. We review two instances to give the reader a taste of the subject. Let G be a connected and compact Lie group. In 1994, Grojnowski sketched the construction of a G-equivariant elliptic cohomology EG , based on a complex elliptic curve of the form Cq = C/Λ ∼ = C× /q Z ; more generally, the construction can be used to give a theory for the universal curve over the complex upper halfplane (Grojnowski’s paper is now available [Gro07]). In the case of the circle, Greenlees [Gre05] has given a complete construction of a rational S 1 -equivariant elliptic spectrum. Note also that Jacob Lurie has obtained analogous and sharper results about equivariant elliptic cohomology, in the context of his derived elliptic curves.

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The functor EG takes its values in sheaves of OMG -modules, where MG is the complex abelian variety MG = (Tˇ ⊗Z Cq )/W. Note that the completion of MG at the origin is spf E(BG); in general one has ∼ EG (X)∧ 0 = E(EG ×G X), where E is the non-equivariant elliptic cohomology associated to Cq . Grojnowski points out that a construction of Looijenga [Loo76] (see also [And03, §5]) associates to a class c ∈ H 4 (BG) a line bundle A(c) over MG . Thus if X is any G-space, then we can form the OMG -module def

EG (X)c = EG (X) ⊗ A(c) This EG (X)-module is a twisted form of EG (X). 12.1. Representations of loop groups. Already the case of a point is interesting: one learns that twisted equivariant elliptic cohomology carries the characters of representations of loop groups. Suppose that G is a simple and simply connected Lie group, such as SU (d) or Spin(2d). Then H 4 (BG) ∼ = Z. We then have the following result, due independently to the first author [And00] (who learned it from Grojnowski) and, in a much more precise form involving derived equivariant elliptic cohomology, Jacob Lurie. Proposition 12.1. Let G be a simple and simply connected compact Lie group, and let φ ∈ H 4 (BG; Z) ∼ = Z. The character of a representation of the loop group LG of level φ is a section of A(φ), and the Kac character formula shows that we have an isomorphism Rφ (LG) ∼ = Γ(EG (∗)φ ) after tensoring with Z((q)). It is fun to compare this result to the work of Freed, Hopkins, and Teleman (for example [FHT]), who show that Rφ (LG) is the twisted G-equivariant K-theory of G. Thus we have a map ΓEG (∗)φ → KG (G)φ . This map is an instance of the relationship between elliptic cohomology and the orbifold K-theory of the free loop space. We hope to provide a more extensive discussion in the future. 12.2. The equivariant sigma orientation. Let T be the circle group, and suppose that V /X is a T-equivariant vector bundle with structure group G (in this section we suppose that G = Spin(2d) or G = SU (d)). Let P/X be the associated principal bundle. Then EG×T (P ) ∼ = ET (X) is a sheaf of EG (∗) = OMG -algebras, and so we can twist ET (X) by A(c) for c ∈ H 4 (BG) ∼ = Z. Let c be the generator corresponding to c2 if G = SU (d) or the “half Pontrjagin class” λ if G = Spin(2d). Note that c determines a Borel equivariant class cT .

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In [And03, AG], the authors show first of all that the twist ET (X) ⊗ A(c) depends only on the equivariant degree-four class cT (V ) ∈ HT4 (X; Z), and so we may define ET (X)cT (V ) = ET (X) ⊗ A(c). Second, they show that the Weierstrass sigma function leads to an isomorphism (12.2) ET (X)c (V ) ∼ = ET (X V ); T

this is an analytic and equivariant form of the isomorphism (8.3). In the case that cT (V ) = 0 we conclude that ET (X) ∼ = ET (X V );

(12.3)

this is the T-equivariant sigma orientation in this context. More precisely, we have the following. Proposition 12.4. Let V /X be an S 1 -equivariant SU vector bundle. Let ) ∈ HT4 (X) be the equivariant second Chern class of V . Let ET denote Grojnowski’s or Greenlees’s T-equivariant elliptic cohomology, associated to the the complex analytic elliptic curve C. Then there is a canonical isomorphism ∼ ET (X) T = ET (X V ),

cT2 (V

c2 (V )

natural in V /X. In particular if V0 and V1 are two such bundles with cT2 (V0 ) = cT2 (V1 ), and W = V 0 − V1 , then there is a canonical isomorphism ET (X) ∼ = ET (X W ). Remark 12.5. In [And03] the author constructs the T-equivariant sigma orientation in Grojnowski’s equivariant elliptic cohomology, for Spin and SU bundles. The construction was motivated by the Proposition stated above, which however was given as Conjecture 1.14. In [AG] the authors construct the T-equivariant sigma orientation for Greenlees’s equivariant elliptic cohomology, for T-equivariant SU -bundles. Proposition 12.4 appears there as Theorem 11.17. It should not be difficult to adapt the methods of these two papers to the case of Spin bundles. Remark 12.6. The careful reader will note that in [AG] we show how to twist ET∗ (X) by cT2 (V ) ∈ HT4 (X; Z): we do not there discuss twisting by general elements of HT4 (X; Z). The construction of such general twists of ET (X) is the subject of on-going work of the first author and Bert Guillou. Remark 12.7. Lurie has obtained similar and sharper results for the elliptic cohomology associated to a derived elliptic curve. In particular he can construct the sigma orientation and twists by HT4 (X; Z). References [ABGHR] [ABS64] [AG]

Matthew Ando, Andrew J. Blumberg, David Gepner, Michael Hopkins, and Charles Rezk. Units of ring spectra and Thom spectra, arxiv:0810.4535v3. Michael F. Atiyah, Raoul Bott, and Arnold Shapiro. Clifford modules. Topology, 3 suppl. 1:3–38, 1964. Matthew Ando and J. P. C. Greenlees. Circle-equivariant classifying spaces and the rational equivariant sigma genus, http://arxiv.org/abs/0705.2687v2. Submitted.

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[AHR]

Matthew Ando, Michael J. Hopkins, and Charles Rezk. Multiplicative orientations of KO and of the spectrum of topological modular forms, http://www.math.uiuc.edu/˜mando/papers/koandtmf.pdf. Preprint. [And00] Matthew Ando. Power operations in elliptic cohomology and representations of loop groups. Trans. Amer. Math. Soc., 352(12):5619–5666, 2000. [And03] Matthew Ando. The sigma orientation for analytic circle-equivariant elliptic cohomology. Geometry and Topology, 7:91–153, 2003, arXiv:math.AT/0201092. [AS04] Michael Atiyah and Graeme Segal. Twisted K-theory. Ukr. Mat. Visn., 1(3):287–330, 2004, arXiv:math/0407054. [Ati69] M. F. Atiyah. Algebraic topology and operators in Hilbert space. In Lectures in Modern Analysis and Applications. I, pages 101–121. Springer, Berlin, 1969. [BG75] J. C. Becker and D. H. Gottlieb. The transfer map and fiber bundles. Topology, 14:1– 12, 1975. [BCMMS] P. Bouwknegt and A. L. Carey and V. Mathai and M. K. Murray and D. Stevenson. Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys., 228:17–49, 2002, http://arxiv.org/abs/hep-th/0106194. [BM00] Peter Bouwknegt and Varghese Mathai. D-branes, B-fields and twisted K-theory. J. High Energy Phys., (3):Paper 7, 11, 2000, http://arxiv.org/abs/hep-th/0002023v3. [CW] Alan L. Carey and Bai-Ling Wang. Thom isomorphism and push-forward map in twisted K-theory, arxiv:math/0507414. [DK70] P. Donovan and M. Karoubi. Graded Brauer groups and K-theory with local coeffi´ cients. Inst. Hautes Etudes Sci. Publ. Math., (38):5–25, 1970. [EKMM96] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in stable homotopy theory, volume 47 of Mathematical surveys and monographs. American Math. Society, 1996. [FHT] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman. Twisted K-theory and Loop Group Representations. arXiv:math.AT/0312155. [FW99] Daniel S. Freed and Edward Witten. Anomalies in string theory with D-branes. Asian J. Math., 3(4):819–851, 1999. [Gre05] J. P. C. Greenlees. Rational S 1 -equivariant elliptic cohomology. Topology, 44(6):1213– 1279, 2005. [Gro07] Ian Grojnowski. Delocalized equivariant elliptic cohomology. In Elliptic cohomology: geometry, applications, and higher chromatic analogues, volume 342 of London Mathematical Society Lecture Notes. Cambridge University Press, 2007. [Hop02] M. J. Hopkins. Algebraic topology and modular forms. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press, arXiv:math.AT/0212397. [Hov01] M. Hovey. Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra, 165(1):63–127, 2001. [HSS00] Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra. J. Amer. Math. Soc., 13(1):149–208, 2000. [Joa04] Michael Joachim. Higher coherences for equivariant K-theory. In Structured ring spectra, volume 315 of London Math. Soc. Lecture Note Ser., pages 87–114. Cambridge Univ. Press, Cambridge, 2004. [Joy02] A. Joyal. Quasi-categories and Kan complexes. J. Pure Appl. Algebra, 175(1-3):207– 222, 2002. Special volume celebrating the 70th birthday of Professor Max Kelly. [Kho10] Mehdi Khorami. A universal coefficient theorem for twisted K-theory. arXiv:math.AT/10014790 [KS04] Igor Kriz and Hisham Sati. M-theory, type IIA superstrings, and elliptic cohomology. Adv. Theor. Math. Phys., 8(2):345–394, 2004, http://arxiv.org/abs/hepth/0404013v3. [Lew91] L. G. Lewis, Jr. Is there a convenient category of spectra? J. Pure Appl. Algebra, 73:233–246, 1991. [LMSM86] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure. Equivariant stable homotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. [Loo76] Eduard Looijenga. Root systems and elliptic curves. Inventiones Math., 38, 1976.

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Jacob Lurie. Derived algebraic geometry I: stable ∞-categories, arXiv:math.CT/0608040. [DAGII] Jacob Lurie. Derived algebraic geometry II: noncommutative algebra, arXiv:math.CT/0702229. [DAGIII] Jacob Lurie. Derived algebraic geometry III: commutative algebra, arXiv:math.CT/0703204. [HTT] Jacob Lurie. Higher Topos Theory. AIM 2006 -20, arXiv:math.CT/0608040. [May72] J. P. May. The geometry of iterated loop spaces. Springer-Verlag, Berlin, 1972. Lectures Notes in Mathematics, Vol. 271. [May74] J. P. May. E∞ spaces, group completions, and permutative categories. In New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pages 61–93. London Math. Soc. Lecture Note Ser., No. 11. Cambridge Univ. Press, London, 1974. [MQRT77] J. P. May. E∞ ring spaces and E∞ ring spectra. Springer-Verlag, Berlin, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave, Lecture Notes in Mathematics, Vol. 577. [MM02] M. A. Mandell and J. P. May. Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc., 159(755), 2002. [MMSS01] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley. Model categories of diagram spectra. Proc. London Math. Soc. (3), 82(2):441–512, 2001. [MS06] J. P. May and J. Sigurdsson. Parametrized homotopy theory, volume 132 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. [Qui68] Daniel G. Quillen. The geometric realization of a Kan fibration is a Serre fibration. Proc. Amer. Math. Soc., 19:1499–1500, 1968. [Ros89] Jonathan Rosenberg. Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A, 47(3):368–381, 1989. [Sat] Hisham Sati. Geometric and topological structures related to M-branes. These proceedings. [Wal06] Robert Waldm¨ uller. Products and push-forwards in parametrised cohomology theories. PhD thesis. [Wan08] Bai-Ling Wang. Geometric cycles, index theory and twisted K-homology. J. Noncommut. Geom., 2(4):497–552, 2008. [DAGI]

Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana IL 61801, USA E-mail address: [email protected] Department of Mathematics, University of Texas, Austin, TX 78703 E-mail address: [email protected] Department of Mathematics, The University of Illinois at Chicago, Chicago IL 60607, USA E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681758

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Division Algebras and Supersymmetry I John C. Baez and John Huerta Abstract. Supersymmetry is deeply related to division algebras. For example, nonabelian Yang–Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6, or 10. The same is true for the Green–Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this, in turn, is the existence of normed division algebras in dimensions two less, namely 1, 2, 4 and 8: the real numbers, complex numbers, quaternions and octonions. Here we provide a self-contained account of how this works.

1. Introduction There is a deep relation between supersymmetry and the four normed division algebras: the real numbers R, the complex numbers C, the quaternions H, and the octonions O. This is visible in the study of superstrings, supermembranes, and supergravity, but perhaps most simply in supersymmetric Yang–Mills theory. In any dimension, we may consider a Yang–Mills field coupled to a massless spinor transforming in the adjoint representation of the gauge group. These fields are described by the Lagrangian: 1 1 / A ψ. L = − F, F  + ψ, D 4 2 Here A is a connection on a bundle with semisimple gauge group G, F is the / A is the covariant Dirac operator curvature of A, ψ is a g-valued spinor field, and D associated with A. It is well-known that this theory is supersymmetric if and only if the dimension of spacetime is 3, 4, 6, or 10. Our goal here is to present a selfcontained proof of the ‘if’ part of this result, based on the theory of normed division algebras. This result goes back to the work of Brink, Schwarz, and Sherk [3] and others. The book by Green, Schwarz and Witten [10] contains a standard proof based on the properties of Clifford algebras in various dimensions. But Evans [7] has shown that the supersymmetry of L in dimension n + 2 implies the existence of a normed division algebra of dimension n. Conversely, Kugo and Townsend [12] showed how spinors in dimension 3, 4, 6, and 10 derive special properties from the normed division algebras R, C, H and O. They formulated a supersymmetric model c 2010 American Mathematical John C. Baez and John Society Huerta

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in 6 dimensions using the quaternions, H. They also speculated about a similar formalism in 10 dimensions using the octonions, O. Shortly after Kugo and Townsend’s work, Sudbery [17] used division algebras to construct vectors, spinors and Lorentz groups in Minkowski spacetimes of dimensions 3, 4, 6, and 10. He then refined his construction with Chung [4], and with Manogue [13] he used these ideas to give an octonionic proof of the supersymmetry of the above Lagrangian in dimension 10. This proof was later simplified by Manogue, Dray and Janesky [5]. In the meantime, Schray [14] applied the same tools to the superparticle. All this work has made it quite clear that normed division algebras explain why the above theory is supersymmetric in dimensions 3, 4, 6, and 10. Technically, what we need to check for supersymmetry is that δL is a total divergence with respect to the supersymmetry transformation δA

=

·ψ

δψ

=

1 2F

for any constant spinor field . (We explain the notation here later; we assume no prior understanding of supersymmetry or normed division algebras.) A calculation that works in any dimension shows that δL = tri ψ + divergence where tri ψ is a certain expression depending in a trilinear way on ψ and linearly on . So, the marvelous fact that needs to be understood is that tri ψ = 0 in dimensions 3, 4, 6, and 10, thanks to special properties of the normed division algebras R, C, H and O. Indeed, this fact is responsible for supersymmetry, not only for Yang– Mills fields in these dimensions, but also for superstrings! The same term tri ψ shows up as the obstruction to supersymmetry in the Green–Schwarz Lagrangian for classical superstrings [9, 10]. So, the vanishing of this term deserves to be understood: clearly, simply, and in as many ways as possible. Unfortunately, many important pieces of the story are scattered throughout the literature. The treatment of Deligne and Freed [6] is self-contained, and it uses normed division algebras, but it does not use ‘purely equational reasoning’: it proves tri ψ = 0 by first showing that the double cover of the Lorentz group acts transitively on the set of nonzero spinors in dimensions 3, 4, 6, and 10. While this geometrical argument is beautiful and insightful, a purely equational approach has its own charm. The line of work carried out by Fairlie, Manogue, Sudbery, Dray, and collaborators [5, 8, 13, 14] has shown that the equation tri ψ = 0 can be derived from the complete antisymmetry of another trilinear expression, the ‘associator’ [a, b, c] = (ab)c − a(bc) in the normed division algebra. Our desire here is to merely present this argument as clearly as we can. So, here we present an equational proof that tri ψ = 0 in dimensions 3, 4, 6, and 10, based on the complete antisymmetry of the associator for the normed division algebras K = R, C, H and O. In Section 2 we review the properties of normed division algebras that we will need. In Section 3 we start by recalling how to interpret vectors as 2 × 2 hermitian matrices with entries in K, and spinors as elements of K2 . We then use this language to describe the basic operations involving

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vectors, spinors and scalars. These include an operation that takes two spinors ψ and φ and forms a vector ψ · φ, and an operation that takes a vector A and a spinor ψ and forms a spinor Aψ. In Section 4 we prove the fundamental identity that holds only in Minkowski spaces of dimensions 3, 4, 6 and 10: (ψ · ψ)ψ = 0. Following Schray [14], we call this the ‘3-ψ’s rule’. In Section 5 we introduce a little superalgebra, and explain why we should treat K as an ‘odd’, or ‘fermionic’, super vector space. In Section 6 we formulate pure super-Yang–Mills theory in terms of normed division algebras, completely avoiding the use of gamma matrices. We explain how the term tri ψ arises as the obstruction to supersymmetry in this theory. Finally, we use the 3-ψ’s rule to prove that tri ψ = 0 in dimensions 3, 4, 6 and 10. 2. Normed Division Algebras By a classic theorem of Hurwitz [11], there are only four normed division algebras: the real numbers, R, the complex numbers, C, the quaternions, H, and the octonions, O. These algebras have dimension 1, 2, 4, and 8. For an overview of this subject, including a Clifford algebra proof of Hurwitz’s theorem, see [1]. Here we introduce the bare minimum of material needed to reach our goal. A normed division algebra K is a (finite-dimensional, possibly nonassociative) real algebra equipped with a multiplicative unit 1 and a norm | · | satisfying: |ab| = |a||b| for all a, b ∈ K. Note this implies that K has no zero divisors. We will freely identify R1 ⊆ K with R. In all cases, this norm can be defined using conjugation. Every normed division algebra has a conjugation operator—a linear operator ∗ : K → K satisfying a∗∗ = a,

(ab)∗ = b∗ a∗

for all a, b ∈ K. Conjugation lets us decompose each element of K into real and imaginary parts, as follows: a + a∗ a − a∗ Re(a) = , Im(a) = . 2 2 Conjugating changes the sign of the imaginary part and leaves the real part fixed. We can write the norm as √ √ |a| = aa∗ = a∗ a. This norm can be polarized to give an inner product on K: (a, b) = Re(ab∗ ) = Re(a∗ b). The algebras R, C and H are associative. The octonions O are not. Yet they come close: the subalgebra generated by any two octonions is associative. Another way to express this fact uses the associator: [a, b, c] = (ab)c − a(bc), a trilinear map K ⊗ K ⊗ K → K. A theorem due to Artin [15] states that for any algebra, the subalgebra generated by any two elements is associative if and only if the associator is alternating (that is, completely antisymmetric in its three arguments). An algebra with this property is thus called alternative. The octonions

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O are alternative, and so of course are R, C and H: for these three the associator simply vanishes! In what follows, our calculations make heavy use of the fact that all four normed division algebras are alternative. Besides this, the properties we require are: Proposition 1. The associator changes sign when one of its entries is conjugated. Proof. Since the subalgebra generated by any two elements is associative, and real elements of K lie in every subalgebra, [a, b, c] = 0 if any one of a, b, c is real. It follows that [a, b, c] = [Im(a), Im(b), Im(c)], which yields the desired result.  Proposition 2. The associator is purely imaginary. Proof. Since (ab)∗ = b∗ a∗ , a calculation shows [a, b, c]∗ = −[c∗ , b∗ , a∗ ]. By alternativity this equals [a∗ , b∗ , c∗ ], which in turn equals −[a, b, c] by the above proposition. So, [a, b, c] is purely imaginary.  For any square matrix A with entries in K, we define its trace tr(A) to be the sum of its diagonal entries. This trace lacks the usual cyclic property, because K is noncommutative, so in general tr(AB) = tr(BA). Luckily, taking the real part restores this property: Proposition 3. Let a, b, and c be elements of K. Then Re((ab)c) = Re(a(bc)) and this quantity is invariant under cyclic permutations of a, b, and c. Proof. Proposition 2 implies that Re((ab)c) = Re(a(bc)). For the cyclic property, it then suffices to prove Re(ab) = Re(ba). Since (a, b) = (b, a) and the inner product is defined by (a, b) = Re(ab∗ ) = Re(a∗ b), we see: Re(ab∗ ) = Re(b∗ a). The desired result follows upon substituting b∗ for b.



Proposition 4. Let A, B, and C be k × ,  × m and m × k matrices with entries in K. Then Re tr((AB)C) = Re tr(A(BC)) and this quantity is invariant under cyclic permutations of A, B, and C. We call this quantity the real trace Re tr(ABC). Proof. This follows from the previous proposition and the definition of the trace.  The reader will have noticed three trilinears in this section: the associator [a, b, c], the real part Re((ab)c), and the real trace Re tr(ABC). This is no coincidence, as they all relate to the star of the show, tri ψ. In fact: tri ψ = Re tr(ψ † ( · ψ)ψ). for some suitable matrices ψ † ,  · ψ and ψ. Of course, we have not yet said how to construct these. We turn to this now.

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3. Vectors, Spinors and Intertwiners It is well-known [1, 12, 17] that given a normed division algebra K of dimension n, one can construct (n + 2)-dimensional Minkowski spacetime as the space of 2 × 2 hermitian matrices with entries in K, with the determinant giving the Minkowski metric. Spinors can then be described as elements of K2 . Our goal here is to provide self-contained proofs of these facts, and then develop all the basic operations involving vectors, spinors and scalars using this language. To begin, let K[m] denote the space of m × m matrices with entries in K. Given A ∈ K[m], define its hermitian adjoint A† to be its conjugate transpose: A† = (A∗ )T . We say such a matrix is hermitian if A = A† . Now take the 2 × 2 hermitian matrices:    t+x y : t, x ∈ R, y ∈ K . h2 (K) = y∗ t−x This is an (n + 2)-dimensional real vector space. Moreover, the usual formula for the determinant of a matrix gives the Minkowski norm on this vector space:   t+x y − det = −t2 + x2 + |y|2 . y∗ t−x We insert a minus sign to obtain the signature (n + 1, 1). Note this formula is unambiguous even if K is noncommutative or nonassociative. It follows that the double cover of the Lorentz group, Spin(n + 1, 1), acts on h2 (K) via determinant-preserving linear transformations. Since this is the ‘vector’ representation, we will often call h2 (K) simply V . The Minkowski metric g: V ⊗ V → R is given by g(A, A) = − det(A). There is also a nice formula for the inner product of two different vectors. This involves the trace reversal of A ∈ h2 (K), introduced by Schray [14] and defined as follows: A˜ = A − (trA)1. ˜ Note we indeed have tr(A) = −tr(A). Also note that     −t + x y t+x y ˜ =⇒ A= A= y∗ t−x −t − x y∗ so trace reversal is really time reversal. Moreover: Proposition 5. For any vectors A, B ∈ V = h2 (K), we have ˜ = − det(A)1 AA˜ = AA and

1 ˜ = 1 Re tr(AB) ˜ = g(A, B) Re tr(AB) 2 2 Proof. We check the first equation by a quick calculation. Taking the real trace and dividing by 2 gives 1 ˜ = 1 Re tr(AA) ˜ = − det(A) = g(A, A). Re tr(AA) 2 2 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Then we use the polarization identity, which says that two symmetric bilinear forms that give the same quadratic form must be equal.  Next we consider spinors. As real vector spaces, the spinor representations S+ and S− are both just K2 . However, they differ as representations of Spin(n + 1, 1). To construct these representations, we begin by defining ways for vectors to act on spinors: γ : V ⊗ S+ → S − A ⊗ ψ → Aψ. and γ˜ : V ⊗ S− → S+ ˜ A ⊗ ψ → Aψ. We can also think of these as maps that send elements of V to linear operators: γ: γ˜ :

V V

→ Hom(S+ , S− ), → Hom(S− , S+ ).

Here a word of caution is needed: since K may be nonassociative, 2 × 2 matrices with entries in K cannot be identified with linear operators on K2 in the usual way. They certainly induce linear operators via left multiplication: LA (ψ) = Aψ. Indeed, this is how γ and γ˜ turn elements of V into linear operators: γ(A) = LA , γ˜ (A) = LA˜ . However, because of nonassociativity, composing such linear operators is different from multiplying the matrices: LA LB (ψ) = A(Bψ) = (AB)ψ = LAB (ψ). Since vectors act on elements of S+ to give elements of S− and vice versa, they map the space S+ ⊕ S− to itself. This gives rise to an action of the Clifford algebra Cliff(V ) on S+ ⊕ S− : Proposition 6. The vectors V = h2 (K) act on the spinors S+ ⊕S− = K2 ⊕K2 via the map Γ : V → End(S+ ⊕ S− ) given by

 Aψ). Γ(A)(ψ, φ) = (Aφ,

Furthermore, Γ(A) satisfies the Clifford algebra relation: Γ(A)2 = g(A, A)1 and so extends to a homomorphism Γ : Cliff(V ) → End(S+ ⊕ S− ), i.e. a representation of the Clifford algebra Cliff(V ) on S+ ⊕ S− . Proof. Suppose A ∈ V and Ψ = (ψ, φ) ∈ S+ ⊕ S− . We need to check that Γ(A)2 (Ψ) = − det(A)Ψ. Here we must be mindful of nonassociativity: we have ˜ ˜ Γ(A)2 (Ψ) = (A(Aψ), A(Aφ)).

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DIVISION ALGEBRAS AND SUPERSYMMETRY I

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˜ ˜ involve multiplying Yet it is easy to check that the expressions A(Aψ) and A(Aφ) at most two different nonreal elements of K. These associate, since K is alternative, so in fact ˜ ˜ Γ(A)2 (Ψ) = ((AA)ψ, (AA)φ). To conclude, we use Proposition 5.  The action of a vector swaps S+ and S− , so acting by vectors twice sends S+ to itself and S− to itself. This means that while S+ and S− are not modules for the Clifford algebra Cliff(V ), they are both modules for the even part of the Clifford algebra, generated by products of pairs of vectors. The group Spin(n + 1, 1) lives in this even part: as is well-known, it is generated by products of pairs of unit vectors in V : that is, vectors A with g(A, A) = ±1. As a result, S+ and S− are both representations of Spin(n + 1, 1). While we will not need this in what follows, one can check that: • When K = R, S+ ∼ = S− is the Majorana spinor representation of Spin(2, 1). • When K = C, S+ ∼ = S− is the Majorana spinor representation of Spin(3, 1). • When K = H, S+ and S− are the Weyl spinor representations of Spin(5, 1). • When K = O, S+ and S− are the Majorana–Weyl spinor representations of Spin(9, 1). This counts as a consistency check, because these are precisely the kinds of spinor representations that go into pure super-Yang–Mills theory. But it is important to note that the differences between these spinor representations are irrelevant to our argument. What matters is how they are the same—they can all be defined on K2 . Now that we have representations of Spin(n + 1, 1) on V , S+ and S− , we need to develop the Spin(n + 1, 1)-equivariant maps that relate them. Ultimately, to define the Lagrangian for pure super-Yang–Mills theory, we need: • An invariant pairing: −, − : S+ ⊗ S− → R. • An equivariant map that turns pairs of spinors into vectors: · : S± ⊗ S± → V. Another name for an equivariant map between group representations is an ‘intertwining operator’. As a first step, we show that the action of vectors on spinors is itself an intertwining operator: Proposition 7. The maps γ:

V ⊗ S+ A⊗ψ

→ S− → Aψ

and

V ⊗ S− → S+ ˜ A ⊗ ψ → Aψ are equivariant with respect to the action of Spin(n + 1, 1). γ˜ :

Proof. Both γ and γ˜ are restrictions of the map Γ : V ⊗ (S+ ⊕ S− ) → S+ ⊕ S− , so it suffices to check that Γ is equivariant. In fact, we will do this not just for the action of Spin(n + 1, 1), but for the larger group Pin(n + 1, 1), the subgroup of

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JOHN C. BAEZ AND JOHN HUERTA

Cliff(V ) generated by unit vectors in V . This group acts on V with unit vectors acting by conjugation on V ⊆ Cliff(V ). It acts on S+ ⊕ S− with a unit vector B acting by Γ(B). Both of these representations of Pin(n + 1, 1) restrict to the usual representations of Spin(n + 1, 1). Thus, we compute: Γ(BAB −1 )Γ(B)Ψ = Γ(B)(Γ(A)Ψ). Here it is important to note that the conjugation BAB −1 is taking place in the associative algebra Cliff(V ), not in the algebra of matrices. This equation says that Γ is indeed Pin(n + 1, 1)-equivariant, as claimed.  Now we exhibit the key tool: the pairing between S+ and S− : Proposition 8. The pairing −, − :

S+ ⊗ S − ψ⊗φ

→ →

R Re(ψ † φ)

is invariant under the action of Spin(n + 1, 1). Proof. Given A ∈ V , we use the fact that the associator is purely imaginary to show that       ˜ † (Aψ) = Re (φ† A)(Aψ) ˜ ˜ Re (Aφ) . = Re φ† (A(Aψ)) As in the proof of the Clifford relation, it is easy to check that the column vector ˜ A(Aψ) involves at most two nonreal elements of K and equals g(A, A)ψ. So: ˜ γ (A)φ, γ(A)ψ = g(A, A)ψ, φ. In particular when A is a unit vector, acting by A swaps the order of ψ and φ and changes the sign at most. −, − is thus invariant under the group in Cliff(V ) generated by products of pairs of unit vectors, which is Spin(n + 1, 1).  With this pairing in hand, there is a manifestly equivariant way to turn a pair of spinors into a vector. Given ψ, φ ∈ S+ , there is a unique vector ψ · φ whose inner product with any vector A is given by g(ψ · φ, A) = ψ, γ(A)φ. Similarly, given ψ, φ ∈ S− , we define ψ · φ ∈ V by demanding g(ψ · φ, A) = ˜ γ (A)ψ, φ for all A ∈ V . This gives us maps S ± ⊗ S± → V which are manifestly equivariant. On the other hand, because S± = K2 and V = h2 (K), there is also a naive way to turn a pair of spinors into a vector using matrix operations: just multiply the column vector ψ by the row vector φ† and then take the hermitian part: ψφ† + φψ † ∈ h2 (K), or perhaps its trace reversal: † + φψ † ∈ h (K). ψφ 2

In fact, these naive guesses match the manifestly equivariant approach described above:

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DIVISION ALGEBRAS AND SUPERSYMMETRY I

Proposition 9. The maps · : S± ⊗ S± → V are given by: ·:

S + ⊗ S+

→

ψ⊗φ

→

V † + φψ † ψφ

·:

S − ⊗ S− → V ψ⊗φ → ψφ† + φψ † . These maps are equivariant with respect to the action of Spin(n + 1, 1). Proof. First suppose ψ, φ ∈ S+ . We have already seen that the map · : S+ ⊗ S+ → V is equivariant. We only need to show that this map has the desired form. We start by using some definitions: g(ψ · φ, A) = ψ, γ(A)φ = Re(ψ † (Aφ)) = Re tr(ψ † Aφ). We thus have

g(ψ · φ, A) = Re tr(ψ † Aφ) = Re tr(φ† Aψ), where in the last step we took the adjoint of the inside. Applying the cyclic property of the real trace, we obtain g(ψ · φ, A) = Re tr(φψ † A) = Re tr(ψφ† A). Averaging gives 1 Re tr((ψφ† + φψ † )A). 2 On the other hand, Proposition 5 implies that 1  · φ)A). g(ψ · φ, A) = Re tr((ψ 2 Since both these equations hold for all A, we must have g(ψ · φ, A) =

 ψ · φ = ψφ† + φψ † . Doing trace reversal twice gets us back where we started, so † + φψ † ψ · φ = ψφ

as desired. A similar calculation shows that if ψ, φ ∈ S− , then ψ·φ = ψφ† +φψ † . Map g: V ⊗ V → R γ : V ⊗ S+ → S −



Division algebra notation Index notation 1 ˜ A μ Bμ 2 Re tr(AB) γμ A μ ψ

γ˜ : V ⊗ S− → S+

Aψ ˜ Aψ

γ˜μ Aμ ψ

· : S+ ⊗ S+ → V

† + φψ † ψφ

ψγ μ φ

· : S− ⊗ S − → V

ψφ† + φψ †

ψ˜ γμφ

Re(ψ † φ) ψφ −, − : S+ ⊗ S− → R Table 1. Division algebra notation vs. index notation

We can summarize our work so far with a table of the basic bilinear maps involving vectors, spinors and scalars. Table 1 shows how to translate between division algebra notation and something more closely resembling standard physics notation.

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JOHN C. BAEZ AND JOHN HUERTA

In this table the adjoint spinor ψ denotes the spinor dual to ψ under the pairing −, −. The gamma matrix γ μ denotes a Clifford algebra generator acting on S+ , while γ˜μ denotes the same element acting on S− . Of course γ˜ is not standard physics notation; the standard notation for this depends on which of the four cases we are considering: R, C, H or O. 4. The 3-ψ’s Rule Now we prove the fundamental identity that makes supersymmetry tick in dimensions 3, 4, 6, and 10. This identity was dubbed the ‘3-ψ’s rule’ by Schray [14]. The following proof is based on an argument in the appendix of the paper by Dray, Janesky and Manogue [5]. Note that it is really the alternative law, rather than the normed division algebra axioms, that does the job: Theorem 10. Suppose ψ ∈ S+ . Then (ψ · ψ)ψ = 0. Similarly, if φ ∈ S− , then (φ · φ)φ = 0. Proof. Suppose ψ ∈ S+ . By definition, † )ψ = 2(ψψ † − tr(ψψ † )1)ψ. (ψ · ψ)ψ = 2(ψψ It is easy to check that tr(ψψ † ) = ψ † ψ, so (ψ · ψ)ψ = 2((ψψ † )ψ − (ψ † ψ)ψ). Since ψ † ψ is a real number, it commutes with ψ: (ψ · ψ)ψ = 2((ψψ † )ψ − ψ(ψ † ψ)). Since K is alternative, every subalgebra of K generated by two elements is associative. Since ψ ∈ K2 is built from just two elements of K, the right-hand side vanishes. The proof of the identity for φ ∈ S− is similar.  It will be useful to state this result in a somewhat more elaborate form. To save space we only give this version for spinors in S+ , though an analogous result holds for spinors in S− : Theorem 11. Define a map T:

S + ⊗ S+ ⊗ S+ ψ⊗φ⊗χ

→ S− → (ψ · φ)χ + (φ · χ)ψ + (χ · ψ)φ.

Then T = 0. Proof. It is easy to check that ψ · φ = φ · ψ for all ψ, φ ∈ S+ , so the map T is completely symmetric in its three arguments. Just as any symmetric bilinear form B(x, y) can be recovered from the corresponding quadratic form B(x, x) by polarization, so too can any symmetric trilinear form be recovered from the corresponding cubic form. Since T (ψ, ψ, ψ) = 0 by Theorem 10, it follows that T = 0.  To see how this theorem is the key to supersymmetry for super-Yang–Mills theory, we need a little superalgebra.

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5. Superalgebra So far we have used normed division algebras to construct a number of algebraic structures: vectors as elements of h2 (K), spinors as elements of K2 , and the various bilinear maps involving vectors, spinors, and scalars. However, to describe supersymmetry, we also need superalgebra. Specifically, we need anticommuting spinors. Physically, this is because spinors are fermions, so we need them to satisfy anticommutation relations. Mathematically, this means that we will do our algebra in the category of ‘super vector spaces’, SuperVect, rather than the category of vector spaces, Vect. A super vector space is a Z2 -graded vector space V = V0 ⊕ V1 where V0 is called the even or bosonic part, and V1 is called the odd or fermionic part. Like Vect, SuperVect is a symmetric monoidal category [2]. It has: • Z2 -graded vector spaces as objects; • Grade-preserving linear maps as morphisms; • A tensor product ⊗ that has the following grading: if V = V0 ⊕ V1 and W = W0 ⊕ W1 , then (V ⊗ W )0 = (V0 ⊗ W0 ) ⊕ (V1 ⊗ W1 ) and (V ⊗ W )1 = (V0 ⊗ W1 ) ⊕ (V1 ⊗ W0 ); • A braiding BV,W : V ⊗ W → W ⊗ V defined as follows: v ∈ V and w ∈ W are of grade p and q, then BV,W (v ⊗ w) = (−1)pq w ⊗ v. The braiding encodes the ‘the rule of signs’: in any calculation, when two odd elements are interchanged, we introduce a minus sign. In what follows we treat the normed division algebra K as an odd super vector space. This turns out to force the spinor representations S± to be odd and the vector representation V to be even, as follows. There is an obvious notion of direct sums for super vector spaces, with (V ⊕ W )0 = V0 ⊕ W0 ,

(V ⊕ W )1 = V1 ⊕ W1

and also an obvious notion of duals, with (V ∗ )0 = (V0 )∗ ,

(V ∗ )1 = (V1 )∗ .

We say a super vector space V is even if it equals its even part (V = V0 ), and odd if it equals its odd part (V = V1 ). Any subspace U ⊆ V of an even (resp. odd) super vector space becomes a super vector space which is again even (resp. odd). We treat the spinor representations S± as super vector spaces using the fact that they are the direct sum of two copies of K. Since K is odd, so are S+ and S− . Since K2 is odd, so is its dual. This in turn forces the space of linear maps from K2 to itself, End(K2 ) = K2 ⊗ (K2 )∗ , to be even. This even space contains the 2 × 2 matrices K[2] as the subspace of maps realized by left multiplication: K[2] A

→ End(K2 ) → LA .

K[2] is thus even. Finally, this forces the subspace of hermitian 2 × 2 matrices, h2 (K), to be even. So, the vector representation V is even. All this matches the usual rules in physics, where spinors are fermionic and vectors are bosonic.

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JOHN C. BAEZ AND JOHN HUERTA

6. Super-Yang–Mills Theory We are now ready to give a division algebra interpretation of the pure superYang–Mills Lagrangian 1 1 / A ψ L = − F, F  + ψ, D 4 2 and use this to prove its supersymmetry. For simplicity, we shall work over Minkowski spacetime, M . This allows us to treat all bundles as trivial, sections as functions, and connections as g-valued 1-forms. At the outset, we fix an invariant inner product on g, the Lie algebra of a semisimple Lie group G. We shall use the following standard tools from differential geometry to construct L, none of which need involve spinors or division algebra technology: • A connection A on a principal G-bundle over M . Since the bundle is trivial we think of this connection as a g-valued 1-form. • The exterior covariant derivative dA = d + [A, −] on g-valued p-forms. • The curvature F = dA + 12 [A, A], which is a g-valued 2-form. • The usual pointwise inner product F, F  on g-valued 2-forms, defined using the Minkowski metric on M and the invariant inner product on g. We also need the following spinorial tools. Recall from the preceding section that S+ and S− are odd objects in SuperVect. So, whenever we switch two spinors, we introduce a minus sign. • A g-valued section ψ of a spin bundle over M . Note that this is, in fact, just a function: ψ : M → S± ⊗ g. We call the collection of all such functions Γ(S± ⊗ g). / A derived from the connection A. Of • The covariant Dirac operator D course, / A : Γ(S± ⊗ g) → Γ(S∓ ⊗ g) D and in fact, / A = ∂/ + A. D • A bilinear pairing −, − : Γ(S+ ⊗ g) ⊗ Γ(S− ⊗ g) → C ∞ (M ) built pointwise using our pairing −, − : S+ ⊗ S− → R and the invariant inner product on g. The basic fields in our theory are a connection on a principal G-bundle, which we think of as a g-valued 1-form: A : M → V ∗ ⊗ g. and a g-valued spinor field, which we think of as a S+ ⊗ g-valued function on M : ψ : M → S+ ⊗ g. All our arguments would work just as well with S− replacing S+ .

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To show that L is supersymmetric, we need to show δL is a total divergence when δ is the following supersymmetry transformation: ·ψ 1 δψ = F 2 where  is an arbitrary constant spinor field, treated as odd, but not g-valued. By a supersymmetry transformation we mean that computationally we treat δ as a derivation. So, it is linear: δA

=

δ(αf + βg) = αδf + βδg where α, β ∈ R, and it satisfies the product rule: δ(f g) = δ(f )g + f δg. For a more formal definition of ‘supersymmetry transformation’ see [6]. The above equations require further explanation. The dot in  · ψ denotes an operation that combines the spinor  with the g-valued spinor ψ to produce a g-valued 1-form. We build this from our basic intertwiner · : S+ ⊗ S+ → V. We identify V with V

∗

using the Minkowski inner product g, obtaining · : S+ ⊗ S+ → V ∗ .

Then we tensor both sides with g. This gives us a way to act by a spinor field on a g-valued spinor field to obtain a g-valued 1-form. We take the liberty of also denoting this with a dot: · : Γ(S+ ) ⊗ Γ(S+ ⊗ g) → Ω1 (M, g). We also need to explain how the 2-form F acts on the constant spinor field . Using the Minkowski metric, we can identify differential forms on M with sections of the Clifford algebra bundle over M : Ω∗ (M ) ∼ = Cliff(M ). Using this, differential forms act on spinor fields. Tensoring with g, we obtain a way for g-valued differential forms like F to act on spinor fields like  to give g-valued spinor fields like F . Let us now apply the supersymmetry transformation to each term in the Lagrangian. First, the bosonic term: Proposition 12. The bosonic term has: δF, F  = 2(−1)n+1 ψ, ( dA F ) + divergence. Proof. By the symmetry of the inner product, we get: δF, F  = 2F, δF . Using the handy formula δF = dA δA, we have: F, δF  = F, dA δA. Now the adjoint of the operator dA is dA , up to a pesky sign: if ν is a g-valued (p − 1)-form and μ is a g-valued p-form, we have μ, dA ν = (−1)dp+d+1+s  dA μ, ν + divergence

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78 14

JOHN C. BAEZ AND JOHN HUERTA

where d is the dimension of spacetime and s is the signature, i.e., the number of minus signs in the diagonalized metric. It follows that F, δF  = F, dA δA = (−1)n  dA F, δA + divergence where n is the dimension of K. By the definition of δA, we get  dA F, δA =  dA F,  · ψ. Now we can use division algebra technology to show:

 1  dA F,  · ψ = Re tr ( dA F )(ψ † + ψ† ) = −ψ, ( dA F ), 2 using the cyclic property of the real trace in the last step, and introducing a minus sign in accordance with the sign rule. Putting everything together, we obtain the desired result.  Even though this proposition involved the bosonic term only, division algebra technology was still a useful tool in its proof. This is even more true in the next proposition, which deals with the the fermionic term: Proposition 13. The fermionic term has: / A ψ = ψ, D / A (F ) + tri ψ + divergence δψ, D where tri ψ = ψ, ( · ψ)ψ. Proof. It is easy to compute: / A ψ = δψ, D / A ψ + ψ, δ D / A ψ + ψ, D / A δψ. δψ, D / A = δA =  · ψ, and thus see that the penultimate term is the Now we insert δ D trilinear one: tri ψ = ψ, ( · ψ)ψ. So, let us concern ourselves with the remaining terms: / A ψ + ψ, D / A δψ. δψ, D A computation using the product rule shows that the divergence of the 1-form ψ · φ / A ψ + ψ, D / A φ, where the minus sign on the first term arises is given by −φ, D from using the sign rule with these odd spinors. In the terms under consideration, / A onto δψ: we can use this identity to move D / A ψ + ψ, D / A δψ = 2ψ, D / A δψ + divergence. δψ, D Substituting δψ = 12 F , we obtain the desired result.



Using these two propositions, it is immediate that 1 1 / A ψ δL = − δF, F  + δψ, D 4 2 1 1 1 / A (F ) + tri ψ + divergence. (−1)n ψ, ( dA F ) + ψ, D = 2 2 2 n+1 / A (F ) = (−1) All that remains to show is that D ( dA F ) . Indeed, Snygg shows (Eq. 7.6 in [16]) that for an ordinary, non-g-valued p-form F / ) = (dF ) + (−1)d+dp+s ( d F ) ∂(F

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79 15

where d is the dimension of spacetime and s is the signature. This is easily generalized to covariant derivatives and g-valued p-forms: / A (F ) = (dA F ) + (−1)d+dp+s ( dA F ). D In particular, when F is the curvature 2-form, the first term vanishes by the Bianchi identity dA F = 0, and we are left with: / A (F ) = (−1)n+1 ( dA F ) D where n is the dimension of K. We have thus shown: Proposition 14. Under supersymmetry transformations, the Lagrangian L has:

1 tri ψ + divergence. 2 The above result actually holds in every dimension, though our proof used division algebras and was thus adapted to the dimensions of interest: 3, 4, 6, and 10. The next result is where division algebra technology becomes really crucial: δL =

Proposition 15. For Minkowski spacetimes of dimensions 3, 4, 6, and 10, tri ψ = 0. Proof. At each point, we can write ψ= ψ a ⊗ ga , where ψ a ∈ S+ and ga ∈ g. When we insert this into tri ψ, we see that tri ψ = ψ a , ( · ψ b )ψ c  ga , [gb , gc ]. Since ga , [gb , gc ] is totally antisymmetric, this implies tri ψ = 0 for all  if and only if the part of ψ a , ( · ψ b )ψ c  that is antisymmetric in a, b and c vanishes for all . Yet these spinors are odd; for even spinors, we require the part of ψ a , ( · ψ b )ψ c  that is symmetric in a, b and c to vanish for all . Now let us bring in some division algebra technology to remove our dependence on . While we do this, let us replace ψ a with ψ, ψ b with φ, and ψ c with χ to lessen the clutter of indices. Substituting in the formulas from Table 1, we have † + φ† )χ) ψ, ( · φ)χ = Re(ψ † (φ

= Re tr(ψ † (φ† + φ† − † φ − φ† )χ) = , (ψ · χ)φ, where again we have employed the cyclic symmetry of the real trace, along with the identity: tr(φ† + φ† ) = Re tr(φ† + φ† ) = φ†  + † φ. This real quantity commutes and associates in any expression. So, if we seek to show that the part of ψ, ( · φ)χ that is totally symmetric in ψ, φ and χ vanishes for all , it is equivalent to show the totally symmetric part of (φ · χ)ψ vanishes. And since the dot operation in φ · χ is symmetric, this follows immediately from our main result, Theorem 11.  Acknowledgements. We thank Geoffrey Dixon, Tevian Dray, and Corinne Manogue for helpful conversations and correspondence. We also thank An Huang, Theo Johnson-Freyd and David Speyer for catching some errors. This work was partially supported by an FQXi grant.

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References 1. J. C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205. Also available as arXiv:math/0105155. 2. J. C. Baez and M. Stay, Physics, topology, logic and computation: a Rosetta Stone, to appear in New Structures For Physics, ed. Bob Coecke. Also available as arXiv:0903.0340. 3. L. Brink, J. Schwarz and J. Scherk, Supersymmetric Yang–Mills theory, Nucl. Phys. B121 (1977), 77–92. 4. K.-W. Chung and A. Sudbery, Octonions and the Lorentz and conformal groups of tendimensional space-time, Phys. Lett. B 198 (1987), 161–164. 5. T. Dray, J. Janesky and C. A. Manogue, Octonionic hermitian matrices with non-real eigenvalues, Adv. Appl. Clifford Algebras 10 (2000), 193–216. Also available as arXiv:math/0006069. 6. P. Deligne et al, eds., Quantum Fields and Strings: A Course for Mathematicians, Volume 1, Amer. Math. Soc., Providence, Rhode Island, 1999. 7. J. M. Evans, Supersymmetric Yang–Mills theories and division algebras, Nucl. Phys. B298 (1988), 92–108. Also available as http://ccdb4fs.kek.jp/cgi-bin/img index?8801412. 8. D. B. Fairlie and C. A. Manogue, A parameterization of the covariant superstring, Phys. Rev. D36 (1987), 475–479. 9. M. Green and J. Schwarz, Covariant description of superstrings, Phys. Lett. B136 (1984), 367–370. 10. M. Green, J. Schwarz and E. Witten, Superstring Theory, Volume 1, Cambridge U. Press, Cambridge, 1987. Appendix 4.A: Super Yang–Mills theories, pp. 244–247. Section 5.1.2: The supersymmetric string action, pp. 253–255. ¨ 11. A. Hurwitz, Uber die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. G¨ ottingen (1898), 309–316. 12. T. Kugo and P. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B221 (1983), 357–380. Also available at http://ccdb4fs.kek.jp/cgi-bin/img index?198301032. 13. C. A. Manogue and A. Sudbery, General solutions of covariant superstring equations of motion, Phys. Rev. D 12 (1989), 4073–4077. 14. J. Schray, The general classical solution of the superparticle, Class. Quant. Grav. 13 (1996), 27–38. Also available as arXiv:hep-th/9407045. 15. R. D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995. 16. J. Snygg, Clifford Algebra: a Computational Tool for Physicists, Oxford U. Press, Oxford, 1997. 17. A. Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17 (1984), 939–955. Department of Mathematics, University of California, Riverside, CA 92521 USA E-mail address: [email protected] Department of Mathematics, University of California, Riverside, CA 92521 USA E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681759

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

K-homology and D-branes Paul Baum

1. Introduction K-homology is the dual theory to K-theory. In algebraic geometry [14] [7], the K-homology of a (possibly singular) projective variety X is the Grothendieck group of coherent algebraic sheaves on X. In topology, there are three ways to define K-homology. First, K-homology is the homology theory determined by the Bott spectrum. Second, K-homology is the group of geometric K-cycles introduced by Baum-Douglas [6]. Third, using functional analysis, K-homology is the group of abstract elliptic operators as in the work of M. F. Atiyah [1], Brown-DouglasFillmore [15], and G. Kasparov [21]. The D-branes of string theory [31] are twisted geometric K-cycles which are endowed with some additional structure. The charge of a D-brane is the element in the twisted K-homology of spacetime determined by the underlying twisted Kcycle of the D-brane. Essentially, the Baum-Douglas theory [6] was rediscovered in terms of constraints on open strings. The aim of this expository note is to briefly describe this development. Thanks go to the referee and to Serge Ballif for helpful comments and technical assistance. 2. K-cycles and K-homology Let X be a locally compact Hausdorff topological space. For example, X can be a locally finite CW-complex. A K-cycle on X is a triple (M, E, ϕ) such that: • M is a compact Spinc manifold (without boundary). • E is a C vector bundle on M . • ϕ : M → X is a continuous map from M to X. On the collection of all K-cycles on X impose the equivalence relation generated by the three elementary moves: • bordism • direct sum-disjoint union • vector bundle modification PB was partially supported by an NSF grant. c Mathematical 0000 (copyright Society holder) c 2010 American

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Precise definitions of these three elementary moves are as follows. Isomorphism. Two K-cycles (M, E, ϕ), (M  , E  , ϕ ) on X are isomorphic if there is a diffeomorphism f : M → M  such that f preserves the Spinc structures on M and M  , the C vector bundles E, f ∗ (E  ) are isomorphic, and the diagram / M M5 55   ϕ 5  ϕ X f

commutes. Bordism. Two K-cycles (M0 , E0 , ϕ0 ) and (M1 , E1 , ϕ1 ) are bordant if there exists a triple (W, E, Ψ) such that W is a compact Spinc manifold with boundary, E is a C vector bundle on W , Ψ : W → X is a continuous map from W to X and (∂W, E|∂W, Ψ|∂W ) ∼ = (M0 , E0 , ϕ0 ) ∪ (−M1 , E1 , ϕ1 ). Note that here ∂W is given the Spinc structure that it receives from W and −M1 denotes M1 with its Spinc structure reversed. ∪ is disjoint union. Direct sum-disjoint union. Let (M, E, ϕ) be a K-cycle on X and let E  be a C vector bundle on M , then (M, E, ϕ) ∪ (M, E  , ϕ) ∼ (M, E ⊕ E  , ϕ). Vector bundle modification. Let (M, E, ϕ) be a K-cycle on X and let F be a Spinc vector bundle on M . Assume that F has even dimensional fibers. Then (M, E, ϕ) ∼ (S(F ⊕ θ 1 ), β ⊗ ρ∗ E, ϕ ◦ ρ). Here θ 1 is the trivial R line bundle on M (θ 1 = M × R), S(F ⊕ θ 1 ) is the unit sphere bundle of F ⊕ θ 1 , and ρ : S(F ⊕ θ 1 ) → M is the projection of S(F ⊕ θ 1 ) onto M . Since F has even dimensional fibers, S(F ⊕ θ 1 ) is a sphere bundle over M with even dimensional spheres as fibers. S(F ⊕ θ 1 ) is a Spinc manifold as it is given the Spinc structure resulting from the Spinc structure of M and the Spinc structure of F . β is the Thom isomorphism C vector bundle on S(F ⊕ θ 1 ) determined by the Spinc structure of F . When restricted to any fiber of ρ : S(F ⊕ θ 1 ) → M , β is the Bott generator vector bundle of that even dimensional sphere. Denote the collection of all K-cycles on X by {(M, E, ϕ)}. The K-homology of X, denoted K∗ (X), is K∗ (X) := {(M, E, ϕ)}/ ∼ . K∗ (X) is an abelian group. Addition is disjoint union. (M, E, ϕ) + (M  , E  , ϕ ) = (M ∪ M  , E ∪ E  , ϕ ∪ ϕ ). The negative of (M, E, ϕ) is (−M, E, ϕ). As above, −M is M with its Spinc structure reversed. The zero element of K∗ (X) is given by any (M, E, ϕ) which bounds. With j = 0, 1 let Kj (X) be the subgroup of K∗ (X) given by all (M, E, ϕ) such that every connected component of M has its dimension congruent to j modulo 2. Then: K∗ (X) = K0 (X) ⊕ K1 (X)

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Let (M, E, ϕ) be a K-cycle on X such that every connected component of M has its dimension congruent to j modulo 2. DE denotes the Dirac operator of M tensored with E. DE yields an element of the Kasparov [21] K-homology group KK j (C(M ), C). [DE ] ∈ KK j (C(M ), C). The map of C ∗ algebras C0 (X) → C(M ) given by ϕ : M → X induces a homomorphism of abelian groups KK j (C(M ), C) → KK j (C0 (X), C) As usual [29] C(M ) is the C ∗ algebra of all continuous complex-valued functions on M , and C0 (X) is the C ∗ algebra of all continuous complex-valued vanishing-atinfinity functions on X. Denote by ϕ∗ [DE ] the element of KK j (C0 (X), C) obtained by applying this homomophism to [DE ] ∈ KK j (C(M ), C) Then (M, E, ϕ) → ϕ∗ [DE ] is a homomophism of abelian groups mapping Kj (X) to KK j (C0 (X), C). Denote this by η : Kj (X) → KK j (C0 (X), C). The Kasparov K-homology of X , KK j (C0 (X), C), is not a compactly supported theory. By a slight abuse of notation, KKcj (C0 (X), C) will denote the direct limit over all compact subsets Δ of X of KK j (C(Δ), C), KKcj (C0 (X), C) :=

lim

−→ Δ⊂X Δ compact

KK j (C(Δ), C).

Thus KKc∗ (C0 (X), C) is Kasparov K-homology with compact supports. For any compact subset Δ of X, the inclusion Δ ⊂ X gives a homomorphism of abelian groups KK j (C(Δ), C) → KK j (C0 (X), C). These fit together to yield a homomorphism of abelian groups : KKcj (C0 (X), C) → KK j (C0 (X), C). It is immediate that η : Kj (X) → KK j (C0 (X), C)). factors through KKcj (C0 (X), C), so we obtain : η : Kj (X) → KKcj (C0 (X), C). Theorem (P. Baum + R. Douglas [6], P. Baum + N. Higson + T. Schick [9], P. Baum + N. Higson + T. Schick [10], P. Baum + H. Oyono-Oyono + T. Schick [11]). Let X be a locally finite CW complex. Then η : Kj (X) → KKcj (C0 (X), C) is an isomorphism of abelian groups. j = 0, 1. Remark. For X any CW complex, there is a natural isomorphism of K∗ (X) to the K-homology of X defined, as in homotopy theory, via the Bott spectrum. Denote homotopy theory K-homology by K∗h (X). To explicitly describe the isomorphism K∗h (X) ∼ = K∗ (X) it will be convenient to recall the

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Lemma 1. Let B be a Banach algebra with unit 1B . Then the open unit ball in B centered at 1B consists of invertible elements. For a proof see any introductory text on Banach Algebras, e.g. [22]. To define the isomorphism of abelian groups K∗h (X) −→ K∗ (X) proceed as follows. Let H be a separable (but not finite dimensional) Hilbert space. L(H) denotes the C ∗ algebra of all bounded operators T : H → H. Within L(H) there is the open set F red(H) of all Fredholm operators. According to [2] and [19], F red(H) can be taken to be Z × BU. So starting with a continuous map f : S n −→ (X × F red(H))/(X × I) a K-cycle on X must be constructed. Here S n is the n-sphere, I is the identity operator of H and (X ×F red(H))/(X ×I) is X ×F red(H) with X ×I collapsed to a point. This collapsing creates the basepoint, denoted p0 , in (X ×F red(H))/(X ×I). The map f is assumed to send the basepoint of S n to p0 . Within S n set Σ = f −1 (p0 )

Choose an open set U in S n such that U contains Σ and U has a smooth boundary.

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Thus S n − U is a (codimension zero) smooth submanifold with boundary in S n . Set Ω = S n − U and consider the restriction of f to Ω. f : Ω −→ (X × F red(H))/(X × I). Composing with the projections of X × F red(H) onto X and F red(H) then gives well-defined continuous maps fX : Ω −→ X fF red(H) : Ω −→ F red(H). Let M be the manifold obtained by ”doubling” Ω along ∂Ω. Thus M is formed by taking the disjoint union of two copies of Ω and then identifying the two copies of ∂Ω, M = Ω ∪∂Ω Ω. M is a closed manifold and is a π manifold , i.e. the tangent bundle T M is stably trivial. Hence a fortiori M is a compact Spinc manifold without boundary. Let ϕ : M −→ X be the composition M = Ω ∪∂Ω Ω −→ Ω −→ X where the first map is the standard map of a ”doubled” space back to the original space and the second map is fX . Now use fF red(H) and the above lemma to define a continuous map Ψ : M = Ω ∪∂Ω Ω −→ F red(H). On the first copy of Ω, Ψ is fF red(H) . The normal bundle ν of ∂Ω in the second copy of Ω is trivial. So there is the emdedding ∂Ω × [0, 1] → second copy of Ω. The open set U above can be chosen small enough so that fF red(H) maps ∂U = ∂Ω to the open unit ball in L(H) centered at I. The above lemma then applies to give the evident linear homotopy from the restrriction of fF red(H) to ∂Ω to the constant map sending all of ∂Ω to I. Ψ on ∂Ω × [0, 1] is this homotopy. On the remainder of the second copy of Ω, Ψ is the constant map sending every point to I. Since F red(H) is Z × BU , Ψ determines an element in the K-theory of M . Denote this K-theory element by E − F where E and F are C vector bundles on M . Then the required K-cycle on X is (M, E, ϕ)  (−M, F, ϕ) where  is disjoint union and −M is M with its Spinc structure reversed. The inverse homomorphism of abelian groups K∗h (X) ←− K∗ (X) is defined as follows. Given a K-cycle (M, E, ϕ) on X, embed M in Rn with even codimension. On the normal bundle of the embedding, the pull-backs of the two 1 2 -Spin vector bundles associated to the normal bundle become isomorphic (via Clifford multiplication) off the zero section of the normal bundle. Similarly when the two 12 -Spin vector bundles are tensored wiuth the pull-back of E. Since F red(H) is Z × BU this produces a continuous map from the normal bundle to F red(H)

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which maps the exterior of the unit ball bundle to I. Combining this with the projection of the unit ball bundle onto M yields a continuous map f : S n −→ (M × F red(H))/(M × I) ϕ : M → X and the identity map of F red(H) to itself give the evident map (M × F red(H))/(M × I) −→ (X × F red(H))/(X × I) The required map f : S n −→ (X × F red(H))/(X × I) is then the composition of these two. 3. String Theory and D-branes In string theory [16], [27], [26] [13] [24] a spacetime X is fixed. Often X is a Spin manifold of dimension 10, although there are other possibilities for X. Spacetime X comes equipped with its B-field and H-flux. The B-field, denoted B, is a locally defined 2-form and the H-flux, denoted H, is a globally defined closed 3-form on X with H = dB. Note that since B is only locally defined, the de Rham cohomology class determined by H can be non-zero. This de Rham cohomology class is required to satisfy a condition which essentially (i.e. up to a normalizing constant) asserts that the de Rham cohomology class of H is in the integral cohomology group H 3 (X, Z). Hence spacetime X comes equipped with an element in H 3 (X, Z). An elementary particle in spacetime X is either a 1-sphere S 1 = {(t1 , t2 ) ∈ R2 | t21 + t22 = 1} or a unit interval [0, 1] = {t ∈ R | 0 ≤ t ≤ 1} which moves in time within X sweeping out a two dimensional manifold or manifold with boundary known as the worldsheet of the particle. The S 1 case is the case of “closed strings” and the [0, 1] case is the case of “open strings”. In the open string case, the two endpoints of the interval [0, 1] are required to remain on a sub-manifold of X. This sub-manifold of X is known as the worldvolume of the D-brane giving the constraint condition on the open string. A D-brane, however, consists of more than just its worldvolume. If M denotes the worldvolume of a D-brane, then on M (due to the mixing of states in quantum mechanics [26]) a C vector bundle E appears known as the Chan-Paton bundle. Also, a D-brane will have additional structure. M will be something like a Spinc manifold, a connection will be given for the Chan-Paton bundle E etc. Consider Type II superstring theory on a spacetime X with all background supergravity form fields turned off. In this special case, spacetime X is a Spin manifold of dimension 10 and B and H are both zero. In this setting, [25] a first approximation to a definition of D-brane is that a D-brane is a triple (M, E, ϕ) such that: • M is a compact Spinc manifold (without boundary). • E is a C vector bundle on M . • ϕ : M → X is an embedding of M into X. In other words, a D-brane is a K-cycle (M, E, ϕ) such that ϕ : M → X is an embedding of M into X. The element in K∗ (X) determined by a D-brane (M, E, ϕ) is the charge of the D-brane. Each of the three elementary moves:

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• bordism • direct sum-disjoint union • vector bundle modification used to define K∗ (X) now takes on a physical meaning. For example, “direct sumdisjoint union” is closely related to “gauge symmetry enhancement for coincident branes” [25], [30] and “vector bundle modification” connects to the ”dielectric effect” [25], [23]. Two D-branes have the same charge if and only if it is possible to pass from one D-brane to the other by a finite sequence of these three elementary moves. Now consider Type II superstring theory where the background supergravity form fields are not turned off. The de Rham cohomology class of the H-flux might be non-zero. If so, this creates an anomaly and the D-branes must satisfy [18] an anomaly cancellation condition. The appropriate anomaly cancellation is achieved by using twisted K-homology, which will now be introduced. 4. Compact Operator Vector Bundles Let H be a separable (but not finite dimensional) Hilbert space. U(H) is the group of unitary operators U : H → H. U(H) is topologized by the weak operator topology. U(H) is a topological group, and as a topological space U(H) is contractible. S 1  K(Z, 1) where K(Z, n) denotes an n-th Eilenberg-MacLane space for Z i.e. K(Z, n) is a topological space, having the homotopy type of a CW complex, whose n-th homotopy group is Z and all other homotopy groups are zero.  Z, j = n, πj K(Z, n) = 0, j = n. The homotopy sequence of the fibration S 1 −→ U(H) −→ P U(H) implies that P U(H)  K(Z, 2). From this we obtain a fibration P U(H) −→ EP U(H) −→ K(Z, 3) This is the universal principal fibration with P U(H) as structure group. K(H) denotes the set of all compact operators T : H → H. U(H) acts on K(H) by conjugation : U(H) × K(H) −→ K(H) (U, T ) −→ U T U ∗ = U T U −1 . This action factors through P U(H), yielding an action of P U(H) on K(H) P U(H) × K(H) −→ K(H). When combined with the above universal principal P U(H) bundle, this action of P U(H) on K(H) gives a fibration K(H) −→ EP U(H) ×P U(H) K(H) −→ K(Z, 3). In this fibration each fiber is a C ∗ algebra which is (non-canonically) isomorphic to K(H).

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A twisted topological space X is a topological space X with a given continuous map α : X −→ K(Z, 3). If X has the homotopy type of a CW complex, then the set of homotopy classes of continuous maps from X to K(Z, 3) is in bijection with H 3 (X, Z) — so one might be tempted to define a twisted topological space as a topological space X together with a given element of H 3 (X, Z). It might, in fact, be possible to proceed this way — but a much greater degree of mathematical precision is achieved by taking the twisting structure (as above) to be a continuous map α : X −→ K(Z, 3). This point will be explained below. If X is a twisted topological space, then the given twisting map α : X −→ K(Z, 3) can be used to pull back the fibration K(H) −→ EP U(H) ×P U(H) K(H) −→ K(Z, 3). Thus a fibration K(H) −→ T −→ X is obtained with base space X. Each fiber is a C ∗ algebra which is (non-canonically) isomorphic to K(H). If X is Hausdorff and locally compact, consider the C ∗ algebra, C0α (X) consisting of all continuous vanishing-at-infinity sections of the fibration K(H) −→ T −→ X. The twisted K-theory of X [3] [4] [20] is, by definition, the K-theory of the C ∗ algebra C0α (X). As usual [29], this K-theory is denoted K∗ C0α (X). The twisted K-homology of X [28] (with compact supports), denoted KKcj (C0α (X), C), is KKcj (C0α (X), C) :=

lim

−→ Δ⊂X Δ compact

KK j (C α (Δ), C).

Here the direct limit is taken over all compact subsets Δ of X, and C α (Δ) is the C ∗ algebra consisting of all continuous sections defined on the compact set Δ of the fibration K(H) −→ T −→ X. Remark. Let α : X −→ K(Z, 3) and γ : X −→ K(Z, 3) be two continuous maps from X to K(Z, 3). If α and γ are homotopic, then K∗ C0α (X) and K∗ C0γ (X) are isomorphic. The isomorphism depends on the choice of homotopy from α to γ and thus is not canonical. So if a twisted topological space were defined as a topological space X together with a given element of H 3 (X, Z), then the twisted Ktheory [3] [4] would not be a well-defined (i.e. functorial) invariant of X. Similarly for the twisted K-homology. Hence the abelian group of which the charge of a Dbrane is an element (and therefore also the charge itself) would not be well-defined.

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5. Twisted K-cycles Let X be a twisted topological space with given twisting map α : X −→ K(Z, 3). M denotes a compact oriented manifold (without boundary) for which a continuous map ν : M −→ BSO has been fixed such that ν is a classifying map for the (oriented) stable normal bundle of M . BSO = lim BSO(n). n→∞

Recall that the third Stiefel-Whitney class determines (up to homotopy) a continuous map π : BSO −→ K(Z, 3) Fix once and for all such a continuous map π. With M, ν as above, a twisted K-cycle for X is a triple (M, E, ϕ) such that • E is a C vector bundle on M . • ϕ : M → X is a continuous map from M to X. • The diagram ν / M BSO ϕ

 X

π

α

 / K(Z, 3)

commutes. Let W3 (M ) be the third (integral) Stiefel-Whitney class of M and let [α] ∈ H 3 (X, Z) be the element in the third (integral) cohomology group of M determined by the twisting map α : X −→ K(Z, 3). Commutativity of the diagram implies ϕ∗ ([α]) + W3 (M ) = 0. This is the Freed-Witten anomaly cancellation condition for D-branes in Type II superstring theory [18]. These D-branes [31] are twisted K-cycles for spacetime X which are endowed with some extra structure. In particular, the charge of a D-brane is the element in the twisted K-homology of spacetime X determined by the underlying twisted K-cycle of the D-brane. Consider the special case when the twisting map α : X −→ K(Z, 3) is the constant map which sends every point of X to one point of K(Z, 3). Recall that for n ≥ 3 π1 (SO(n)) = Z/2Z. Spin(n) is the unique non-trivial two-fold cover of SO(n), i.e. Spin(n) is the universal covering group of SO(n). Spinc (n) := S 1 ×Z/2Z Spin(n). Thus, by definition, there is a short exact sequence of compact connected Lie groups 1 −→ S 1 −→ Spinc (n) −→ SO(n) −→ 1. Passing to classifying spaces this gives a fibration BS 1 −→ BSpinc (n) −→ BSO(n). Since BS 1 = K(Z, 2), a standard topological construction yields a fibration BSpinc (n) −→ BSO(n) −→ K(Z, 3)

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in which the map BSO(n) −→ K(Z, 3) is the map determined by the third StiefelWhitney class. Letting n go to infinity, this gives a fibration BSpinc −→ BSO −→ K(Z, 3). We may assume that the map π : BSO −→ K(Z, 3) fixed above is the projection of BSO onto K(Z, 3). If the twisting map α : X −→ K(Z, 3) is the constant map, then commutativity of the diagram ν

M ϕ

/ BSO π

 X

α

 / K(Z, 3)

implies that ν : M −→ BSO maps M to BSpinc . This is tantamount to giving a Spinc structure for M . So when the twisting map α : X −→ K(Z, 3) is the constant map, twisted K-cycles coincide with ordinary (i.e. untwisted) Kcycles as defined above. Remark. If the twisting map α : X −→ K(Z, 3) is only homotopic to a constant map and the diagram ν

M ϕ

/ BSO π

 X

α

 / K(Z, 3)

is only commutative up to homotopy, then a Spinc structure for M is not unambiguously determined. Homotopy-commutativity of the diagram implies that the Freed-Witten anomaly cancellation condition ϕ∗ ([α]) + W3 (M ) = 0 is valid, and α null-homotopic implies [α] = 0 Therefore, W3 (M ) = 0, so M is Spinc -able (i.e. M admits a Spinc structure), but there is insufficient information for a Spinc structure on M to be definitely determined. So again it is mathematically more precise to use maps and true commutativity of diagrams rather than homotopy classes of maps and homotopycommutative diagrams. In [28] this issue is dealt with by requiring that a homotopy of α ◦ ϕ to π ◦ ν be part of the given data. In the present note, however, the point

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of view is that the diagrams are required to truly commute at the level of spaces and maps. Suppose now that the twisting map α : X −→ K(Z, 3) is not the constant map. With α fixed, consider the collection of all the twisted K-cycles (M, E, ϕ). Proceeding as in the untwisted case, impose on this collection the equivalence relation generated by the three elementary moves: • bordism • direct sum-disjoint union • vector bundle modification With addition given by disjoint union, there are then two abelian groups: Kjα (X)

j = 0, 1

c

If X is a Spin or Spin manifold, then the given Spin or Spinc structure for X determines Poincar´e duality isomorphisms ∼ Kj+ C α (X) K α (X) = j = 0, 1. j

and

0

KKcj (C0α (X), C) ∼ = Kj+ C0α (X)

j = 0, 1.

Here is the dimension of X modulo 2. Combining these two isomorphisms gives an isomorphism Kjα (X) ∼ j = 0, 1 = KKcj (C0α (X), C) This last isomorphism (unlike the two preceding isomorphisms) does not depend on the choice of Spin or Spinc structure for X — which gives plausibility to Conjecture. Let X be a locally finite CW complex with given twisting map α : X −→ K(Z, 3). Then, (as in the untwisted case) there is a natural isomorphism of abelian groups Kjα (X) ∼ j = 0, 1. = KKcj (C0α (X), C) Outline of Proof. Let (M, E, ϕ) be a twisted K-cycle for X. With notation as above, consider the commutative diagram M

ν

ϕ

 X

/ BSO π

α

 / K(Z, 3)

Since ν is a classifying map for the (oriented) stable normal bundle of M , [π ◦ ν] = −W3 (M ) ∈ H 3 (M, Z). This implies that M has a fundamental class, denoted {M } ∈ KK ∗ (C π◦ν (M ), C). Commutativity of the diagram is precisely what is needed to define a homomorphism of abelian groups ϕ∗ : KK j (C π◦ν (M ), C) −→ KKcj (C0α (X), C). The conjectured isomorphism Kjα (X) −→ KKcj (C0α (X), C)

j = 0, 1

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will then be (M, E, ϕ) → ϕ∗ (E ∩ {M }) where E ∩ {M } is the cap product of E ∈ K 0 (M ) with {M } ∈ KK ∗ (C π◦ν (M ), C).  This conjecture will be taken up in [12]. In future versions of string theory, it is possible that spacetime X might not be a Spin or Spinc manifold (e.g. spacetime might be an orbifold [17] [24] [13] , rather than a manifold). If so, then this conjecture will be relevant to developing the theory. Appendix: K-theory versus K-homology If spacetime X is a Spin or Spinc manifold , then (as noted above) there is the Poincar´e duality isomorphism Kjα (X) ∼ = Kj+ C0α (X)

j = 0, 1

= dim(X) mod 2.

Thus the twisted K-theory, K∗ C0α (X) of X could be used as the abelian group for which the charge of a D-brane is an element. It seems more natural (and closer to the basic geometry-physics) to use twisted K-homology. Here’s a low-dimensional classical example. Let X be the torus S 1 × S 1 . H1 (X, Z) is a free abelian group on two generators a, b indicated below :

Consider an embedded S 1 in X, which winds 8 times in the a direction and 1 time in the −b direction.

The “charge” of this “D-brane” is 8a − b and it seems more natural to view this as an element of the homology group H1 (X, Z) rather than to invoke Poincar´e duality and take this to be an element of the cohomology group H 1 (X, Z). In many low-dimensional examples (such as this one) the Chern character maps K-homology –K-theory to ordinary integral homology – cohomology and hence gives an isomorphism of K-homology –K-theory to ordinary integral homology – cohomology.

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K-HOMOLOGY AND D-BRANES

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Some interesting examples of K-homology are calculated in [25]. In general, as observed in [27] [5], K-homology – K-theory is conceptually simpler than ordinary homology – cohomology. It is possible that in future versions of string theory spacetime X might have singularities [17] [24] [13] and thus Poincar´e duality might not be valid. If so, then a decision will have to be made on K-theory versus K-homology. In extending Riemann-Roch [7] [8] [6] from non-singular projective algebraic varieties to projective algebraic varieties which may have singularities, such a decision had to be made — and it was K-homology which, at the end of the day, played the more fundamental role.

References [1] Michael Atiyah, Global theory of elliptic operators. 1970 Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) pp. 21–30 Univ. of Tokyo Press, Tokyo. [2] Micahl Atiyah, K-theory, W. A. Benjamin Inc. New York, 1967. [3] Michael Atiyah and Graeme Segal, Twisted K-theory. Ukr. Mat. Visn. 1 (2004), no. 3, 287– 330; translation in Ukr. Math. Bull. 1 (2004), no. 3, 291–334. [4] Michael Atiyah and Graeme Segal, Twisted K-theory and cohomology. Inspired by S. S. Chern, 5–43, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006. [5] Paul Baum, Dirac operator and K-theory for discrete groups. to appear in proceedings of conference in memory of Raoul Bott, Montreal, June 2008. [6] Paul Baum and Ronald G. Douglas, K homology and index theory. In Operator algebras and applications, Part I (Kingston, Ont., 1980), volume 38 of Proc. Sympos. Pure Math., pages 117–173. Amer. Math. Soc., Providence, R.I., 1982. [7] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties. Publ. Math. IHES 45 (1975), 101–167. [8] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch and topological Ktheory for singular varieties, Acta Math 143 (1979), 155-192. [9] Paul Baum, Nigel Higson, and Thomas Schick, On the equivalence of geometric and analytic K-homology. Pure Appl. Math. Q. 3 (2007), no. 1, part 3, 1–24. [10] Paul Baum, Nigel Higson, and Thomas Schick, A geometric description of equivariant Khomology for proper actions, to appear in volume in honor of Alain Connes’ 60th birthday, Clay Mathematics Proceedings. [11] Paul Baum, Herve Oyono-Oyono, and Thomas Schick, Equivariant geometric K-homology for compact Lie groups actions, to appear in Abhandlungen aus dem Mathematischen Seminar der Universit¨ at Hamburg. [12] Paul Baum, Alan Carey, and Bai-Ling Wang, Geometric and analytic twisted K-homology. In preparation. [13] Katrin Becker, Melanie Becker, and John H. Schwarz, String theory and M-theory : A modern introduction, Cambridge University Press, Cambridge, 2007. [14] Armand Borel and Jean-Pierre Serre, Le th´eor` eme de Riemann-Roch, Bull. Soc. math. France, 86 (1958), 97–136. [15] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of C ∗ -algebras and K-homology. Ann. of Math. (2) 105 (1977), no. 2, 265–324. [16] Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Ed. Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison and Edward Witten. Ameri. Math. Soc., Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Vol. 1: xxii+723 pp.; Vol. 2: pp. i–xxiv and 727–1501. [17] L. Dixon, J. A. Harvey, C. Vafa, and Edward Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985), no. 4, 678–686. [18] Daniel S. Freed, Edward Witten, Anomalies in string theory with D-branes. Asian J. Math. 3 (1999), no. 4, 819–851. [19] Klaus J¨ anich. Vektorraumb¨ undel und der Raum der Fredholm-Operatoren. Math. Ann. 161 1965 129–142.

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[20] M. Karoubi, Twisted K-theory—old and new. K-theory and noncommutative geometry, 117– 149, EMS Ser. Congr. Rep., Eur. Math. Soc., Zrich, 2008. [21] G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology. (Russian) (Russian) ; translated from Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 796–838 Math. USSR-Izv. 9 (1975), no. 4, 751–792 (1976). [22] Lynn H. Loomis, An introduction to Abstract Harmonic Analysis, D. Van Nostrand Company Inc., New York, 1953. [23] R. C. Myers, Dielectric-Branes. J. High Energy Phys. 9912, 022 (1999). [24] Joseph Polchinski, String theory. Vol I : An introduction to the bosonic string, and, String theory. Vol II : Superstring theory and beyond. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge 2005 (reprint of the 2003 edition). [25] Rui M. G. Reis, Richard J. Szabo, Geometric K-homology of flat D-branes. Comm. Math. Phys. 266 (2006), no. 1, 71–122. [26] Jonathan Rosenberg, Topology, C ∗ -Algebras, and String Duality, CBMS Number 111, Amer. Math. Soc. 2009. [27] Graeme Segal, Topological structures in string theory. Topological methods in the physical sciences (London, 2000). R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), no. 1784, 1389–1398. [28] Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology. J. Noncommut. Geom. 2 (2008), no. 4, 497–552. [29] N. E. Wegge-Olson, K-theory and C ∗ -algebras: A friendly approach, Oxford Science Publications, Oxford Univ. Press, New York, 1993. [30] Edward Witten, Bound states of strings and p-branes. Nuclear Phys. B 460 (1996), no. 2, 335–350. [31] Edward Witten, D-branes and K-theory. J. High Energy Phys. 1998, no. 12, Paper 19, 41 pp. (electronic).

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http://dx.doi.org/10.1090/pspum/081/2681760

Proceedings of Symposia in Pure Mathematics Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Riemann-Roch and Index Formulae in Twisted K-theory Alan L. Carey and Bai-Ling Wang A BSTRACT. In this paper, we establish the Riemann-Roch theorem in twisted K-theory extending our earlier results. We also give a careful summary of twisted geometric cycles explaining in detail some subtle points in the theory. As an application, we prove a twisted index formula and show that D-brane charges in Type I and Type II string theory are classified by twisted KO-theory and twisted K-theory respectively in the presence of B-fields as proposed by Witten.

C ONTENTS 1. Introduction 2. Twisted K-theory: Preliminary Review 3. Twisted K-homology 4. The Chern Character in Twisted K-theory 5. Thom Classes and Riemann-Roch Formula in Twisted K-theory 6. The Twisted Index Formula 7. Mathematical Definition of D-branes and D-brane Charges References

96 97 103 111 119 125 127 130

1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. Twisted K-theory, twisted K-homology, twisted Riemann-Roch, twisted index theorem, D-brane charges. c c 0000 (copyright holder) 2010 American Mathematical Society

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1. Introduction 1.1. String geometry. We begin by giving a short discussion of the physical background. Readers uninterested in this motivation may move to the next subsection. In string theory D-branes were proposed as a mechanism for providing boundary conditions for the dynamics of open strings moving in space-time. Initially they were thought of as submanifolds. As D-branes themselves can evolve over time one needs to study equivalence relations on the set of D-branes. An invariant of the equivalence class is the topological charge of the D-brane which should be thought of as an analogue of the Dirac monopole charge as these D-brane charges are associated with gauge fields (connections) on vector bundles over the D-brane. These vector bundles are known as Chan-Paton bundles. In [MM] Minasian and Moore made the proposal that D-brane charges should take values in K-groups and not in the cohomology of the space-time or the D-brane. However, they proposed a cohomological formula for these charges which might be thought of as a kind of index theorem in the sense that, in general, index theory associates to a K-theory class a number which is given by an integral of a closed differential form. In string theory there is an additional field on space-time known as the H-flux which may be thought of as a global closed three form. Locally it is given by a family of ‘two-form potentials’ known as the B-field. Mathematically we think of these B-fields as defining a degree three integral ˇ Cech class on the space-time, called a ‘twist’. Witten [Wit], extending [MM], gave a physical argument for the idea that D-brane charges should be elements of K-groups and, in addition, proposed that the D-brane charges in the presence of a twist should take values in twisted K-theory (at least in the case where the twist is torsion). The mathematical ideas he relied on were due to Donovan and Karoubi [DK]. Subsequently Bouwknegt and Mathai [BouMat] extended Witten’s proposal to the non-torsion case using ideas from [Ros]. A geometric model (that is, a ‘string geometry’ picture) for some of these string theory constructions and for twisted K-theory was proposed in [BCMMS] using the notion of bundle gerbes and bundle gerbe modules. Various refinements of twisted K-theory that are suggested by these applications are also described in the article of Atiyah and Segal [AS1] and we will need to use their results here.

1.2. Mathematical results. From a mathematical perspective some immediate questions arise from the physical input summarised above. When there is no twist it is well known that K-theory provides the main topological tool for the index theory of elliptic operators. One version of the Atiyah-Singer index theorem due to Baum-Higson-Schick [BHS] establishes a relationship between the analytic viewpoint provided by elliptic differential operators and the geometric viewpoint provided by the notion of geometric cycle introduced in the fundamental paper of Baum and Douglas [BD2]. The viewpoint that geometric cycles in the sense of [BD2] are a model for D-branes in the untwisted case is expounded in [RS, RSV, Sz]. Note that in this viewpoint D-branes are no longer submanifolds but the images of manifolds under a smooth map. It is thus tempting to conjecture that there is an analogous picture of D-branes as a type of geometric cycle in the twisted case as well. More precisely we ask the question of whether there is a way to formulate the notion of ‘twisted geometric cycle’ (cf [BD1] and [BD2]) and to prove an index theorem in the spirit of [BHS] for twisted K-theory. This precise question was answered in the positive in [Wa]. It is important to emphasise that string geometry ideas from [FreWit] played a key role in finding the correct way to generalise [BD1].

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

973

Our purpose here in the present paper is threefold. First, we explain the results in [Wa] (see Section 3) in a fashion that is more aligned to the string geometry viewpoint. Second, we prove an analogue of the Atiyah-Hirzebruch Riemann-Roch formula in twisted K-theory by extending the results and approach of [CMW]. An interesting by-product of our approach in Section 5 is a discussion of the Thom class in twisted K theory. Third, in Section 6 we prove an index formula using our twisted Riemann-Roch theorem. It will be clear from our approach to this twisted index theory that our twisted geometric cycles provide a geometric model for D-branes and we give details in Section 7. Our main new results are stated as two theorems, Theorem 5.3 (twisted RiemannRoch) and Theorem 6.1 (the index pairing). We remark that the Minasian-Moore formula [MM] arises from the fact that the index pairing they discuss may be regarded as a quadratic form on K-theory. In the twisted index formula that we establish, the pairing is asymmetric and may be thought of as a bilinear form, from which there is no obvious way to extract a twisted analogue of the Minasian-Moore formula. Nevertheless we interpret our results in terms of the physics language in Section 7 explaining the link to Witten’s original ideas on D-brane charges. Acknowledgements. We thank the Australian Research Council, the Hausdorff Institute for Mathematics and the participants and organisers of the CBMS Conference on Topology, C*-algebras, and String Duality for their respective contributions to the writing of this paper. 2. Twisted K-theory: Preliminary Review 2.1. Twisted K-theory: topological and analytic definitions. We begin by reviewing the notion of a ‘twisting’. Let H be an infinite dimensional, complex and separable Hilbert space. We shall consider locally trivial principal P U (H)-bundles over a paracompact Hausdorff topological space X, the structure group P U (H) is equipped with the norm topology. The projective unitary group P U (H) with the topology induced by the norm topology on U (H) (Cf. [Kui]) has the homotopy type of an Eilenberg-MacLane space K(Z, 2). The classifying space of P U (H), denoted BP U (H), is a K(Z, 3). The set of isomorphism classes of principal P U (H)-bundles over X is given by (Proposition 2.1 in [AS1]) homotopy classes of maps from X to any K(Z, 3) and there is a canonical identification [X, BP U (H)] ∼ = H 3 (X, Z). A twisting of complex K-theory on X is given by a continuous map α : X → K(Z, 3). For such a twisting, we can associate a canonical principal P U (H)-bundle Pα through the usual pull-back construction from the universal P U (H) bundle denoted by EK(Z, 2), as summarised by the diagram: (2.1)

/ EK(Z, 2)

Pα  X

α

 / K(Z, 3).

We will use P U (H) as a group model for a K(Z, 2). We write Fred(H) for the connected component of the identity of the space of Fredholm operators on H equipped with the norm topology. There is a base-point preserving action of P U (H) given by the conjugation action of U (H) on Fred(H): (2.2)

P U (H) × Fred(H) −→ Fred(H).

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ALAN L. CAREY AND BAI-LING WANG

The action (2.2) defines an associated bundle over X which we denote by Pα (Fred) = Pα ×P U(H) Fred(H) We write {ΩnX Pα (Fred) = Pα ×P U(H) Ωn Fred} for the fiber-wise iterated loop spaces. D EFINITION 2.1. The (topological) twisted K-groups of (X, α) are defined to be   K −n (X, α) := π0 Cc (X, ΩnX Pα (Fred)) , the set of homotopy classes of compactly supported sections (meaning they are the identity operator in Fred off a compact set) of the bundle of Pα (Fred). Due to Bott periodicity, we only have two different twisted K-groups K 0 (X, α) and K (X, α). Given a closed subspace A of X, then (X, A) is a pair of topological spaces, and we define relative twisted K-groups to be 1

K ev/odd (X, A; α) := K ev/odd (X − A, α). Take a pair of twistings α0 , α1 : X → K(Z, 3), and a map η : X × [1, 0] → K(Z, 3) which is a homotopy between α0 and α1 , represented diagrammatically by α0

 

X

η  

K(Z, 3). >

α1

Then there is a canonical isomorphism Pα0 ∼ = Pα1 induced by η. This canonical isomorphism determines a canonical isomorphism on twisted K-groups (2.3)

η∗ : K ev/odd (X, α0 )

∼ =

/ K ev/odd (X, α1 ),

This isomorphism η∗ depends only on the homotopy class of η. The set of homotopy classes of maps between α0 and α1 is labelled by [X, K(Z, 2)]. Recall the first Chern class isomorphism Vect1 (X) ∼ = [X, K(Z, 2)] ∼ = H 2 (X, Z) where Vect1 (X) is the set of equivalence classes of complex line bundles on X. We remark that the isomorphisms induced by two different homotopies between α0 and α1 are related through an action of complex line bundles. Let K be the C ∗ -algebra of compact operators on H. The isomorphism P U (H) ∼ = Aut(K) via the conjugation action of the unitary group U (H) provides an action of a K(Z, 2) on the C ∗ -algebra K. Hence, any K(Z, 2)-principal bundle Pα defines a locally trivial bundle of compact operators, denoted by Pα (K) = Pα ×P U(H) K. Let C0 (X, Pα (K)) be the C ∗ -algebra of sections of Pα (K) vanishing at infinity. Then C0 (X, Pα (K) is the (unique up to isomorphism) stable separable complex continuoustrace C ∗ -algebra over X with Dixmier-Douday class [α] ∈ H 3 (X, Z) (here we identify ˇ the Cech cohomology of X with its singular cohomology, cf [Ros] and [AS1]). T HEOREM 2.2. ( [AS1] and [Ros]) The topological twisted K-groups K ev/odd (X, α) are canonically isomorphic to analytic K-theory of the C ∗ -algebra C0 (X, Pα (K)) K ev/odd (X, α) ∼ = Kev/odd (C0 (X, Pα (K))) where the latter group is the algebraic K-theory of C0 (X, Pα (K)), defined to be   lim π1 GLk (C0 (X, Pα (K))) . −→ k→∞

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Note that the algebraic K-theory of C0 (X, Pα (K)) is isomorphic to Kasparovs KKtheory ([Kas1] and [Kas2]) KK ev/odd (C, C0 (X, Pα (K)). It is important to recognise that these groups are only defined up to isomorphism by the Dixmier-Douady class [α] ∈ H 3 (X, Z). To distinguish these two equivalent definitions of twisted K-theory if needed, we will write ev/odd

Ktop

(X, α)

ev/odd Kan (X, α)

and

for the topological and analytic twisted K-theories of (X, α) respectively. Twisted Ktheory is a 2-periodic generalized cohomology theory: a contravariant functor on the category consisting of pairs (X, α), with the twisting α : X → K(Z, 3), to the category of Z2 -graded abelian groups. Note that a morphism between two pairs (X, α) and (Y, β) is a continuous map f : X → Y such that β ◦ f = α. 2.2. Twisted K-theory for torsion twistings. There are some subtle issues in twisted K-theory and to handle these we have chosen to use the language of bundle gerbes, connections and curvings as explained in [Mur]. We explain first the so-called ‘lifting bundle gerbe’ Gα [Mur] associated to the principal P U (H)-bundle π : Pα → X and the central extension (2.4)

1 → U (1) −→ U (H) −→ P U (H) → 1. [2]

This is constructed by starting with π : Pα → X, forming the fibre product Pα which is a groupoid π1 / [2] Pα = Pα ×X Pα π2 / Pα with source and range maps π1 : (y1 , y2 ) → y1 and π2 : (y1 , y2 ) → y2 . There is an [2] obvious map from each fiber of Pα to P U (H) and so we can define the fiber of Gα over [2] a point in Pα by pulling back the fibration (2.4) using this map. This endows Gα with a groupoid structure (from the multiplication in U (H)) and in fact it is a U (1)-groupoid [2] extension of Pα . A torsion twisting α is a map α : X → K(Z, 3) representing a torsion class in H 3 (X, Z). Every torsion twisting arises from a principal P U (n)-bundle Pα (n) with its classifying map X → BP U (n), or a principal P U (H)-bundle with a reduction to P U (n) ⊂ P U (H). For a torsion twisting α : X → BP U (n) → BP U (H), the corresponding lifting bundle gerbe Ga (2.5)

Gα  Pα (n)[2]

π1 π2

// P (n) α π

 M ∼ Pα (n)  P U (n) ⇒ Pα (n) (as a groupoid) and the central is defined by Pα (n)[2] = extension 1 → U (1) −→ U (n) −→ P U (n) → 1.

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There is an Azumaya bundle associated to Pα (n) arising naturally from the P U (n) action on the n × n matrices. We denote this associated Azumaya bundle by Aα . An Aα -module is a complex vector bundle E over M with a fiberwise Aα action Aσ ×M E −→ E. ∗

The C -algebra of continuous sections of Aα , vanishing at infinity if X is non-compact, is Morita equivalent to a continuous trace C ∗ -algebra C0 (X, Pα (K)). Hence there is an isomorphism between K 0 (X, α) and the K-theory of the bundle modules of Aa . There is an equivalent definition of twisted K-theory using bundle gerbe modules (Cf. [BCMMS] and [CW1]). A bundle gerbe module E of Gα is a complex vector bundle E over Pα (n) with a groupoid action of Gα , i.e., an isomorphism φ : Gα ×(π2 ,p) E −→ E where Gα ×(π2 ,π) E is the fiber product of the source π2 : Gα → Pα (n) and p : E → Pα (n) such that (1) p ◦ φ(g, v) = π1 (g) for (g, v) ∈ Gα ×(π2 ,p) E, and π1 is the target map of Gα . (2) φ is compatible with the bundle gerbe multiplication m : Ga ×(π2 ,π1 ) Gα → Gα , which means φ ◦ (id × φ) = φ ◦ (m × id). Note that the natural representation of U (n) on Cn induces a Gα bundle gerbe module Sn = Pα (n) × Cn . Here we use the fact that Gα = Pα (n)  U (n) ⇒ Pα (n) (as a groupoid). Similarly, the dual representation of U (n) on Cn induces a G−α bundle gerbe module Sn∗ = Pα (n)×Cn . Note that Sn∗ ⊗Sn ∼ = π ∗ Aα descends to the Azumaya bundle Aα . Given a Gα bundle gerbe module E of rank k, then as a P U (n)-equivariant vector bundle, Sn∗ ⊗ E descends to an Aα -bundle over M . Conversely, given an Aα -bundle E over M , Sn ⊗π∗ Aα π ∗ E defines a Gα bundle gerbe module. These two constructions are inverse to each other due to the fact that S ∗ ⊗ (Sn ⊗π∗ A π ∗ E) ∼ = (S ∗ ⊗ Sn ) ⊗π∗ A π ∗ E ∼ = π ∗ Aα ⊗π∗ A π ∗ E ∼ = π ∗ E. n

α

n

α

α

Therefore, there is a natural equivalence between the category of Gα bundle gerbe modules and the category of Aα bundle modules, as discussed in [CW1]. In summary, we have the following proposition. P ROPOSITION 2.3. ([BCMMS][CW1]) For a torsion twisting α : X → BP U (n) → BP U (H), twisted K-theory K 0 (X, α) has another two equivalent descriptions: (1) the Grothendieck group of the category of Gα bundle gerbe modules. (2) the Grothendieck group of the category of Aσ bundle modules. One important example of torsion twistings comes from real oriented vector bundles. Consider an oriented real vector bundle E of even rank over X with a fixed fiberwise inner product. Denote by νE : X → BSO(2k) the classifying map of E. The following twisting o(E) := W3 ◦ νE : X −→ BSO(2k) −→ K(Z, 3), will be called the orientation twisting associated to E. Here W3 is the classifying map of the principal BU(1)-bundle BSpinc (2k) → BSO(2k). Note that the orientation twisting o(E) is null-homotopic if and only if E is K-oriented.

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P ROPOSITION 2.4. Given an oriented real vector bundle E of even rank over X with an orientation twisting o(E), then there is a canonical isomorphism K 0 (X, o(E)) ∼ = K 0 (X, W3 (E)) where K 0 (X, W3 (E)) is the K-theory of the Clifford modules associated to the bundle Cliff (E) of Clifford algebras. P ROOF. Denote by Fr the frame bundle of V , the principal SO(2k)-bundle of positively oriented orthonormal frames, i.e., E = Fr ×ρ2n R2k , where ρn is the standard representation of SO(2k) on Rn . The lifting bundle gerbe associated to the frame bundle and the central extension 1 → U (1) −→ Spinc (2k) −→ SO(2k) → 1 is called the Spinc bundle gerbe GW3 (E) of E, whose Dixmier-Douady invariant is given by the integral third Stiefel-Whitney class W3 (E) ∈ H 3 (X, Z). The canonical representation of Spinc (2k) gives a natural inclusion Spinc (2k) ⊂ U (2k ) which induces a commutative diagram U (1) =

 U (1) =

 U (1)

/ Spinc (2k)

/ SO(2k)

 / U (2k )

 / P U (2k )

 / U (H)

 / P U (H).

This provides a reduction of the principal P U (H)-bundle Po(E) . The associated bundle of Azumaya algebras is in fact the bundle of Clifford algebras, whose bundle modules are called Clifford modules ([BGV]). Hence, there exists a canonical isomorphism between K 0 (X, o(E)) and the K-theory of the Clifford modules associated to the bundle Cliff (E).  2.3. Twisted K-theory: general properties. Twisted K-theory satisfies the following properties whose proofs are rather standard for a 2-periodic generalized cohomology theory ([AS1] [CW1] [Kar] [Wa]). (Note that when we write (X, A) for a pair of spaces we assume A ⊂ X.) (I) (The homotopy axiom) If two morphisms f, g : (Y, B) → (X, A) are homotopic through a map η : (Y × [0, 1], B × [0, 1]) → (X, A), written in terms of the following homotopy commutative diagram f

(Y, B) g

tt tt tt

 u} (X, A)

η

α

/ (X, A) tt tt α

 / K(Z, 3),

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then we have the following commutative diagram K ev/odd (X, A; α) SSS kk SSS g∗ k kkk SSS k k SSS k k ∗ k f SS) ukkk η∗ ev/odd ev/odd / K (Y, B; α ◦ f ) K (Y, B; α ◦ g). Here η∗ is the canonical isomorphism induced by the homotopy η. (II) (The exact axiom) For any pair (X, A) with a twisting α : X → K(Z, 3), there exists the following six-term exact sequence K 0 (X, A; α) O

/ K 0 (X, α)

/ K 0 (A, α|A )

K 1 (A, α|A ) o

K 1 (X, α) o

 K 1 (X, A; α)

here α|A is the composition of the inclusion and α. (III) (The excision axiom) Let (X, A) be a pair of spaces and let U ⊂ A be a subspace such that the closure U is contained in the interior of A. Then the inclusion ι : (X −U, A−U ) → (X, A) induces, for all α : X → K(Z, 3), an isomorphism K ev/odd (X, A; α) −→ K ev/odd (X − U, A − U ; α ◦ ι). (IV) (Multiplicative property) Let α, β : X → K(Z, 3) be a pair of twistings on X. Denote by α + β the new twisting defined by the following map1 (2.6)

α+β :

X

(α,β)

/ K(Z, 3) × K(Z, 3)

m

/ K(Z, 3),

where m is defined as follows BP U (H) × BP U (H) ∼ = B(P U (H) × P U (H)) −→ BP U (H), for a fixed isomorphism H ⊗ H ∼ = H. Then there is a canonical multiplication (2.7)

K ev/odd (X, α) × K ev/odd (X, β) −→ K ev/odd (X, α + β), which defines a K 0 (X)-module structure on twisted K-groups K ev/odd (X, α). (V) (Thom isomorphism) Let π : E → X be an oriented real vector bundle of rank k over X, then there is a canonical isomorphism, for any twisting α : X → K(Z, 3), K ev/odd (X, α + oE ) ∼ = K ev/odd (E, α ◦ π),

(2.8)

with the grading shifted by k(mod 2). (VI) (The push-forward map) For any differentiable map f : X → Y between two smooth manifolds X and Y , let α : Y → K(Z, 3) be a twisting. Then there is a canonical push-forward homomorphism   f!K : (2.9) K ev/odd X, (α ◦ f ) + of −→ K ev/odd (Y, α), with the grading shifted by n mod(2) for n = dim(X) + dim(Y ). Here of is the orientation twisting corresponding to the bundle T X ⊕ f ∗ T Y over X. 1In terms of bundles of projective Hilbert space, this operation corresponds to the Hilbert space tenrsor product, see [AS1].

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

1039

(VII) (Mayer-Vietoris sequence) If X is covered by two open subsets U1 and U2 with a twisting α : X → K(Z, 3), then there is a Mayer-Vietoris exact sequence / K 1 (U1 ∩ U2 , α12 )

K 0 (X, α) O K 0 (U1 , α1 ) ⊕ K 0 (U2 , α2 ) o

/ K 1 (U1 , α1 ) ⊕ K 1 (U2 , α2 )  K 1 (X, α)

K 0 (U1 ∩ U2 , α12 ) o

where α1 , α2 and α12 are the restrictions of α to U1 , U2 and U1 ∩U2 respectively. 3. Twisted K-homology Complex K-theory, as a generalized cohomology theory on a CW complex, is developed by Atiyah-Hirzebruch using complex vector bundles. It is representable in the sense that there exists a classifying space Z × BU (∞), where BU (∞) = limk BU (k), such that −→ K 0 (X) = [X, Z × BU (∞)] for any finite CW complex X. The classifying space for complex K-theory is referred to as the BU (∞)-spectrum with even term Z × BU (∞) and odd term U (∞). They are also called the ‘complex K-spectra’ in the literature. The advantage of using spectra is that there is a natural definition of a homology theory associated to a classifying space of each generalized cohomology theory. Hence, the topological K-homology of a CW complex X, dual to complex K-theory, is defined by the following stable homotopy groups top Kev (X) = lim π2k (BU (∞) ∧ X + ) −→ k→∞

and top (X) = lim π2k+1 (BU (∞) ∧ X + ). Kodd −→ k→∞

Here X + is the space X with one point added as a based point, and the wedge product of two based CW complexes (X, x0 ) and (Y, y0 ) is defined to be X ×Y . X ∧Y = (X × {y0 } ∪ {x0 } × Y ) All the properties of K-homology, as a generalized homology theory, can be obtained in a natural way see for example in [Swi]. There are two other equivalent definitions of K-homology, called analytic K-homology developed by Kasparov, and geometric Khomology by Baum and Douglas. We now give a brief review of these two definitions. Kasparov’s analytic K-homology KK ev/odd (C(X), C) is generated by unitary equivalence classes of (graded) Fredholm modules over C(X) modulo an operator homotopy an relation ([Kas1] and [HigRoe]). For brevity we will use the notation Kev/odd (X) for this an K-homology. A cycle for K0 (X), also called a Z2 -graded Fredholm module, consists of a triple (φ0 ⊕, φ1 , H0 ⊕ H1 , F ), where • φi : C(X) → B(Hi ) is a representation of C(X) on a separable Hilbert space Hi ; • F : H0 → H1 is a bounded operator such that φ1 (a)F − F φ0 (a),

φ0 (a)(F ∗ F − Id)

φ1 (a)(F F ∗ − Id)

are compact operators for all a ∈ C(X). A cycle for K1an (X), also called a trivially graded or odd Fredholm module, consists of a pair (φ, F ), where

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104 10

ALAN L. CAREY AND BAI-LING WANG

• φ : C(X) → B(H) is a representation of C(X) on a separable Hilbert space H; • F is a bounded self-adjoint operator on H such that φ(a)F − F φ(a),

φ(a)(F 2 − Id)

are compact operators for all a ∈ C(X). In [BD1] and [BD2], Baum and Douglas gave a geometric definition of K-homology geo (X) (respectively using what are now called geometric cycles. The basic cycles for Kev geo Kodd (X)) are triples (M, ι, E) consisting of even-dimensional (resp. odd-dimensional) closed smooth manifolds M with a given Spinc structure on the tangent bundle of M together with a continuous map ι : M → X and a complex vector bundle E over M . The equivalence relation on the set of all cycles is generated by the following three steps (see [BD1] for details): (i) Bordism. (ii) Direct sum and disjoint union. (iii) Vector bundle modification. geo Addition in Kev/odd (X) is given by the disjoint union operation of geometric cycles. Baum-Douglas in [BD2] showed that the Atiyah-Singer index theorem is encoded in the following commutative diagram

(3.1)

top Kev/odd (X) NNN p p ∼ NNN∼ =ppp = NNN pp p NN& p xp μ geo an / Kev/odd (X) Kev/odd (X)

where μ is the assembly map assigning an abstract Dirac operator an ι∗ ([D /E M ]) ∈ Kev/odd (X)

to a geometric cycle (M, ι, E). For a paracompact Hausdorff space X with a twisting α : X → K(Z, 3), all these three versions of twisted K-homology were studied in [Wa]. They are called there the twisted topological, analytic and geometric K-homologies, and denoted respectively by top geo an Kev/odd (X, α), Kev/odd (X, α) and Kev/odd (X, α). Our first task in this Section is to review these three definitions, see [Wa] for greater detail. 3.1. Topological and analytic definitions of twisted K-homology. Let X be a CW complex (or paracompact Hausdorff space) with a twisting α : X → K(Z, 3). Let Pα be the corresponding principal K(Z, 2)-bundle. Any base-point preserving action of a K(Z, 2) on a space defines an associated bundle by the standard construction. In particular, as a classifying space of complex line bundles, K(Z, 2) acts on the complex K-theory spectrum K representing the tensor product by complex line bundles, where Kev = Z × BU (∞),

Kodd = U (∞).

Denote by Pα (K) = Pα ×K(Z,2) K the bundle of based K-theory spectra over X. There is a section of Pα (K) = Pα ×K(Z,2) K defined by taking the base points of each fiber. The image of this section can be identified with X and we denote by Pα (K)/X the quotient space of Pα (K) obtained by collapsing the image of this section.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

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The stable homotopy groups of Pα (K)/X by definition give the topological twisted top (X, α). (There are only two due to Bott periodicity of K.) K-homology groups Kev/odd Thus we have   top Kev (X, α) = lim π2k Pα (BU (∞))/X −→ k→∞

and

  top Kodd (X, α) = lim π2k+1 Pα (BU (∞))/X . −→ k→∞

Here the direct limits are taken by the double suspension     πn+2k Pα (BU (∞))/X −→ πn+2k+2 Pα (S 2 ∧ BU (∞))/X and then followed by the standard map   πn+2k+2 Pα (S 2 ∧ BU (∞))/X

b∧1/

  πn+2k+2 Pα (BU (∞) ∧ BU (∞))/X

m

  / πn+2k+2 Pα (BU (∞))/X

where b : R2 → BU (∞) represents the Bott generator in K 0 (R2 ) ∼ = Z, m is the base point preserving map inducing the ring structure on K-theory. For a relative CW-complex (X, A) with a twisting α : X → K(Z, 3), the relative top version of topological twisted K-homology, denoted Kev/odd (X, A, α), is defined to be top (X/A, α) where X/A is the quotient space of X obtained by collapsing A to a Kev/odd point. Then we have the following exact sequence top Kodd (X, A; α) O

top Kodd (X, α) o

/ K top (A, α|A ) ev

top Kodd (A, α|A ) o

/ K top (X, α) ev  top Kev (X, A; α)

and the excision properties top top Kev/odd (X, B; α) ∼ = Kev/odd (A, A − B; α|A )

for any CW-triad (X; A, B) with a twisting α : X → K(Z, 3). A triple (X; A, B) is A CW-triad if X is a CW-complex, and A, B are two subcomplexes of X such that A ∪ B = X. For the analytic twisted K-homology, recall that Pα (K) is the associated bundle of an compact operators on X. Analytic twisted K-homology, denoted by Kev/odd (X, α), is defined to be   an Kev/odd (X, α) := KK ev/odd C0 (X, Pα (K)), C , Kasparov’s Z2 -graded K-homology of the C ∗ -algebra C0 (X, Pα (K)). For a relative CW-complex (X, A) with a twisting α : X → K(Z, 3), the relative an an version of analytic twisted K-homology Kev/odd (X, A, α) is defined to be Kev/odd (X − A, α). Then we have the following exact sequence an (X, A; α) Kodd O

an Kodd (X, α) o

/ K an (A, α|A ) ev

an Kodd (A, α|A ) o

an / Kev (X, α)

 an Kev (X, A; α)

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106 12

ALAN L. CAREY AND BAI-LING WANG

and the excision properties an an Kev/odd (X, B; α) ∼ (A, A − B; α|A ) = Kev/odd

for any CW-triad (X; A, B) with a twisting α : X → K(Z, 3). T HEOREM 3.1. (Theorem 5.1 in [Wa]) There is a natural isomorphism top an Φ : Kev/odd (X, α) −→ Kev/odd (X, α)

for any smooth manifold X with a twisting α : X → K(Z, 3). The proof of this theorem requires Poincar´e duality between twisted K-theory and twisted K-homology (we describe this duality in the next theorem), and the isomorphism (Theorem 2.2) between topological twisted K-theory and analytic twisted K-theory. Fix an isomorphism H ⊗ H ∼ = H which induces a group homomorphism U (H) × U (H) −→ U (H) whose restriction to the center is the group multiplication on U (1). So we have a group homomorphism P U (H) × P U (H) −→ P U (H) which defines a continuous map, denoted m∗ , of CW-complexes BP U (H) × BP U (H) −→ BP U (H). As BP U (H) is identified as K(Z, 3), we may think of this as a continuous map taking K(Z, 3) × K(Z, 3) to K(Z, 3), which can be used to define α + oX . There are natural isomorphisms from twisted K-homology (topological resp. analytic) to twisted K-theory (topological resp. analytic) of a smooth manifold X where the twisting is shifted by α → α + oX where τ : X → BSO is the classifying map of the stable tangent space and α+oX denotes the map X → K(Z, 3), representing the class [α] + W3 (X) in H 3 (X, Z). T HEOREM 3.2. Let X be a smooth manifold with a twisting α : X → K(Z, 3). There exist isomorphisms ev/odd top Kev/odd (X, α) ∼ = Ktop (X, α + oX ) and

an ev/odd (X, α) ∼ (X, α + oX ) Kev/odd = Kan

with the degree shifted by dim X(mod 2). Analytic Poincar´e duality was established in [EEK] and [Tu], and topological Poincar´e duality was established in [Wa]. Theorem 3.1 and the exact sequences for a pair (X, A) imply the following corollary. C OROLLARY 3.3. There is a natural isomorphism top an Φ : Kev/odd (X, A, α) −→ Kev/odd (X, A, α)

for any smooth manifold X with a twisting α : X → K(Z, 3) and a closed submanifold A ⊂ X. R EMARK 3.4. In fact, Poincar´e duality as in Theorem 3.2 holds for any compact Riemannian manifold W with boundary ∂W and a twisting α : W → K(Z, 3). This duality takes the following form ev/odd top Kev/odd (W, α) ∼ = Ktop (W, ∂W, α + oW )

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

and

107 13

an ev/odd (W, α) ∼ (X, ∂X, α + oW ) Kev/odd = Kan

with the degree shifted by dim W (mod 2). From this, we have a natural isomorphism top an Φ : Kev/odd (X, A, α) −→ Kev/odd (X, A, α)

for any CW pair (X, A) with a twisting α : X → K(Z, 3) using the Five Lemma. 3.2. Geometric cycles and geometric twisted K-homology. Let X be a paracompact Hausdorff space and let α : X −→ K(Z, 3) be a twisting over X. D EFINITION 3.5. Given a smooth oriented manifold M with a classifying map ν of its stable normal bundle then we say that M is an α-twisted Spinc manifold over X if M is equipped with an α-twisted Spinc structure, that means, a continuous map ι : M → X such that the following diagram / BSO vv η v v v ι W3 vv v   w vv / X α K(Z, 3), M

ν

commutes up to a fixed homotopy η from W3 ◦ ν and α ◦ ι. Such an α-twisted Spinc manifold over X will be denoted by (M, ν, ι, η). P ROPOSITION 3.6. M admits an α-twisted Spinc structure if and only if there is a continuous map ι : M → X such that ι∗ ([α]) + W3 (M ) = 0. If ι is an embedding, this is the anomaly cancellation condition obtained by Freed and Witten in [FreWit]. 

P ROOF. This is clear.

A morphism between α-twisted Spinc manifolds (M1 , ν1 , ι1 , η1 ) and (M2 , ν2 , ι2 , η2 ) is a continuous map f : M1 → M2 where the following diagram (3.2)

M1 C ν1 CC f CC CC C! M2 ι1

! / BSO vv η2 v vv ι2 W3 vv v  #  w vv / K(Z, 3) X α ν2

is a homotopy commutative diagram such that (1) ν1 is homotopic to ν2 ◦ f through a continuous map ν : M1 × [0, 1] → BSO; (2) ι2 ◦ f is homotopic to ι1 through continuous map ι : M1 × [0, 1] → X; (3) the composition of homotopies (α ◦ ι) ∗ (η2 ◦ (f × Id)) ∗ (W3 ◦ ν) is homotopic to η1 . Two α-twisted Spinc manifolds (M1 , ν1 , ι1 , η1 ) and (M2 , ν2 , ι2 , η2 ) are called isomorphic if there exists a diffeomorphism f : M1 → M2 such that the above holds. If the identity map on M induces an isomorphism between (M, ν1 , ι1 , η1 ) and (M, ν2 , ι2 , η2 ), then these two α-twisted Spinc structures are called equivalent.

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ALAN L. CAREY AND BAI-LING WANG

Orientation reversal in the Grassmannian model defines an involution r : BSO −→ BSO. Choose a good cover {Vi } of M and hence a trivialisation of the universal bundle over BSO(n) with transition functions gij : Vi ∩ Vj −→ SO(n). Let g˜ij : Vi ∩ Vj −→ Spinc (n) be a lifting of gij . Then {cijk }, obtained from g˜ij g˜jk = cijk g˜ik , defines [W3 ] ∈ H 3 (BSO, Z). Let h be the diagonal matrix with the first (n − 1) diagonal entries 1 and the last entry −1. Then {hgij h−1 } are the transition functions for the universal bundle over BSO(n) with the opposite orientation. Note that {h˜ gij h−1 } is a lifting of −1 {hgij h }, which leaves {cijk } unchanged. We have [W3 ] = [W3 ◦ r] ∈ H 3 (BSO, Z). Hence there is a homotopy connecting W3 and W3 ◦ r. (It is unique up to homotopy as H 2 (BSO, Z) = 0). Given an α-twisted Spinc manifold (M, ν, ι, η), let −M be the same manifold with the orientation reversed. Then the homotopy commutative diagram r / BSO / BSO t ww t η w tt ww w t ι W3 t t t ww  v~ tt  w ww / X W3 α K(Z, 3) o

M

ν

determines a unique equivalence class of α-twisted Spinc structure on −M , called the opposite α-twisted Spinc structure, simply denoted by −(M, ν, ι, η). D EFINITION 3.7. A geometric cycle for (X, α) is a quintuple (M, ι, ν, η, [E]) where [E] is a K-class in K 0 (M ) and M is a smooth closed manifold equipped with an α-twisted Spinc structure (M, ι, ν, η). Two geometric cycles (M1 , ι1 , ν1 , η1 , [E1 ]) and (M2 , ι,2 ν2 , η2 , [E2 ]) are isomorphic if there is an isomorphism f : (M1 , ι1 , ν1 , η1 ) → (M2 , ι2 , ν2 , η2 ), as α-twisted Spinc manifolds over X, such that f! ([E1 ]) = [E2 ]. Let Γ(X, α) be the collection of all geometric cycles for (X, α). We now impose an equivalence relation ∼ on Γ(X, α), generated by the following three elementary relations: (1) Direct sum - disjoint union If (M, ι, ν, η, [E1 ]) and (M, ι, ν, η, [E2 ]) are two geometric cycles with the same α-twisted Spinc structure, then (M, ι, ν, η, [E1 ]) ∪ (M, ι, ν, η, [E2 ]) ∼ (M, ι, ν, η, [E1 ] + [E2 ]). (2) Bordism Given two geometric cycles (M1 , ι1 , ν1 , η1 , [E1 ]) and (M2 , ι2 , ν2 , η2 , [E2 ]), if there exists a α-twisted Spinc manifold (W, ι, ν, η) and [E] ∈ K 0 (W ) such that ∂(W, ι, ν, η) = −(M1 , ι1 , ν1 , η1 ) ∪ (M2 , ι2 , ν2 , η2 ) and ∂([E]) = [E1 ] ∪ [E2 ]. Here −(M1 , ι1 , ν1 , η1 ) denotes the manifold M1 with the opposite α-twisted Spinc structure.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

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(3) Spinc vector bundle modification Suppose we are given a geometric cycle (M, ι, ν, η, [E]) and a Spinc vector bundle V over M with even dimensional fibers. Denote by R the trivial rank one real vector bundle. Choose a Riemannian metric on V ⊕ R, let ˆ = S(V ⊕ R) M ˆ ) of be the sphere bundle of V ⊕ R. Then the vertical tangent bundle T v (M ˆ admits a natural Spinc structure with an associated Z2 -graded spinor bundle M ˆ → M the projection which is K-oriented. Then SV+ ⊕ SV− . Denote by ρ : M ˆ , ι ◦ ρ, ν ◦ ρ, η ◦ ρ, [ρ∗ E ⊗ S + ]). (M, ι, ν, η, [E]) ∼ (M V D EFINITION 3.8. Denote by K∗geo (X, α) = Γ(X, α)/ ∼ the geometric twisted Khomology. Addition is given by disjoint union - direct sum relation. Note that the equivalence relation ∼ preserves the parity of the dimension of the underlying α-twisted Spinc manifold. Let K0geo (X, α) (resp. K1geo (X, α) ) the subgroup of K∗geo (X, α) determined by all geometric cycles with even (resp. odd) dimensional α-twisted Spinc manifolds. (1) If M , in a geometric cycle (M, ι, ν, η, [E]) for (X, α), is a R EMARK 3.9. compact manifold with boundary, then [E] has to be a class in K 0 (M, ∂M ). (2) If f : X → Y is a continuous map and α : Y → K(Z, 3) is a twisting, then there is a natural homomorphism of abelian groups geo geo f∗ : Kev/odd (X, α ◦ f ) −→ Kev/odd (Y, α)

sending [M, ι, ν, η, E] to [M, f ◦ ι, ν, η, E]. (3) Let A be a closed subspace of X, and α be a twisting on X. A relative geometric cycle for (X, A; α) is a quintuple (M, ι, ν, η, [E]) such that (a) M is a smooth manifold (possibly with boundary), equipped with an αtwisted Spinc structure (M, ι, ν, η); (b) if M has a non-empty boundary, then ι(∂M ) ⊂ A; (c) [E] is a K-class in K 0 (M ) represented by a Z2 -graded vector bundle E over M , or a continuous map M → BU (∞). The relation ∼ generated by disjoint union - direct sum, bordism and Spinc vector bundle modification is an equivalence relation. The collection of relative geometric cycles, modulo the equivalence relation is denoted by geo Kev/odd (X, A; α).

There exists a natural homomorphism, called the assembly map geo an (X, α) → Kev/odd (X, α) μ : Kev/odd

whose definition (which we will now explain) requires a careful study of geometric cycles. Given a geometric cycle (M, ι, ν, η, [E]), equip M with a Riemannian metric. Denote by Cliff (T M ) the bundle of complex Clifford algebras of T M over M . The algebra of sections, C(M, Cliff (T M )), is Morita equivalent to C(M, τ ∗ BSpinc (K)). Hence, we have a canonical isomorphism K an (M, W3 ◦ τ ) ∼ = KK ev/odd (C(M, Cliff (M )), C) ev/odd

with the degree shift by dim M (mod 2). Applying Kasparov’s Poincar´e duality (Cf. [Kas2]) KK ev/odd (C, C(M )) ∼ = KK ev/odd (C(M, Cliff (M )), C),

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we obtain a canonical isomorphism an P D : K 0 (M ) ∼ (M, oM ), = Kev/odd an with the degree shift by dim M (mod 2). The fundamental class [M ] ∈ Kev/odd (M, oM ) an is the Poincar´e dual of the unit element in K 0 (M ). Note that [M ] ∈ Kev (M, oM )) if M an is even dimensional and [M ] ∈ Kodd (M, oM ) if M is odd dimensional. The cap product an an ∩ : Kev/odd (M, oM ) ⊗ K 0 (M ) −→ Kev/odd (M, oM )

is defined by the Kasparov product. We remark that Poincar´e duality is given by the cap product of the fundamental K-homology class [M ] an [M ]∩ : K 0 (M ) ∼ (M, oM ). = Kev/odd Choose an embedding ik : M → Rn+k and take the resulting normal bundle νM . The natural isomorphism T M ⊕ νM ⊕ νM ∼ = Rn+k ⊕ νM c and the canonical Spin structure on νM ⊕ νM define a canonical homotopy between the orientation twisting oM of T M and the orientation twisting oνM of νM . This canonical homotopy defines an isomorphism I∗ : K an (M, oM ) ∼ (3.3) = K an (M, oν ). ev/odd

ev/odd

M

c

Given an α-twisted Spin manifold (M, ν, ι, η) over X, the homotopy η induces an isomorphism ν ∗ BSpinc ∼ = ι∗ Pα as principal K(Z, 2)-bundles on M . Hence there is an isomorphism η∗ ∼ =

ν ∗ BSpinc (K)

/ ι∗ Pα (K)

as bundles of C ∗ -algebras on M . This isomorphism determines a canonical isomorphism between the corresponding continuous trace C ∗ -algebras ∼ C(M, ι∗ Pα (K)). C(M, ν ∗ BSpinc (K)) = Hence, we have a canonical isomorphism an an η∗ : Kev/odd (M, oνM ) ∼ (M, α ◦ ι). = Kev/odd

(3.4)

Now we can define the assembly map as μ(M, ι, ν, η, [E]) = ι∗ ◦ η∗ ◦ I∗ ([M ] ∩ [E]) an in Kev/odd (X, α).

Here ι∗ is the natural push-forward map in analytic twisted K-homology.

geo (X, α) → T HEOREM 3.10. (Theorem 6.4 in [Wa]) The assembly map μ : Kev/odd is an isomorphism for any smooth manifold X with a twisting α : X → K(Z, 3). an Kev/odd (X, α)

top (X, α) → The proof follows by establishing the existence of a natural map Ψ : Kev such that the following diagram

K0geo (X, α)

top (X, α) Kev/odd PPP 7n PPP Ψn n PP n∼ ∼ n = PPPP = wn ' μ geo an / Kev/odd (X, α) Kev/odd (X, α) ∼ =

commutes. All the maps in the diagram are isomorphisms.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

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R EMARK 3.11. The equivalence between the geometric twisted K-homology and the analytic twisted K-theory holds for any CW pair (X, A). We will return to this in a separate paper. an an C OROLLARY 3.12. Kev/odd (X, α) ∼ (X, −α). = Kev/odd

P ROOF. By the Brown representation theorem ([Swi]), there is a continuous map ˆi : K(Z, 3) → K(Z, 3) (unique up to homotopy as H 2 (K(Z, 3), Z) = 0) such that [ˆi ◦ α] = −[α] ∈ H 3 (X, Z) for any map α : X → K(Z, 3). Then we have [ˆi ◦ W3 ] = −[W3 ] ∈ H 3 (BSO, Z). As [W3 ] is 2-torsion, we know that [ˆi ◦ W3 ] = −[W3 ] = [W3 ]. Therefore, there is a homotopy η0 connecting ˆi◦W3 and W3 , that is, the following diagram is homotopy commutative y W3 yy y yy W3 y yη  x yy 0 ˆi K(Z, 3) BSO

 / K(Z, 3).

Note that the homotopy class of η0 as a homotopy connecting W3 and ˆi ◦ W3 is unique due to the fact that H 2 (BSO, Z) = 0. Given an α-twisted Spinc manifold (M, ι, ν, η), then the following homotopy commutative diagram / BSO y W3 ww yy y η w w yy w ι W3 y yη ww w  x yy 0 ˆ  w ww i / X α K(Z, 3) M

ν

 / K(Z, 3)

defines a unique (due to H 2 (BSO, Z) = 0) equivalence class of (−α)-twisted Spinc structures. Here −α = ˆi◦α. We denote by ˆi(M, ι, ν, η) this (−α)-twisted Spinc manifold. Obviously,   ˆi ˆi(M, ι, ν, η) = (M, ι, ν, η). an an (X, α) ∼ (X, −α) is induced by the involution ˆi on The isomorphism Kev/odd = Kev/odd geometric cycles. 

4. The Chern Character in Twisted K-theory In this Section, we will review the Chern character map in twisted K-theory on smooth manifolds developed in [CMW] using gerbe connections and curvings. For the topological and analytic definitions, see [AS2] and [MatSte] respectively. Recently, Gomi and Terashima in [GoTe] gave another construction of a Chern character for twisted K-theory using a notion of connection on a finite-dimensional approximation of a twisted family of Fredholm operators developed by Gomi ([Gomi].

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4.1. Twisted Chern character. For a fibration π ∗ : Y → X, let Y [p] denote the pth fibered product. There are projection maps πi : Y [p] → Y [p−1] which omit the ith element for each i = 1 . . . p. These define a map δ : Ωq (Y [p−1] ) → Ωq (Y [p] )

(4.1) by (4.2)

δ(ω) =

p 

(−1)i πi∗ (ω).

i=1 2

Clearly δ = 0. In fact, the δ-cohomology of this complex vanishes identically, hence, the sequence 0

/ Ωq (X)

π∗

/ Ωq (Y ) · · ·

δ

/ Ωq (Y [p−1] )

δ

/ Ωq (Y [p] )

/ ···

is exact. Returning now to our particular example, a bundle gerbe connection on Pα is a unitary [2] connection θ on the principal U (1)-bundle Gα over Pα which commutes with the bundle gerbe product. A bundle gerbe connection θ has curvature Fθ ∈ Ω2 (Pα[2] ) satisfying δ(Fθ ) = 0. There exists a two-form ω on Pα such that Fθ = π2∗ (ω) − π1∗ (ω). Such an ω is called a curving for the gerbe connection θ. The choice of a curving is not unique, the ambiguity in the choice is precisely the addition of the pull-back to Pα of a two-form on X. Given a choice of curving ω, there is a unique closed three-form on β on X satisfying dω = π ∗ β. We denote by α ˇ = (Gα , θ, ω) β √ 2π −1 is a de Rham representative for the Dixmier-Douady class [α]. We shall call α ˇ the differential twisting, as it is the twisting in differential twisted K-theory (Cf. [CMW]). The following theorem is established in [CMW]. the lifting bundle gerbe Gα with the connection θ and a curving ω. Moreover H =

T HEOREM 4.1. Let X be a smooth manifold, π : Pα → X be a principal P U (H) bundle over X whose classifying map is given by α : X −→ K(Z, 3). Let α ˇ = (Gα , θ, ω) be a bundle gerbe connection θ and a curving ω on the lifting bundle gerbe Gα . There is a well-defined twisted Chern character Chαˇ : K ∗ (X, α) −→ H ev/odd (X, d − H). Here the groups H ev/odd (X, d − H) are the twisted cohomology groups of the complex of differential forms on X with the coboundary operator given by d − H. The twisted Chern character is functorial under the pull-back. Moreover, given another differential twisting α ˇ + b = (Gα , θ, ω + π ∗ b) for a 2-form b on X, Chα+b = Chαˇ · exp( ˇ

b √ ). 2π −1

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

113 19

P ROOF. Choose a good open cover {Vi } of X such that Pα → X has trivializing sections φi over each Vi with transition functions gij : Vi ∩ Vj −→ P U (H) satisfying φj = φi gij . Define {σijk } by gˆij gˆjk = gˆik σijk for a lift of gij to gˆij : Vi ∩ Vj → U (H). [2] Note that the pair (φi , φj ) defines a section of Pα over Vi ∩ Vj . The connection θ can be pulled back by (φi , φj ) to define a 1-form Aij on Vi ∩ Vj and the curving ω can be pulled-back by the φi to define two-forms Bi on Vi . Then the differential twisting defines the triple {(σijk , Aij , Bi )}

(4.3)

which is a degree two smooth Deligne cocycle. Now we explain in some detail the twisted Chern characters in both the odd and even case following [CMW]. The even case: As a model for the K 0 classifying space, we choose Fred, the space of bounded self-adjoint Fredholm operators with essential spectrum {±1} and otherwise discrete spectra, with a grading operator Γ which anticommutes with the given family of Fredholm operators. A twisted K-class in K 0 (X, α) can be represented by f : Pα → Fred, a P U (H)equivariant family of Fredholm operators. We can select an open cover {Vi } of X such that on each Vi there is a local section φi : Vi → Pα and for each i the Fredholm operators f (φi (x)), x ∈ Vi have a gap in the spectrum at both ±λi = 0. Then over Vi we have a finite rank vector bundle Ei defined by the spectral projections of the operators f (φi (x)) corresponding to the interval [−λi , λ]. Passing to a finer cover {Ui } if necessary, we may assume that Ei is a trivial vector bundle over Ui of rank ni . Choosing a trivialization of Ei gives a Z2 graded parametrix qi (an inverse up to finite rank operators) of the family f ◦ φi . In the index zero sector the operator qi (x)−1 is defined as the direct sum of the restriction of f (φi (x)) to the orthogonal complement of Ei in H and an isomorphism between the vector bundles Ei+ and Ei− . Clearly then f (φi (x))qi (x) = 1 modulo rank ni operators. In the case of nonzero index one defines a parametrix as a graded invertible operator qi such that f (φi (x))qi (x) = sn modulo finite rank operators, with sn a fixed Fredholm operator of index n equal to the index of f (φi (x)). On the overlap Uij we have a pair of parametrices qi and qj of families of f ◦ φi and f ◦ φj respectively. These are related by an invertible operator fij which is of the form 1+ a finite rank operator, −1 gˆij qj (x)ˆ gij = qi (x)fij (x). The conjugation on the left hand side by gˆij comes from the equivariance relation gij (x). f (φj (x)) = f (φi (x)gij (x)) = gˆij (x)−1 f (φi (x))ˆ ˇ cocycle relation needed to define a The system {fij } does not quite satisfy the Cech principal bundle, because of the different local sections φi : Ui → Pσ involved. Instead, we have on Uijk −1 −1 −1 −1 gˆjk qk gˆjk = qj fjk = (ˆ gij qi fij gˆij )fjk = gˆjk (ˆ gik qi fik gˆik )ˆ gjk . −1 −1 Using the relation gˆjk gˆik = σijk gˆij , we get −1 −1 −1 gˆjk (ˆ gik qi fik gˆik )ˆ gjk = gˆij qi fik gˆij −1 multiplying the last equation from right by gˆij and from the left by qi−1 gˆij one gets the twisted cocycle relation −1 fij (ˆ gij fjk gˆij ) = fik ,

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ALAN L. CAREY AND BAI-LING WANG

which is independent of the choice of the lifting gˆij . For simplicity, we will just write the above twisted cocycle relation as −1 fij (gij fjk gij ) = fik .

(4.4)

This twisted cocycle relation (4.4) actually defines an untwisted cocycle relation for {(gij , fij )} in the twisted product G = P U (H)  GL(∞), where the group P U (H) acts on the group GL(∞) of invertible 1+ finite rank operators by conjugation. Thus the product in G is given by (g, f ) · (g  , f  ) = (gg  , f (gf  g −1 )). The cocycle relation for the pairs {(gij , fij )} then encodes both the cocycle relation for the transition functions {gij } of the P U (H) bundle over X and the twisted cocycle relation (4.4). In summary, this cocycle {(gij , fij )} defines a principal G bundle over X. The classifying space BG is a fiber bundle over K(Z, 3). The fiber at each point in K(Z, 3) is homeomorphic (but not canonically so) to the space Fred of graded Fredholm operators; to set up the isomorphism one needs a choice of element in each fiber. Given a principal P U (H)-bundle Pα over X defined by α : X → K(Z, 3), the even twisted K-theory K 0 (X, Pα ) is the set of homotopy classes of maps X → BG covering the map α. Next we construct the twisted Chern character from a connection ∇ on a principal G bundle over X associated to the cocycle {(gij , fij )}. Locally, on a good open cover {Ui } ˆ of the of X we can lift the connection to a connection taking values in the Lie algebra g central extension U (H) × GL(∞) of G. Denote by Fˆ∇ the curvature of this connection. On the overlaps {Uij } the curvature satisfies a twisted relation ∗ c, Fˆ∇,j = Ad(gij ,fij )−1 Fˆ∇,i + gij

where c is the curvature of the canonical connection θ on the principal U (1)-bundle U (H) → P U (H). Since the Lie algebra u(∞) ⊕ C is an ideal in the Lie algebra of U (H)  GL(∞),  the projection F∇,i of the curvature Fˆ∇,i onto this subalgebra transforms in the same way ˆ as F under change of local trivialization. It follows that for a P U (H)-equivariant map f : Pα → Fred, we can define a twisted Chern character form of f as 

chαˇ (f, ∇) = eBi tr eF∇,i /2πi ,

(4.5)

over Vi . Here the trace is well-defined on gl(∞) and on the center C it is defined as the coefficient of the unit operator. Note that chαˇ (f, ∇) is globally defined and (d−H)-closed (d − H)chαˇ (f, ∇) = 0, and depends on the differential twisting α ˇ = (Gα , θ, ω). Let f0 and f1 be homotopic, and ∇0 and ∇1 be two connections on the principal G bundle over X, we have a Chern-Simons type form CS((f0 , ∇0 ), (f1 , ∇1 )), well-defined modulo (d − H)-exact forms, such that (4.6)

chαˇ (f1 , ∇1 ) − chαˇ (f0 , ∇0 ) = (d − H)CS((f0 , ∇0 ), (f1 , ∇1 )).

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

115 21

The proof follows directly from the local computation using (4.5). Hence, the (d − H)cohomology class of chαˇ (f, ∇) does not depend choices of a connection ∇ on the principal G bundle over X, and depends only on the homotopy class of f . We denote the (d − H)cohomology class of chαˇ (f, ∇) by Chαˇ ([f1 ]) which is a natural homomorphism Chαˇ : K 0 (X, α) −→ H ev (X, d − H). From (4.5), we have = Chαˇ · exp( Chα+b ˇ

b √ ). 2π −1

for a differential twisting α ˇ + b = (Gα , θ, ω + π ∗ b). The odd case: The odd case is a little easier. First, as a model for the K 1 classifying space, we choose U (∞) = limn U (n), the stabilized unitary group. Let Θ be the universal −→ odd character form on U (∞) defined by the canonical left invariant u(∞)-valued form on U (∞). Let H = H+ ⊕ H− be a polarized Hilbert space and let Ures = Ures (H) denotes the group of unitary operators in H with Hilbert-Schmidt off-diagonal blocks. The conjugation action of U (H+ ) × U (H− ) on Ures defines an action of P U0 (H) = P (U (H+ ) × U (H− )) on Ures . Note that the classifying space of Ures is U (∞). Define H = P U0 (H)  Ures . Then given a principal P U0 (H)-bundle Pα over X defined by α : X → K(Z, 3), the odd twisted K-theory K 1 (X, α) is the set of homotopy classes of maps X → BH covering the map α. These are represented by P U0 (H)-equivariant maps f : Pα → U (∞). With respect to trivializing sections φi over each Vi . Then eBi (f ◦ φi )∗ Θ is a globally defined and (d − H)-closed differential form on X. This defines the odd version of the twisted Chern character Chαˇ : K 1 (X, α) −→ H odd (X, d − H).  4.2. Differential twisted K-theory. Recall that the Bockstein exact sequence in complex K-theory for any finite CW complex: (4.7)

K 0 (X) O 1 KR/Z (X) o

ch

0 / KR/Z (X)

/ H ev (X, R)

H odd (X, R) o

ch

 K 1 (X)

∗ (X) is K-theory with R/Z-coefficients as in [Kar1] and [Ba]. where KR/Z Analogously, in twisted K-theory, given a smooth manifold X with a twisting α : X → K(Z, 3), upon a choice of a differential twisting

α ˇ = (Gα , θ, ω)

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ALAN L. CAREY AND BAI-LING WANG

lifting α, we have the corresponding Bockstein exact sequence in twisted K-theory

(4.8)

K 0 (X, α) O

Chα ˇ

H odd (X, d − H) o

1 KR/Z (X, α) o

0 / KR/Z (X, α) .

/ H ev (X, d − H)

Chα ˇ

 K 1 (X, α)

0 1 (X, α) and KR/Z (X, α) are subgroups of differential twisted K-theory, respecHere KR/Z 0 1 ˇ (X, α ˇ (X, α ˇ ) and K ˇ ) (see [CMW] for the detailed construction). Here we give tively K another equivalent construction of differential twisted K-theory. ˇ 0 (X, α Fix a choice of a connection ∇ on a principal G bundle over X. Then K ˇ ) is the abelian group generated by pairs

{(f, η)}, modulo an equivalence relation, where f : Pα → Fred is a P U (H)-equivariant map and η is an odd differential form modulo (d − H)-exact forms. Two pairs (f0 , η0 ) and (f1 , η1 ) are called equivalent if and only if η1 − η0 = CS((f1 , ∇), (f0 , ∇)). The differential Chern character form of f is given by chαˇ (f, ∇) − (d − H)η which defines a homomorphism ˇ 0 (X, α chαˇ : K ˇ ) −→ Ωev 0 (X, d − H), ˇ0 where Ωev ˇ ) → Ωev (X). The kernel of chαˇ is ˇ : K (X, α 0 (X, d − H) is the image of chα 1 isomorphic to KR/Z (X, α). ˇ 1 (X, α Similarly, we define the odd differential twisted K-theory K ˇ ) with the differential Chern character form homomorphism ˇ 1 (X, α chαˇ : K ˇ ) −→ Ωodd 0 (X, d − H). 0 The kernel of chαˇ is isomorphic to KR/Z (X, α). The following commutative diagrams were established in [CMW] relating differential twisted K-theory with twisted K-theory

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

and with the diagram (4.8) 0

H odd (X, d − H)

 1 / KR/Z (X, σ ˇ) NNN NNN NNN NNN NN'  0 /K / ˇ (X, σ K 0 (X, σ) → 0 ˇ)

 Ωodd (X) 0 → odd Ω0 (X, d − H) QQQ QQQ chσ ˇ Q d−H QQQ QQ(  Ωev 0 (X, d − H)

Chσ ˇ

 / H ev (X, d − H)

 0 and 0

H ev (X, d − H)

 0 / KR/Z (X, σ ˇ) OOO OOO OOO OOO OO'  1 / K 1 (X, σ) → 0 /K ˇ (X, σ ˇ)

 Ωev (X) 0 → ev Ω0 (X, d − H) QQQ QQQ chσ ˇ Q d−H QQQQ Q(  Ωodd 0 (X, d − H)

Chσ ˇ

 / H odd (X, d − H)

 0 with exact horizontal and vertial sequences, and exact upper-right and exact lower-left 4term sequences. We expect that these two commutative diagrams uniquely characterize differential twisted K-theory. 4.3. Twisted Chern character for torsion twistings. In this paper, we will only use the twisted Chern character for a torsion twisting and in this case we will give an explicit construction. Let E be a real oriented vector bundle of rank 2k over X with its orientation twisting denoted by o(E) : X → K(Z, 3). The associated lifting bundle Go(E) has a canonical reduction to the Spinc bundle gerbe GW3 (E) . Choose a local trivialization of E over a good open cover {Vi } of X. Then the transition functions gij : Vi ∩ Vj −→ SO(2k).

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ALAN L. CAREY AND BAI-LING WANG

define an element in H 1 (X, SO(2k)) whose image under the Bockstein exact sequence H 1 (X, Spin(2k)) → H 1 (X, SO(2k)) → H 2 (X, Z2 ) is the second Stieffel-Whitney class w2 (E) of E. Denote the differential twisting by w ˇ2 (E) = (GW3 (E) , θ, 0), the Spinc bundle gerbe GW3 (E) with a flat connection θ and a trivial curving. With respect to a good cover {Vi } of X the differential twisting w ˇ2 (E) defines a Deligne cocycle {(αijk , 0, 0)} with trivial local B-fields, here αijk = gˆij gˆjk gˆki where gˆij : Uij → Spinc (2n) is a lift of gij . By Proposition 2.4, a twisted K-class in K 0 (X, o(E)) can be represented by a Clifford bundle, denoted E. Equip E with a Clifford connection, and E with a SO(2k)-connection. Locally, over each Vi we let E|Vi ∼ = Si ⊗ Ei where Si is the local fundamental spinor bundle associated to E|Vi with the standard Clifford action of Cliff (E|Vi ) obtained from the fundamental representation of Spin(2k). Then Ei is a complex vector bundle over Vi with a connection ∇i such that on Vi ∩ Vj Ch(Ei , ∇i ) = Ch(Ej , ∇j ). Hence, the twisted Chern character Chwˇ2 (E) : K 0 (X, o(E)) −→ H ev (X) is given by [E] → {[ch(Ei , ∇i )] = ch(Ei )}. The proof of the following proposition is straightforward. P ROPOSITION 4.2. The twisted Chern character satisfies the following identities (1) Chwˇ2 (E1 ⊕E2 ) ([E1 ⊕ E2 ]) = Chwˇ2 (E1 ) ([E1 ]) + Chwˇ2 (E2 ) ([E2 ]). (2) Chwˇ2 (E1 ⊗E2 ) ([E1 ⊗ E2 ]) = Chwˇ2 (E1 ) ([E1 ])Chwˇ2 (E2 ) ([E2 ]). In the case that E has a Spinc structure whose determinant line bundle is L, there is a canonical isomorphism K 0 (X) −→ K 0 (X, o(E)), given by [V ] → [V ⊗ SE ] where SE is the associated spinor bundle of E. Then we have Chwˇ2 (E) ([V ⊗ SE ]) = e

c1 (L) 2

ch([V ]),

where ch([V ]) is the ordinary Chern character of [V ] ∈ K 0 (X). In particular, when X is an even dimensional Riemannian manifold, and T X is equipped with the Levi-Civita connection, under the identification of K 0 (X, o(E)) with the Grothendieck group of Clifford modules. Then (4.9)

Chwˇ2 (X) ([E]) = ch(E/S)

where ch(E/S) is the relative Chern character of the Clifford module E constructed in Section 4.1 of [BGV].

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

119 25

5. Thom Classes and Riemann-Roch Formula in Twisted K-theory 5.1. The Thom class. Given any oriented real vector bundle π : E → X of rank 2k, E admits a Spinc structure if its classifying map τ : X → BSO(2k) admits a lift τ˜

τ˜

X

u

u

u τ/

u

BSpinc u:

 BSO(2k).

As BSpinc → BSO(2k) is a BU (1)-principal bundle with the classifying map given by W3 : BSO(2k) → K(Z, 3), E admits a Spin structure if W3 ◦ τ : X → K(Z, 3) is null homotopic, and a choice of null homotopy determines a Spinc structure on E. Associated to a Spinc structure s on E, there is canonical K-theoretical Thom class c

0 UEs = [π ∗ S + , π ∗ S − , cl] ∈ Kcv (E)

in the K-theory of E with vertical compact supports. Here S + and S − are the positive and negative spinor bundle over X defined by the Spinc structure on E, and cl is the bundle map π ∗ S + → π ∗ S − given by the Clifford action E on S ± . R EMARK 5.1. (1) The restriction of UEs to each fiber is a generator of K 0 (R2k ), c so a Spin structure on E is equivalent to a K-orientation on E. Note that Thom classes and K-orientation are functorial under pull-backs of Spinc vector bundles. (2) Let s ⊗ L be another Spinc structure on E which differs from s by a complex line bundle p : L → X, then 

UEs = UEs · p∗ ([L]). (3) Let (E1 , s1 ) and (E2 , s2 ) be two Spinc vector bundles over X, p1 and p2 be the projections from E1 ⊕ E2 to E1 and E2 respectively, then s1 s2 ∗ ∗ 0 2 UEs11⊕s ⊕E2 = p1 (UE1 ) · p2 (UE2 ) ∈ Kcv (E1 ⊕ E2 ).

(4) The Thom isomorphism in K-theory for a Spinc vector bundle π : E → X of rank 2k is given by ΦK E :

K 0 (X) −→ a →

K 0 (E) π ∗ (a)UEs .

Here for locally compact spaces, we shall consider only K-theory with compact supports. When X is compact, UEs ∈ K 0 (T X) and the Thom isomorphism ΦK X is the inverse of the push-forward map π! : K 0 (T X) → K 0 (X) associated to the K-orientation of π defined the Spinc structure on E. If an oriented vector bundle E of even rank over X does not admit a Spinc structure, W3 ◦ τ : X → K(Z, 3) is not null homotopic. Thus, W3 ◦ τ defines a twisting on X for K-theory, called the orientation twisting oE . In this Section we will define a canonical Thom class UE ∈ K 0 (E, π ∗ oE )

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ALAN L. CAREY AND BAI-LING WANG

such that a → π ∗ (a) ∪ UE defines the Thom isomorphism K 0 (X, oE ) ∼ = K 0 (E). In fact, a → π ∗ (a) ∪ UE defines the Thom isomorphism (Cf. [CW1]) K 0 (X, α + oE ) ∼ = K 0 (E, α ◦ π) for any twisting α : X → K(Z, 3). Choose a good open cover {Vi } of X such that Ei = E|Vi is trivialized by an isomorphism Ei ∼ = Vi × R2n . This defines a canonical Spinc structure si on each Ei . Denote by UEsii the associated Thom class of (Ei , si ). Then we have ∗ 0 ([Lij ]) ∈ Kcv (Eij ) UEjj = UEsii πij s

where Lij is the difference line bundle over Vij = Vi ∩ Vj defined by sj = si ⊗ Lij on Eij = E|Vij . Recall that these local line bundles {Lij } define a bundle gerbe [Mur] associated to the twisting oE = W3 ◦ τ : X → K(Z, 3) and a locally trivializing cover {Vi } . By the definition of twisted K-theory, {UEsii } defines a twisted K-theory class of E with compact vertical supports and twisting given by π ∗ (oE ) = oE ◦ π : E → K(Z, 3). We denote this canonical twisted K-theory class by 0 (E, π ∗ (oE )). UE ∈ Kcv

When X is compact, then UE ∈ K 0 (E, π ∗ (oE )). One can easily show that the Thom class UE does not depend on the choice of the trivializing cover. Now we can list the properties of the Thom class in twisted K-theory. P ROPOSITION 5.2. (1) If E is equipped with a Spinc structure s, then s defines a canonical isomorphism 0 0 φs : Kcv (E, π ∗ (oE )) −→ Kcv (E)

such that φs (UE ) = UEs . (2) Let f : X → Y be a continuous map and E be an oriented vector bundle of even rank over Y , then Uf ∗ E = f ∗ (UE ). (3) Let E1 and E2 be two oriented vector bundles of even rank over X, p1 and p2 be the projections from E1 ⊕ E2 to E1 and E2 respectively, that is, we have the diagram E1 ⊕ E2

p2

p1

 E1

/ E2 π2

π1

 / X,

then UE1 ⊕E2 = p∗1 (UE1 ) · p∗2 (UE2 ). (4) Let π : E → X be an oriented vector bundle of even rank over a compact space X, the Thom isomorphism in twisted K-theory ([CW1]) 0 ∗ ∼ 0 ΦK E : K (X, α + oE ) = K (E, π (α))

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

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is given by a → π ∗ (a)·UE . Moreover, the push-forward map in twisted K-theory ([CW1]) π! : K 0 (E, π ∗ (α)) −→ K 0 (X, α + oE ) ∗ satisfies π! (π (a) · UE ) = a. P ROOF.

(1) The Spinc structure s defines canonical isomorphism 0 0 φs : Kcv (E, π ∗ (oE )) −→ Kcv (E)

as follows. Given a trivializing cover {Vi } and the canonical Spinc structure si on Ei = E|Vi , we have s|Ei = si ⊗ Li for a complex line bundle πi : Li → Vi . This implies UEs i = UEsii π ∗ ([Li ]). Note that Lij = Li ⊗ L∗j . 0 (E, π ∗ (oE )) is given by a local K-class ai with Any twisted K-class a in Kcv ∗ compact vertical support such that aj = ai πij ([Lij ]), then ai πi∗ ([Li ]) = aj π ∗ ([Lj ]) 0 in Kcv (E|Vij ). This defines the homomorphism φ, which is obviously an isomorphism sending UE to UEs . (2) Choose a good open cover {Vi } of Y . By definition, the Thom class UE is defined by {UEsii } with ∗ ([Lij ]). UEjj = UEsii πij s

Then {f −1 (Vi )} is an open cover of X, and (f ∗ E)|f −1 (Vi ) = f ∗ Ei is trivialized with the canonical Spinc structure f ∗ si , thus ∗

Uff∗ Esii = f ∗ UEsii . This gives Uf ∗ E = f ∗ (UE ). (3) The proof is similar to the proof of (2). (4) From ([CW1]), we know that the Thom isomorphism and the push-forward map in twisted K-theory are both homomorphisms of K 0 (X, α)-modules. There exists an oriented real vector bundle F of even rank such that E ⊕ F = X × R2m for some m ∈ N. Thus, we have / X × R2m E II II IIπ p II II  $ X i

From the construction of the push-forward map in ([CW1]), we see that π! (UE ) = p! ◦ i! (UE ) = p! (UE⊕F ) = 1. As the Thom isomorphism and the push-forward map in twisted K-theory are both homomorphisms of K 0 (X, α)-modules, we get π! (π ∗ (a) · UE ) = a. Note that the Thom isomorphism is inverse to the push-forward map π! , hence, the Thom isomorphism in twisted K-theory K 0 (X, α + oE ) ∼ = K 0 (E, π ∗ (α))

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ALAN L. CAREY AND BAI-LING WANG

is given by a → π ∗ (a) · UE .



5.2. Twisted Riemann-Roch. By an application of the Thom class and Thom isomorphism in twisted K-theory, we will now give a direct proof of a special case of the Riemann-Roch theorem for twisted K-theory. With some notational changes, the argument can be applied to establish the general Riemann-Roch theorem in twisted K-theory. Denote by oX and oY the orientation twistings associated to the tangent bundles πX : T X → X and πY : T Y → Y respectively. T HEOREM 5.3. Given a smooth map f : X → Y between oriented manifolds, assume that dim Y − dim X = 0 mod 2. Then the Riemann-Roch formula is given by     ˆ ) = f∗H Chwˇ (X) (a)A(X) ˆ Chwˇ2 (Y ) f!K (a) A(Y . 2 ˆ ˆ ) are the A-hat classes of X and Y respecand A(Y for any a ∈ K 0 (X, oX ). Here A(X) tively. P ROOF. For simplicity, assume that both X and Y are of even dimension, say 2m and 2n respectively, equipped with a Riemannian metric. We will consider Chern character defects in each of the following three squares (5.1)

K 0 (X, oX ) Chw ˇ 2 (X)

∼ =

/ K 0 (T X) c

(df )K !

ΦH TX ∼ =

 / H ev (T X) c

K

−1

/ K 0 (T Y ) (ΦT Y ) / K 0 (Y, oY ) ∼ =

c

Ch



H ev (X) ΦK TX

ΦK TX

Chw ˇ 2 (Y )

Ch

(df )H ∗

 H −1  / H ev (T Y )(ΦT Y ) / H ev (Y ) c ∼ =

ΦK TY

and are the Thom isomorphisms in twisted K-theory for T X and T Y , where H and Φ are the cohomology Thom isomorphisms for T X and T Y . Then we have ΦH TX TY (1) The push-forward map in twisted K-theory as established in [CW1] f!K : K 0 (X, oX ) → K 0 (Y, oY ) agrees with −1 K ◦ (df )K (ΦK TY ) ! ◦ ΦT X .

(2) The push-forward map in cohomology theory f∗H : H ev (X) → H ev (Y ) is given by −1 H f∗H = (ΦH ◦ (df )H TY ) ∗ ◦ ΦT X . Denote by UTHX and UTHY the cohomological Thom classes for T X and T Y . Then under the pull-back of the zero section, 0∗X (UTHX ) = e(T X) and 0∗Y (UTHY ) = e(T Y ) are the Euler classes for T X and T Y respectively. Let the Pontrjagin classes of πX : T X → X be symmetric polynomials in x21 , · · · , x2m , then m  xk /2 ˆ A(X) = . sinh(xk /2) k=1

The Chern character defect for the left square in (5.1) is given by     H ˆ−1 (X) Ch ΦK (5.2) ˇ2 (X) (a)A T X (a) = ΦT X Chw ˆ is the A-hat class of T X. for any a ∈ K 0 (X, oX ). Here A(X)

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

123 29

To prove (5.2), note that   −1 Ch ΦK (ΦH TX) T X (a)   −1 ∗ Ch(πX (a) · UT X ) Apply Prop. (4.2) = (ΦH TX)   −1 ∗ ∗ w ∗ w ChπX = (ΦH ˇ2 (X) (πX (a)) · ChπX ˇ2 (X) (UT X ) TX)   −1 ∗ −1 ∗ w πX (Chwˇ2 (X) (a)) · ChπX Note that (ΦH = (πX )∗ . = (ΦH ˇ2 (X) (UT X ) TX) TX)   H −1 ∗ w ChπX By the projection formula. = Chw2 (X) (a)(ΦT X ) ˇ2 (X) (UT X ) So the Chern character defect for the square K 0 (X, oX ) Chw ˇ 2 (X)

/ K 0 (T X)

∼ =

c

Ch



ΦH TX

H ev (X) is given by

ΦK TX

∼ =

 / Hcev (T X)

  −1 ev ∗ w D(X) = (ΦH ChπX ˇ2 (X) (UT X ) ∈ H (X). TX)

From the cohomology Thom isomorphism, we have 0∗X ◦ ΦH T X (D(X)) = D(X)e(T X), under the pull-back of the zero section 0X of the tangent bundle T X. Therefore, we have   ∗ w 0∗X ChπX ˇ2 (X) (UT X ) . D(X) = e(T X) By the construction of the Thom class UT X , under the pull-back of the zero section 0X of T X, 0∗X (UT X ) is a twisted K-class in K 0 (X, oX ) and ∗ w 0∗X ChπX ˇ2 (X) (UT X )

Chwˇ2 (X) (0∗X (UT X )) m xk /2 − e−xk /2 ). k=1 (e

= = Thus, (5.2) follows from

m  (exk /2 − e−xk /2 ) = Aˆ−1 (X). D(X) = xk k=1

This implies that the following diagram commutes (5.3)

K 0 (X, oX ) ˆ Chw ˇ 2 (X) (−)·A(X)



H ev (X)

ΦK TX ∼ =

/ K 0 (T X) c ∗ ˆ2 Ch(−)·πX A (X)

ΦH TX ∼ =

 / H ev (T X). c

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124 30

ALAN L. CAREY AND BAI-LING WANG

Similarly, the Chern character defect in K 0 (Y, oY ) Chw ˇ 2 (Y )

∼ =

/ K 0 (T Y ) c Ch



H ev (Y ) is given by

ΦK TY

ΦH TY ∼ =

 / Hcev (T Y )

    H ˆ−1 (Y ) Ch ΦK ˇ2 (Y ) (a)A T Y (a) = ΦT Y Chw

for any a ∈ K 0 (Y, oY ). This implies that the Chern character defect for the right square in (5.1) is given by     −1 −1 Chwˇ2 (Y ) (ΦK (5.4) Ch(c) (c) · Aˆ−1 (Y ) = (ΦH TY ) TY ) for any c ∈ Kc0 (T Y ). Hence, we have the following commutative diagram (5.5)

Kc0 (T Y )

−1 (ΦK TY )

∼ =

/ K 0 (Y, oY )

∗ ˆ2 Ch(−)·πY A (Y )

ˆ Chw ˇ 2 (Y ) (−)·A(Y )

−1   (ΦH TY ) Hcev (T Y ) ∼ / H ev (Y ).

=

The Chern character for the middle square in (5.1) follows from the Riemann-Roch theorem in ordinary K-theory for the K-oriented map df : T X → T Y with the orientation given by canonical Spinc manifolds T X and T Y . Note that the Todd classes of Spinc ∗ ˆ2 manifolds of T X and T Y are given by πX (A (X)) and πY∗ (Aˆ2 (Y )) respectively. This is ∗ ∼ due to two facts, that T (T X) = πX (T X ⊗ C) and that T d(T X ⊗ C) = Aˆ2 (X). So we have    ∗ 2   H ∗ ˆ2 ˆ Ch (df )K (5.6) ! (a) · πY A (Y ) = (df )∗ Ch(a)πX (A (X)) for any a ∈ Kc0 (T X). Hence, the following diagram commutes (5.7)

Kc0 (T X)

(df! )K

∗ ˆ2 Ch(−)·πX A (X)

 Hcev (T X)

/ K 0 (T Y ) c ∗ ˆ2 Ch(−)·πY A (Y )

(df )H ∗

 / Hcev (T Y ).

Putting (5.3), (5.5) and (5.7) together, we get the following commutative diagram K 0 (X, oX ) ˆ Chw ˇ 2 (X) (−)·A(X)



H ev (X) which leads to

f!K

f∗H

/ K 0 (Y, oY ) 

ˆ Chw ˇ 2 (Y ) (−)·A(Y )

/ H ev (Y )

    ˆ ) = f H Chwˇ (X) (a)A(X) ˆ Chwˇ2 (Y ) f!K (a) A(Y ∗ 2

for any a ∈ K 0 (X, oX ). This completes the proof of the Riemann-Roch theorem in twisted K-theory. 

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125 31

With some notational changes, the above argument can be applied to establish the general Riemann-Roch theorem in twisted K-theory. Let f : X → Y a smooth map between oriented manifolds with dim Y − dim X = 0 mod 2. Let α ˇ = (Gα , θ, ω) be a differential twisting which lifts α : Y → K(Z, 3), f ∗ (α) ˇ is the pull-back differential twisting which lifts α ◦ f : X → K(Z, 3). Then we have the following Riemann-Roch formula     ˆ ) = f∗H Chf ∗ α+ ˆ Chαˇ f!K (a) A(Y (5.8) ˇ w ˇ (Y )+f ∗ w ˇ (Y ) (a)A(X) 2

2

∗

for any a ∈ K (X, α ◦ f + oX + f (oY )). In particular, we have the following RiemannRoch formula for a trivial twisting α : Y → K(Z, 3)     ˆ ) = f H Chwˇ (Y )+f ∗ wˇ (Y ) (a)A(X) ˆ Ch f!K (a) A(Y (5.9) ∗ 2 2 0

for any a ∈ K 0 (X, oX + f ∗ (oY )). When Y is a point, then the Riemann-Roch formula (5.9) agrees with the index formula obtained by Murray and Singer in [MS]. When f is K-oriented, and equipped with a Spinc structure whose determinant bundle is L, there is a canonical isomorphism Ψ : K 0 (X) ∼ = K 0 (X, oX + f ∗ (oY )) such that Chwˇ2 (Y )+f ∗ wˇ2 (Y ) (Ψ(a)) = ec1 (L)/2 Ch(a) for any a ∈ K 0 (X). Then the twisted Riemann-Roch formula (5.9) agrees with the Riemann-Roch formula for K-oriented maps as established in [AH] and [BEM] in the presence of H-flux. 6. The Twisted Index Formula In this Section, we establish the index pairing for a closed smooth manifold with a twisting α an K ev/odd (X, α) × Kev/odd (X, α) −→ Z in terms of the local index formula for twisted geometric cycles. T HEOREM 6.1. Let X be a smooth closed manifold with a twisting α : X → K(Z, 3). The index pairing K 0 (X, α) × K0 (X, α) −→ Z is given by    ˆ ) ξ, (M, ι, ν, η, [E]) = Chwˇ (M ) η∗ (ι∗ ξ ⊗ E) A(M 2

M

where ξ ∈ K 0 (X, α), and the geometric cycle (M, ι, ν, η, [E]) defines a twisted Khomology class on (X, α). Here η∗ : K ∗ (M, ι∗ α) ∼ = K ∗ (M, oM ) is an isomorphism, and Chwˇ2 (M ) is the Chern character on K 0 (M, oM ). P ROOF. Recall that the index pairing K 0 (X, α) × K0 (X, α) −→ Z can be defined by the internal Kasparov product (Cf. [Kas3] and [ConSka]) KK(C, C(X, Pα (K))) × KK(C(X, Pα (K)), C) −→ KK(C, C) ∼ = Z, and is functorial in the sense that if f : Y → X is a continuous map and Y is equipped with a twisting α : X → Z then f ∗ b, a = b, f∗ (a) for any a ∈ K0 (Y, f ∗ α) and b ∈ K 0 (X, α).

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ALAN L. CAREY AND BAI-LING WANG

Note that under the assembly map, the geometric cycle (M, ι, ν, η, [E]) is mapped to ι∗ ◦ η∗ ([M ] ∩ [E]), for ξ ∈ K 0 (X, α). Hence, we have ξ, (M, ι, ν, η, [E]) = ξ, ι∗ ◦ η∗ ([M ] ∩ [E]) =

ι∗ ξ, η∗ ([M ] ∩ [E])

=

η∗ (ι∗ ξ ⊗ E), [M ].

an (M, oM ) Here η∗ (ι∗ ξ ⊗ E) ∈ K 0 (M, oM ) and [M ] is the fundamental class in Kev 0 which is Poincar´e dual to the unit element C in K (M ). The index pairing between an K 0 (M, oM ) × Kev (M, oM ) can be written as an K 0 (M, oM ) × Kev (M, oM ) → K 0 (M, oM ) × K 0 (M ) → K 0 (M, oM ) → Z an where the first map is given by the Poincar´e duality Kev (M, oM ) ∼ = K 0 (M ), the middle map is the action of K 0 (M ) on K 0 (M, oM ), and the last map is the push-forward map of  : M → pt. Therefore, we have

η∗ (ι∗ ξ ⊗ E), [M ]   ∗ = K ! η∗ (ι ξ ⊗ E) ⊗ C   ∗ = K ! η∗ (ι ξ ⊗ E) . By twisted Riemann-Roch (Theorem 5.3),   ! η∗ (ι∗ ξ ⊗ E)     ∗ ˆ = H ˇ2 (M ) η∗ (ι ξ ⊗ E) A(M ) ∗ Chw    ˆ ). Chwˇ2 (M ) η∗ (ι∗ ξ ⊗ E) A(M = M

This completes the proof of the twisted index formula.



Note that  : M → pt can be written as ι ◦ X : M → X → pt. Applying the Riemann-Roch Theorem 5.3, we can write the above index pairing as < (M, ι, ν, η, [E]), ξ >    ˆ ) = Chwˇ2 (M ) η∗ (ι∗ ξ ⊗ E) A(M M    ˆ Chwˇ2 (X) ι! (E) ⊗ ξ A(X) = X

where ι! : K (M ) → K (X, −α + oX ) is the push-forward map in twisted K-theory, 0

0

K 0 (X, α) × K 0 (X, −α + oX ) −→ K 0 (X, oX ) is the multiplication map (2.7), and Chwˇ2 (X) : K 0 (X, oX ) −→ H ev (X) is the twisted Chern character (which agrees with the relative Chern character under the identification K 0 (X, oX ) ∼ = K 0 (X, W3 (X)), the K-theory of Clifford modules on X).

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127 33

7. Mathematical Definition of D-branes and D-brane Charges Here we give a mathematical interpretation of D-branes in Type II string theory using the twisted geometric cycles and use the index theorem in the previous Section to compute charges of D-branes. In Type II superstring theory on a manifold X, a string worldsheet is an oriented Riemann surface Σ, mapped into X with ∂Σ mapped to an oriented submanifold M (called a D-brane world-volume, a source of the Ramond-Ramond flux). The theory also has a Neveu-Schwarz B-field classified by a characteristic class [α] ∈ H 3 (X, Z). In physics, the D-brane world volume M carries a gauge field on a complex vector bundle (called the Chan-Paton bundle), so a D-brane is given by a submanifold M of X with a complex bundle E and a connection ∇E . This data actually defines a differential K-class [(E, ∇E )] ˇ in differential K-theory K(M ). When the B-field is topologically trivial, that is [α] = 0, D-brane charge takes values in ordinary K-theory K 0 (X) or K 1 (X) for Type IIB or Type IIA string theory (as explained in [MM][Wit]). For a D-brane M to define a class in the K-theory of X, its normal bundle νM must be endowed with a Spinc structure. Equivalently, the embedding ι : M −→ X is K-oriented so that the push-forward map in K-theory ([AH]) 0 ev/odd (X) ιK ! : K (M ) −→ K

is well-defined, (it takes values in even or odd K-groups depending on the dimension of M ). So the D-brane charge of (ι : M → X, E) is ev/odd ιK (X). ! ([E]) ∈ K

It was proposed in [MM] that the cohomological Ramond-Ramond charge of the D-brane is given by  ˆ QRR (ι : M → X, E) = ch(f!K (E)) A(X) when X is a Spin manifold. A natural K-theoretic interpretation follows from the fact that the modified Chern character isomorphism ev/odd

KQ (X) −→ H ev/odd (X, Q)  ˆ given by mapping a → ch(a) A(X) is an isometry with the natural bilinear parings on ∗ ∗ ev/odd (X, Q). Here the pairing on K(X) is given by the KQ (X) = K (X) ⊗ Q and H index of the Dirac operator      ˆ ˆ ˆ (a, b)K = Index(D / a⊗b ) = ch(a)ch(b)A(X) = ch(a) A(X), . (ch(b) A(X) H X

When the B-field is not topologically trivial, that is [α] = 0, then [α] defines a complex line bundle over the loop space LX, or a stable isomorphism class of bundle gerbes over X. Then in order to have a well-defined worldsheet path integral, Freed and Witten in [FreWit] showed that (7.1)

ι∗ [α] + W3 (νM ) = 0.

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ALAN L. CAREY AND BAI-LING WANG

When ι∗ [α] = 0, that means ι is not K-oriented, then the push-forward map in K-theory ([AH]) 0 ∗ ιK ! : K (M ) −→ K (X) is not well-defined. Witten explained in [Wit] that D-brane charges should take values in a twisted form of K-theory, as supported further by evidence in [BouMat] and [Kap]. In [Wa], the mathematical meaning of (7.1) was discovered using the notion of αtwisted Spinc manifolds for a continuous map α : X −→ K(Z, 3) representing [α] ∈ H (X, Z). When X is Spinc , the datum to describe a D-brane is exgeo actly a geometric cycle for the twisted K-homology Kev/odd (X, α). By Poincar´e duality, we have geo Kev/odd (X, α) ∼ = K 0 (X, α + oX ) 3

with the orientation twisting oX : X → K(Z, 3) trivialized by a choice of a Spinc structure. Hence, K 0 (X, α + oX ) ∼ = K 0 (X, α). For a general manifold X, a submanifold ι : M → X with ι∗ ([α]) + W3 (νM ) = 0, then there is a homotopy commutative diagram / BSO vv η v vv v ι W3 vv   w vv / K(Z, 3), X α νM

M

here νM also denotes a classifying map of the normal bundle, or a classifying map of the bundle T M ⊕ ι∗ T X. This motivates the following definition (see also [CW2]). D EFINITION 7.1. Given a smooth manifold X with a twisting α : X → K(Z, 3), a B-field of (X, α) is a differential twisting lifting α α ˇ = (Gα , θ, ω), which is a (lifting, or local) bundle gerbe Gα with a connection θ and a curving ω. The field strength of the B-field (Gα , θ, ω) is given by the curvature H of α ˇ. A Type II (generalized) D-brane in (X, α) is a complex vector bundle E with a connection ∇E over a twisted Spinc manifold M . The twisted Spinc structure on M is given by the following homotopy commutative diagram together with a choice of a homotopy η (7.2)

/ BSO ww η w ww w ι W3 ww   w ww / K(Z, 3) X α M

νι

where νι is the classifying map of T M ⊕ ι∗ T X. R EMARK 7.2. The twisted Spinc manifold M in Definition 7.1 is the D-brane world volume in Type II string theory. The twisted Spinc structure given in (7.2) implies that D-brane world volume M ⊂ X in Type II string theory satisfies the Freed-Witten anomaly cancellation condition ι∗ [α] + W3 (νM ) = 0.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

129 35

In particular, if the B-field of (X, α) is topologically trivial, then the normal bundle of M ⊂ X is equipped with a Spinc structure given by (7.2). Given a Type II D-brane (M, ι, νι , η, E, ∇E ), the homotopy η induces an isomorphism η∗ : K 0 (M ) → K 0 (M, ι∗ α + oι ). Here oι denotes the orientation twisting of the bundle T M ⊕ ι∗ T X. Note that 0 ∗ en/odd (X, α) ιK ! : K (M, ι α + oι ) −→ K

is the pushforward map (2.9) in twisted K-theory. Hence we have a canonical element in K en/odd (X, α) defined by ιK ! (η∗ ([E])), called the D-brane charge of (M, ι, νι , η, E). We remark that a Type II D-brane (M, ι, νι , η, E, ∇E ) ˇ en/odd (X, α ˇ ). defines an element in differential twisted K-theory K From (7.2), we know that M is an (α + oX )-twisted Spinc manifold as we have the following homotopy commutative diagram / BSO w ww w ww ι W3 ww   w ww X α+oX/ K(Z, 3) ν

M

where ν is the classifying map of the stable normal bundle of M . Together with the following proposition, we conclude that the Type II D-brane charges, in the present of a B-field α ˇ = (Gα , θ, ω), are classified by twisted K-theory K 0 (X, α). P ROPOSITION 7.3. Given a twisting α : X → K(Z, 3) on a smooth manifold X, every twisted K-class in K ev/odd (X, α) is represented by a geometric cycle supported on an (α + oX )-twisted closed Spinc -manifold M and an ordinary K-class [E] ∈ K 0 (M ). For completeness, we also give a definition of Type I D-branes (Cf. [MMS], [RSV] and Section 8 in [Wa]). D EFINITION 7.4. Given a smooth manifold X with a KO-twisting α : X → K(Z2 , 2), a Type I (generalized) D-brane in (X, α) is a real vector bundle E with a connection ∇E over a twisted Spin manifold M . The twisted Spin structure on M is given by the following homotopy commutative diagram together with a choice of a homotopy η (7.3)

/ BSO vv η v vv v w2 ι vv   v~ vv / X α K(Z2 , 2) M

νι

where w2 is the classifying map of the principal K(Z2 , 1)-bundle BSpin → BSO associated to the second Stiefel-Whitney class, η is a homotopy between w2 ◦ νι and α ◦ ι. Here νι is the classifying map of T M ⊕ ι∗ T X.

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ALAN L. CAREY AND BAI-LING WANG

R EMARK 7.5. A Type I D-brane in (X, α) has its support on a manifold M if and only if there is a differentiable map ι : M → X such that ι∗ ([α]) + w2 (νι ) = 0. Here νι denotes the bundle T M ⊕ ι∗ T X. Given a Type I D-brane in (X, α), the push-forward map in twisted KO-theory KO ∗ (M )

η∗ ∼ =

/ KO ∗ (M, α ◦ ι + oι )

ιKO !

/ KO ∗ (X, α)

defines a canonical element in KO ∗ (X, α). Every class in KO ∗ (X, α) can be realized by a Type I (generalized) D-brane in (X, α). Hence, we conclude that the Type I D-brane charges are classified by twisted KO-theory KO ev/odd (X, α). References [ABS] [AH] [AS1] [AS2] [AtiSin1] [AtiSin3] [AtiSin4] [BC] [BD1] [BD2] [BHS] [Ba] [BGV] [BEM] [BouMat] [BCMMS] [BMRS] [CMW] [CW1] [CW2] [ConSka] [DK]

M. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology Vol. 3, Suppl. 1, pp 3-38. M. Atiyah, F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Society, pp. 276-281 (1959). M. Atiyah, G. Segal, Twisted K -theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330; translation in Ukr. Math. Bull. 1 (2004), no. 3, 291–334. ArXiv:math/0407054 M. Atiyah, G. Segal, Twisted K -theory and cohomology. Inspired by S. S. Chern, 5-43, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006. M. Atiyah, I. Singer, The index of elliptic operators. I. Ann. of Math. (2), 87, 1968, 484–530. M. Atiyah, I. Singer, The index of elliptic operators. III. Ann. of Math. (2), 87, 1968, 546-604. M. Atiyah, I. Singer, The index of elliptic operators. IV. Ann. of Math. (2), 93, 1971, 119–138. P. Baum, A. Connes, Geometric K -theory for Lie groups and foliations. Enseign. Math. (2) 46 (2000), no. 1-2, 3–42. P. Baum, R. Douglas, K homology and index theory. Proceedings Symp. Pure Math 38, 117-173. Amer. Math. Soc., Providence, 1982. P. Baum, R. Douglas, Index theory, bordism, and K -homology. Operator Algebras and K-Theory, Contemporary Math. 10, 1-31. Amer. Math. Soc., Providence, 1982. P. Baum, N. Higson, T. Schick, On the equivalence of geometric and analytic K -homology. Pure and Applied Mathematics Quarterly 3(1):1-24, 2007. ArXiv:math/0701484 D. Basu, K -theory with R/Z coefficients and von Neumann algebras. K-Theory 36 (2005), no. 3-4, 327–343. N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Grundlehren Math. Wiss., vol. 298, Springer-Verlag, New York, 1992. P. Bouwknegt, J. Evslin and V. Mathai, T -duality: topology change from H -flux. Comm. Math. Phys. 249 (2004), no. 2, 383–415. P. Bouwknegt, V. Mathai, D-branes, B -fields and twisted K -theory, J. High Energy Phys. 03 (2000) 007, hep-th/0002023. P. Bouwknegt, A. Carey, V. Mathai, M. Murray, D. Stevenson, Twisted K -theory and K -theory of bundle gerbes. Comm. Math. Phys. 228 (2002), no. 1, 17–45. J. Brodzki, V. Mathai, J. Rosenberg, R. Szabo, D-branes, RR-fields and duality on noncommutative manifolds. Comm. Math. Phys. 277 (2008), no. 3, 643–706. A. Carey, J. Mickelsson, B.-L. Wang, Differential Twisted K -theory and Applications. Jour. of Geom. Phys., Vol. 59, no. 5, 632-653, (2009), arXiv:0708.3114. A. Carey, B.-L. Wang, Thom isomorphism and Push-forward map in twisted K -theory. Journal of K-Theory, Volume 1, Issue 02, April 2008, pp 357-393. A. Carey, B.-L. Wang, Fusion of symmetric D-branes and Verlinde rings. Comm. Math. Phys. 277 (2008), no. 3, 577–625. A. Connes, G. Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. Kyoto 20 (1984), 1139-1183. P. Donavan, M. Karoubi, Graded Brauer groups and K -theory with lcoal coefficients, Publ. Math. de IHES, Vol 38, 5-25, 1970.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY

[ES] [EEK] [FH] [FreWit] [Gomi] [GoTe] [HigRoe] [HopSin] [Kar] [Kar1] [Kap] [Kas1] [Kas2] [Kas3] [Kui] [LM] [MMS] [MatSte] [MM] [Mur] [MS] [Pol] [Ros] [RS] [RSV] [Swi] [Sz] [Tu] [Wa] [Wit]

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Evslin, Jarah; Sati, Hisham Can D-branes wrap nonrepresentable cycles? J. High Energy Phys. 2006, no. 10. S. Echterhoff, H. Emerson, H. J. Kim, KK -theoretical duality for proper twisted action. Math. Ann. 340 (2008), no. 4, 839–873. D. Freed, M. J. Hopkins, On Ramond-Ramond fields and K -theory. J. High Energy Phys. no.5, 2000. D. Freed and E. Witten, Anomalies in String Theory with D-Branes, Asian J. Math. 3 (1999), no. 4, 819–851. K. Gomi, Twisted K -theory and finite-dimensional approximation. arXiv:0803.2327. K. Gomi, Y. Terashima, Chern-Weil construction for twisted K -theory. Preprint. N. Higson, J. Roe, Analytic K -homology. Oxford University Press. 2000. M. Hopkins, I. Singer, Quadratic functions in geometry, topology, and M-theory. J. Differential Geom. 70 (2005), no. 3, 329–452. M. Karoubi, Twisted K -theory, old and new. Preprint, math.KT/0701789. M. Karoubi, Homologie cyclique et K -th´eorie. Astrisque No. 149 (1987), 147 pp. A. Kapustin, D-branes in a topologically non-trivial B-field, Adv. Theor. Math. Phys. 4 (2001) 127, hep-th/9909089. G. Kasparov, Topological invariants of elliptic operators I: K -homology. Math. USSR Izvestija 9, 1975, 751-792. G. Kasparov, Equivariant KK -theory and the Novikov conjecture. Invent. Math. 91 (1988) 147201. G. Kasparov, The operator K-functor and extensions of C ∗ -algebras. Izv. Akad. Nauk. SSSR Ser. Mat. 44 (1980), 571-636. N. Kuiper, The homotopy type of the unitary group of Hilbert space. Topology, 3 (1965), 19-30. B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. V. Mathai, M. Murray, D. Stevenson, Type-I D-branes in an H -flux and twisted KO-theory. J. High Energy Phys. 2003, no. 11, 053, 23 V. Mathai and D. Stevenson, Chern character in twisted K -theory: equivariant and holomorphic cases, Commun.Math.Phys. 228, 17-49, 2002. R. Minasian and G. W. Moore, K -theory and Ramond-Ramond charge. JHEP 9711:002 (1997) (arXiv:hep-th/9710230) M. K. Murray Bundle gerbes, J. London Math. Soc. (2) 54, 403–416, 1996. Michael K. Murray and Michael A. Singer, Gerbes, Clifford modules and the index theorem. Ann. Global Anal. Geom. 26 (2004), no. 4, 355–367. J. Polchinski, Dirichlet branes and Ramond-Ramond charges. Phys. Rev. Lett. 75 (1995) 47244727, arXiv:hep-th/9510017 J. Rosenberg, Continuous-trace algebras from the bundle-theoretic point of view. Jour. of Australian Math. Soc. Vol A 47, 368-381, 1989. R. M. G. Reis, R.J. Szabo, Geometric K -Homology of Flat D-Branes, Commun. Math. Phys. 266 (2006), 71122. R. M. G. Reis, R.J. Szabo, A. Valentino, KO-homology and type I string theory arXiv:hepth/0610177 R. Switzer, Algebraic topology - homotopy and homology. Springer-Verlag, Berlin, 1975 R. J. Szabo, D-branes and bivariant K -theory, arXiv:0809.3029. J.-L. Tu, Twisted K -theory and Poincare duality, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1269– 1278. B.-L. Wang, Geometric cycles, Index theory and twisted K -homology. Journal of Noncommutative Geometry 2 (2008), no. 4, 497552. E. Witten, D-branes and K -theory J. High Energy Phys. 1998, no. 12, Paper 19.

M ATHEMATICAL S CIENCES I NSTITUTE , AUSTRALIAN NATIONAL U NIVERSITY, C ANBERRA ACT 0200, AUSTRALIA E-mail address: [email protected] D EPARTMENT OF M ATHEMATICS , AUSTRALIAN NATIONAL U NIVERSITY, C ANBERRA ACT 0200, AUS TRALIA

E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681761

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Noncommutative Principal Torus Bundles via Parametrised Strict Deformation Quantization Keith C. Hannabuss and Varghese Mathai Abstract. In this paper, we initiate the study of a parametrised version of Rieffel’s strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.

Introduction Operator theoretic deformation quantization appeared in quantum physics a long time ago, but was put on a firm footing relatively recently by Rieffel [15] (see the references therein), who called it strict deformation quantization, mainly to distinguish it from formal deformation quantization, where convergence isn’t an issue. His theory has been remarkably successful, giving rise to many examples of noncommutative manifolds, which have become extremely useful both in mathematics and mathematical physics. In a recent paper [3] Echterhoff, Nest, and Oyono-Oyono defined noncommutative principal torus bundles, inspired by fundamental results in [17], as well as the T-duals of certain continuous trace algebras [13, 14]. They also classified all noncommutative principal torus bundles in terms of (noncommutative) fibre products of principal torus bundles and group C∗ -algebras of lattices in simply-connected 2-step nilpotent Lie groups, cf. §5. In this paper, we show that their classification can be neatly understood in terms of a generalization of Rieffel’s strict deformation quantization [15, 16], to the parametrised case that is developed here. More precisely, we generalize to the parametrised case, the recent version of Rieffel’s strict deformation quantization given by Kasprzak [9] based on work of Landstad [11, 12]. More precisely, we give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. Strict deformation quantization theory works with smooth subalgebras, so we start with a section, §1, on smooth subalgebras of C∗ -algebras and a smooth version 2000 Mathematics Subject Classification. 58B34 (81S10, 46L87, 16D90, 53D55). Key words and phrases. Parametrised strict deformation quantization, Noncommutative principal torus bundles, T-duality. Acknowledgements. V.M. thanks the Australian Research Council for support. c Mathematical 0000 (copyright Society holder) c 2010 American

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of the noncommutative torus bundle theory. That is followed in §2 by a summary recalling the ideas of Rieffel’s strict deformation quantization, and then a section, §3, generalising that to a parametrised version. Then we explain in §4, Kasprzak’s recent account of Rieffel’s strict deformation quantization theory based on ideas of Landstad, and extend it to a parametrised version. In §5, after a summary of the relevant parts of the Echterhoff, Nest, and Oyono-Oyono classification of noncommutative principal torus bundles, we explain the connection to parametrised strict deformation quantization theory. We end with a section, §6, containing examples of parametrised strict deformation quantization including the case of principal torus bundles. It also contains a putative definition of noncommutative non-principal torus bundles. There is an appendix containing a discussion about factors of automorphy which is used in the paper. 1. Fibrewise Smooth ∗-bundles We begin by recalling the notion of C ∗ -bundles over X and the special case of noncommutative principal bundles. Then we discuss the fibrewise smoothing of these, which is used in parametrised Rieffel deformation later on. Let X be a locally compact Hausdorff space and let C0 (X) denote the C ∗ algebra of continuous functions on X that vanish at infinity. A C ∗ -bundle A(X) over X in the sense of [3] is exactly a C0 (X)-algebra in the sense of Kasparov [8]. That is, A(X) is a C ∗ -algebra together with a non-degenerate ∗-homomorphism ΦA : C0 (X) → ZM (A(X)), called the structure map, where ZM (A) denotes the center of the multiplier algebra M (A) of A. The fibre over x ∈ X is then A(X)x = A(X)/Ix , where Ix = {Φ(f ) · a; a ∈ A(X) and f ∈ C0 (X) such that f (x) = 0}, and the canonical quotient map qx : A(X) → A(X)x is called the evaluation map at x. Note that this definition does not require local triviality of the bundle, or even for the fibres of the bundle to be isomorphic to one another. Let G be a locally compact group. One says that there is a fibrewise action of G on a C ∗ -bundle A(X) if there is a homomorphism α : G −→ Aut(A(X)) which is C0 (X)-linear in the sense that ∀g ∈ G, a ∈ A(X), f ∈ C0 (X).

αg (Φ(f )a) = Φ(f )(αg (a)),

This means that α induces an action α on the fibre A(X)x for all x ∈ X. The first observation is that if A(X) is a C ∗ -algebra bundle over X with a fibrewise action α of a Lie group G, then there is a canonical smooth ∗-algebra bundle over X. We recall its definition from [2]. A vector y ∈ A(X) is said to be a smooth vector if the map x

G  g −→ αg (y) ∈ A(X) is a smooth map from G to the normed vector space A(X). Then A∞ (X) = {y ∈ A(X) | y is a smooth vector} is a ∗-subalgebra of A(X) which is norm dense in A(X). Since G acts fibrewise on A(X), it follows that A∞ (X) is again a C0 (X)-algebra which is fibrewise smooth. Let T denote the torus of dimension n. The authors of [3] define a noncommutative principal T -bundle (or NCP T -bundle) over X to be a separable C ∗ -bundle

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A(X) together with a fibrewise action α : T → Aut(A(X)) such that there is a Morita equivalence, A(X) α T ∼ = C0 (X, K), as C ∗ -bundles over X, where K denotes the C ∗ -algebra of compact operators. The motivation for calling such C ∗ -bundles A(X) NCP T -bundles arises from a special case of a theorem of Rieffel [17], which states that if q : Y −→ X is a principal T -bundle, then C0 (Y )  T is Morita equivalent to C0 (X, K). If A(X) is a NCP T -bundle over X, then we call A∞ (X) a fibrewise smooth noncommutative principal T -bundle (or fibrewise smooth NCP T -bundle) over X. In this paper, we are able to give a complete classification of fibrewise smooth NCP T -bundles over X via a parametrised version of Rieffel’s theory of strict deformation quantization. 2. Rieffel Deformation Unlike Rieffel’s deformation theory [15, 16], the version which we shall use [11, 12, 9] starts with multipliers, so in this section we shall recapitulate some standard results but in a formulation which suits the later extension to a parametrised theory and the Landstad–Kasprzak approach. In what follows, a Poisson bracket {, } on A is a bilinear form from A to itself, which is a Hochschild 2-cocycle satisfying a couple of additional technical conditions that will not be repeated here, but we refer the reader to §5 in [15]. Definition 2.1 (§5, [15]). Let A be a dense ∗-subalgebra of a C∗ algebra, equipped with a Poisson bracket {, }. A strict deformation quantisation of A in the direction of {, } means an open interval I containing 0 in R, together with associative products  ,  ∈ I, involutions and C∗ -norms on A which for  = 0 are the original product, involution and norm on A, such that: (1) The corresponding field of C∗ -algebras with continuity structure given by the elements of A as constant fields, is a continuous field of C∗ -algebras. (2) For all a, b ∈ A, as  → 0 one has (a  b − ab)/(i) − {a, b} → 0. Typically, one tries to find strict deformation quantizations of Poisson manifolds, thus obtaining interesting noncommutative manifolds. Rieffel’s definition and construction are motivated by Moyal’s product but to link it with Kasprzak’s work it is useful to give the background. Suppose that A is a pre- C∗ -algebra with an action α of a locally compact abelian group V (written additively), and let σ be a multiplier on its Pontryagin dual V , that σ : V × V → T is a borel map, satisfying the cocycle identity σ(ξ, η)σ(ξ + η, ζ) = σ(ξ, η + ζ)σ(η, ζ), for all ξ, η, ζ ∈ V . The group of all such cocycles (or multipliers) is denoted by Z 2 (V , T). Two multipliers σ1 and σ2 are equivalent (or cohomologous) if and only if there is a borel map ρ : V → T such that σ1 (ξ, η)ρ(ξ + η) = σ2 (ξ, η)ρ(ξ)ρ(η), and the equivalence classes form the cohomology group H 2 (V , T). A cocycle equivalent to the constant cocycle V × V → {1} is said to be trivial.

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Recalling that a bicharacter β : V × V → T defines characters βξ1 : η → β(ξ, η) for each fixed ξ, and βη2 : ξ → β(ξ, η) for fixed η, we see that bicharacters always define cocycles, because β(ξ, η)β(ξ + η, ζ) = β(ξ, η)β(ξ, ζ)β(η, ζ) = β(ξ, η + ζ)β(η, ζ). Theorem 2.2 ([10, 5]). Every multiplier on an abelian group V is equivalent to a bicharacter (and so is continuous). Two bicharacters β1 and β2 are equivalent if and only if β = β1 β2−1 a symmetric bicharacter , that is β(ξ, η) = β(η, ξ). If V = 2V then each cohomology class can be represented by a unique antisymmetric bicharacter β, that is β(ξ, η) = β(η, ξ)−1 . We can therefore assume that σ is a bicharacter, and this means that it is actually continuous in each variable. When V = 2V , the element σ(ξ, 12 η)/σ(η, 12 ξ) gives the canonical antisymmetric bicharacter representative of the class containing σ. We note that vector groups V = Rn = 2V , so that each cocycle can be represented by a continuous antisymmetric antisymmetric bicharacter, which must be the exponential exp[iπs(ξ, η)] of a skew-symmetric bilinear form s. These can be identified  with 2 V . There is a similar analysis for lattices L ∼ = Zn , where the bicharacters 2 2 2 (V /L) = V/ L (restrictions modulo those with trivare given by the torus ial restriction), but a torus Tn has only trivial bicharacters β(ξ, η) = 1, due to the following observation. Corollary 2.3. There are no non-trivial bicharacters on a connected compact group V . This follows because the non-trivial multipliers on the dual of an infinite connected compact group (such as V = T2n ) are never invertible, since the dual (e.g. V = Z2n ) is discrete and the two groups are not isomorphic. Theorem 2.4. Given a continuous bicharacter cocycle σ on V and a pre-C ∗ algebra A we may form the ∗-algebra of functions f, g : V → A smooth with respect to the translation automorphisms τu [f ] = f (u + v), with the twisted convolution product  (f ∗ g)(ξ) =

 V

σ(η, ξ − η)f (η)g(ξ − η) dη,

and involution f ∗ (ξ) = σ(ξ, ξ)f (−ξ). Up to isomorphism this algebra depends only on the cohomology class of σ. Proof. The cocycle identity on σ ensures associativity. When σ is an antisymmetric bicharacter the involution reduces to f ∗ (ξ) = f (−ξ). Changing σ to σ(ξ, η)ρ(ξ)ρ(η)ρ(ξ + η)−1 gives the algebra isomorphism f → ρ.f (the pointwise product).



These functions can be Fourier transformed to functions on V  f(v) = ξ(v)f (ξ) dξ,  V

where we assume that the Haar measure is normalised to make the transform unitary, and one has the usual inverse transform.

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Theorem 2.5. The transformed product is   (f ∗ g)(v) = σ(η, ξ)η(u)ξ(w)f(v + u) g (v + w) dudwdξdη. Proof. We calculate that   ξ(v)(f  g)(ξ) dξ (f ∗ g)(v) =  V ξ(v)σ(η, ξη −1 )f (η)g(ξη −1 ) dξdη =  ×V  V  = σ(η, ξ)η(v)f (η)ξ(v)g(ξ) dξdη    V ×V = σ(η, ξ)η(v)f(u)η(u)ξ(v) g (w)ξ(w) dudwdξdη  g (v + w) dudwdξdη, = σ(η, ξ)η(u)ξ(w)f(v + u)

where we replaced u and w by v + u and v + w in the last step.



We now want to connect this transformed product with Rieffel’s deformation. To this end we introduce a bicharacter e on V , which defines a homomorphism e1 : V → V . Rieffel works with a vector group V and e(u, w) = exp(i(u · w)) for some inner product on V . When σ is non-degenerate (that is, σ 1 : V → V is an isomorphism) we can choose e so that e1 is the inverse of σ 1 , but in general we have an automorphism T = σ 1 ◦ e1 : V → V . As a final piece of notation we introduce the adjoint T ∗ with respect to e: e(T ∗ u, w) = e(u, T w). Proposition 2.6. The bicharacters σ and e are related by σ(e1u , e1v ) = e1v (T u) = e(T u, v) for all u, v ∈ V . Suppose that σ is an antisymmetric bicharacter. Then if e is symmetric T = −T ∗ , and if e is antisymmetric T = T ∗ . Proof. By definition we have σ(e1u , e1v ) = e1v (T u) = e(T u, v). Since σ is skew symmetric this gives e(T u, v) = σ(e1u , e1v ) = σ(e1v , e1u )−1 = e(T v, u)−1 = e(−T v, u). When e is symmetric this shows that e(T u, v) = e(u, −T v), so that T = −T ∗ , and when e is antisymmetric T = T ∗ .  Theorem 2.7. Given a non-degenerate bicharacter e on V , set T = σ 1 ◦ e1 : V → V . and e(T ∗ u, w) = e(u, T w). Then   g (v + w) dudw. (f ∗ g)(v) = e(u, w)f(v + T ∗ u) We change the order of integration in our earlier expression for f ∗ g and concentrate on the integrals over V :  σ(η, ξ)η(u)ξ(w) dξdη.  ×V  V

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Since e is nondegenerate we may set η = ev , and then, by definition, we have σ(ev , ξ)ev (u)ξ(w) = ξ(T v)e(v, u)ξ(w). By the Fourier inversion theorem, integration over ξ gives a delta function δ(T v−w). Replacing u by T ∗ u and integrating over v, we now get   ∗ σ(ev , ξ)e(v, T u)ξ(w) dξdv = δ(w − T v)e(T v, u) dv = e(w, u). Up to a multiple, integration over η and v are the same, and with appropriate choices of measure we can ensure that they agree precisely. Then inserting this into the original formula for the product we have   (f ∗ g)(v) = e(u, w)f(v + T ∗ u) g (v + w) dudw.  Theorem 2.8. The Fourier transformed product (f ∗ g) = f  g where  (f  g)(v) = e(u, w)f(v + T ∗ u) g (v + w) dudw. In terms of the translation automorphisms τw [g](v) = g(v + w), we have  (f  g)(v) = e(u, w)τT ∗ u [f](v)τw [ g ](v) dudw. Evaluating at the identity v = 0 gives   (f  g)(0) = e(u, w)τT ∗ u [f](0)τw [ g ](0) dudw. Rieffel noticed that this formula can now be interpreted whenever α defines automorphisms of A, so that one can define  a  b = e(u, w)αT ∗ u [a]αw [b] dudw, for a and b in the algebra. (Our T ∗ is Rieffel’s J.) When both bicharacters σ and e are nondegenerate we can also write this as  a  b = det[T ∗ ]−1 e(T ∗ −1 u, w)αu [a]αw [b] dudw. The above arguments are formal and one must check that the integrals converge. In the standard Moyal theory this is done by working only with Schwarz functions and in the general case one uses the smooth vectors A∞ for the action α, which form a dense Fr´echet subalgebra of A. For vector groups this works particularly smoothly, and one obtains a strict deformation quantisation [15], Theorem 9.3. However, there are technical problems when V = T2n since, as we have seen, there are no nontrivial bicharacters e on V . There are two ways of dealing with this problem. One is by the Kasprzak–Landstad approach of working with the dual crossed product algebra, [12, 9], and the other is Rieffel’s approach of lifting the action of the torus T = V /L (with L a lattice), to the vector group V , [15] Ch 2.

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3. Parametrised Rieffel Deformations An interesting generalisation comes from inserting a parameter. More precisely, we work with a C0 (X)-algebra A, where X is a locally compact Hausdorff space. (That is there is a map C0 (X) → ZM A.) Consider a function σ ∈ Cb (X, Z 2 (V , T)) taking values in the bicharacter cocycles. At each point x ∈ X this defines a multiplier σx , and a map σx1 : V → V . We then form Tx = σx1 ◦ e1 and its adoint Tx∗ with respect to e, where e, e1 are defined just prior to Proposition 2.6. If the image of σ lies in the non-degenerate cocycles we can then form the continuous function x → e(Tx∗ −1 u, v), which acts on A. Theorem 3.1. Given a C0 (X)-algebra A, where X is a locally compact Hausdorff space, and a function σ ∈ Cb (X, Z 2 (V , T)) taking values in the nondegenerate bicharacter cocycles , let Tx = σx1 ◦ e1 and e(Tx∗ u, w) = e(u, Tx w). Then, if T has an inverse in a subalgebra of Cb (X) whose action preserves the (fibrewise) smooth subalgebra A∞ , one has a product  a  b = det[T ∗ ]−1 e(T ∗ −1 u, w)αu [a]αw [b] dudw, defined by the actions of the continuous functions det[T ∗ ]−1 and e(T ∗ −1 u, w) on A. This gives an algebra Aσ with an involution, which inherits a C0 (X)-algebra structure. (The C0 (X)-structure on the algebra is such that for F ∈ C0 (X) we have F.(a  b) = (F.a)  b = a  (F.b).) For vector groups it follows from the definition that the iterated parametrised strict deformation quantization (Aσ1 )σ2 ∼ = Aσ1 σ2 , with the isomorphism defined by the obvious identification map. This follows on writing down the repeated deformation product and evaluating a double integral using Parseval’s formula or the Fourier inversion formula. Alternatively we can note that the bicharacter σ −1 can always be written in Rieffel form, and then the result follows from his. Yet another approach would be to use the equivalence with Kasprzak’s formulation given below, and then to deduce it from his result. In particular, we can undeform Aσ using σ. In this more general context we can generalise Rieffel’s discussion of the action of continuous automorphisms of the group V (which give GL(V ) when V is a vector group), to allow functions S ∈ C ∞ (X, Aut(V )) and using  σ −1 (Su, Sw).(αu [a]αw [b]) dudw. a S b = V ×V

Note that the original automorphisms of V on A are also automorphisms of the deformed algebra, since  αv [a] σ αv [b] = det[T ∗ ]−1 e(T ∗ −1 u, v).(αu+v [a]αw+v [b]) dudw V ×V = det[T ∗ ]−1 e(T ∗ −1 u, v).αv [(αu [a]αw [b]) dudw =

V ×V

αv [a σ b],

since αw commutes with the C0 (X) action. We constructed the deformation as the dual of a twisted crossed product, and the reverse is also true. Given an algebra A with an action of V one can take

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the crossed product A  V with a dual action of V . Looking first at the unparametrised case, when σ is non-degenerate there is a dual multiplier σ  on V 1  defined by σ (u, σ1 η) = η(u), and similarly for e(ξ, e v) = ξ(v), and T = σ 1 ◦ e1 . 1 1 These definitions effectively mean that e is the inverse of e and similarly for σ. We can now deform the crossed product.  (a σ b)(v) = e(ξ, η) αTξ [a] αη [b] dξdη  αη [b](v − u)] dξdηdu = e(ξ, η) αTξ [a](u)αu [  = e(ξ, η)(Tξ)(u)η(v − u)a(u)αu [b(v − u)] dξdηdu  = η( e1 ξ)Tξ)(u)η(v − u)a(u)αu [b(v − u)] dξdηdu. The integral over η gives a delta function concentrated on e1 ξ = u − v, or equivalently where ξ = e1 (u − v), so that the ξ integral then gives  (a σ b)(v) = (Te1 (u − v))(u)a(u)αu [b(v − u)] , du. By definition, we have

Te1 = σ 1 e1 e1 = σ 1 ,

which leads to the reduction   1 σ (u−v))(u)a(u)αu [b(v−u)] , du = σ (u−v, u)a(u)αu [b(v−u)] , du. (aσ b)(v) = ( This is a twisted crossed product with multiplier. There is a similar parametrised version. 4. Landstad–Kasprzak and Rieffel Deformation Building on work of Landstad [11, 12], Kasprzak [9] gives an alternative dual picture of deformation theory. It is useful to give the equivalence with Rieffel deformation explicitly, as Kasprzak omits the details. (The correspondence is not obvious since the algebra elements in Rieffel’s deformation are the same and only the product changes, whereas in Kasprzak’s formulation the deformed and undeformed algebras are distinct fixed point subalgebras of the multiplier algebra of the crossed product, with different actions of V . Smoothness or some equivalent is also needed; Landstad suggests in [12] that it is sufficient to use the Fourier algebra instead of smooth subalgebras.) In the following account we use Rieffel’s notation of α rather than ρ for the automorphisms. Landstad showed in [11] that when a group V acts on an algebra A, the crossed product B = A  V has a coaction which is defined by a homomorphism λ : V → UM B (the unitary multiplier algebra). By integration λ extends to C(V ) → MB. When V is abelian there is also the dual Takai–Takesaki action α  of V , and these  interact by α ξ [λv ] = ξ(v)λv . By Takai–Takesaki duality B α V is isomorphic to A ⊗ K, reconstructing A up to stable equivalence. When B has the Landstad λ as well we can deduce a stronger duality that there is an algebra A with V -action α such that B = A α V . Kasprzak’s idea is that α ξ can be deformed by a cocycle σ for V to a new action α ξσ .

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Theorem 4.1. [9]. Let (B, λ, α ) be as above, and σ a continuous cocycle for 1  V . Setting Uξ = λ(σξ ) there is an action of V on A given by ξ [b]Uξ , α ξσ : b → Ux i∗ α which also satisfies αξσ [λv ] = ξ(v)λv . Corollary 4.2. There is an algebra Aσ and V -action ασ such that the crossed product Aσ ασ V ∼ = B = A α V . The deformed and undeformed algebras can be identified with the subalgebras of MB fixed by the action α  of V . In particular, the undeformed algebra is fixed under the dual group action on the crossed product given by α ξ [a](v) = ξ(v)a(v). The fixed points of this action are distributions concentrated on the group identity v = 0, which make sense as elements of the multiplier algebra. They give an algebra isomorphic to A, and this is just Rieffel’s construction as defined above. For the algebra deformed by σ, Uξ = δσξ1 , and by the covariance property of crossed products the adjoint action of Uξ is the same as the action of ασξ1 . We therefore have α ξσ [a](v) = ασ−1 αξ )[a(v)]] = ξ(x)ασ−1 1 [ 1 [a(v)]. ξ

ξ

Changing variable, the fixed subalgebra, where α ξσ [a] = a, therefore consists of elements a satisfying ασξ1 [a(v)] = ξ(v)a(v). In the notation of previous sections we set ξ = e1u so that σξ1 = T u, and then the condition becomes αT u [a(v)] = e(u, v)a(v). Thus the value of a(v) always lies in a particular eigenspace of the action α. (In particular, when e is an antisymmetric bicharacter a(0) must be in the fixed point algebra of αT .) In other words we can characterise the fixed point algebra elements as the elements whose value at v lies in the relevant spectral subspace ker[αT u − e(u, v)] of the action of α. To get all the eigenspaces we must do a direct integral,  or, for suitably wellbehaved functions (the smooth subalgebra), we set I(a) = a(v) dv. Theorem 4.3. When T is invertible, the product of fixed point algebra elements a and b satisfies I(a ∗ b) = I(a)  I(b). Proof. When T is invertible, the product of fixed point algebra elements is given by  I(a ∗ b) = a(u)αu [b(v − u)] dudv  = a(u)αu [b(v)] dudv  = e(T −1 u, v)a(u)b(v) dudv.

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On the other hand, under the same conditions, and with e symmetric, so that T = −T ∗  (I(a)  I(b)) = det[T ∗ ]−1 e(T ∗ −1 u, v)αu [I(a)]αv [I(b)] dudv  = det[T ∗ ]−1 e(−T −1 u, v)αu [a(y)]αv [b(x)] dxdydudv  = det[T ∗ ]−1 e(T −1 u, −v)e(T −1 u, y)a(y)e(T −1 v, x)b(x) dxdydudv  = det[T ∗ ]−1 e(T −1 u, y − v)a(y)e(T −1 v, x)b(x) dxdydudv. The integral of e(T ∗ −1 u, v − y) over u produces a delta function concentrated on v = −y, and then the v integral gives  (I(a)  I(b)) = e(T −1 y, x)a(y)b(x) dudv  = e(T −1 x, y)a(y)b(x) dudv = I(a ∗ b), showing that I defines a homomorphism from the Kasprzak deformation to the Rieffel deformation.  Standard harmonic analysis shows that this is formally an isomorphism on suitably defined smooth subalgebras. (The inverse map takes an algebra element a and does harmonic analysis of α action setting a(x) to be the component of a such that αy [a(x)] = σ −1 (x, y)a(x).) The same constructions can be carried out for C0 (X)-algebras. 5. Classifying Noncommutative Principal Torus Bundles The noncommutative principal torus bundles of Echterhoff, Nest, and OyonoOyono, whose definition was recalled in Section 1, were classified in [3] and will be outlined in this section. We also give a classification of fibrewise smooth noncommutative principal torus bundles in terms of parametrized strict deformation quantization of ordinary principal torus bundles. By Takai–Takesaki duality A(X) is Morita equivalent to C0 (X, K)  T, so the authors in [3] note that the NCPT-bundles can be classified by up to Morita equivalence by the outer equivalence classes ET (X) of T-actions, and one has the sequence 0 −→ H 1 (X, T ) −→ ET (X) −→ C(X, H 2 (T, T)) −→ 0. This leads to a classification in terms of a principal torus bundle q : Y → X, from H 1 (X, T ), and a map σ ∈ Cb (X, H 2 (T, T)), the equivalence classes of multipliers on the dual group T. These data define a noncommutative torus bundle by forming the fixed point algebra ∗ T [C0 (Y ) ⊗C0 (Z)  C (Hσ ))] with C ∗ (Hσ ) being the bundle of group C ∗ -algebras of the central extensions of T  on C0 (Y ) coming from  := H 2 (T, T) defined by σ(x) at x, the action of C0 (Z) by Z

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the composition σ ◦ q : Y → X → Z and that on C ∗ (Hσ ) from the natural action of a subgroup algebra. The bundle is a classical principal bundle when σ is the constant map to the trivial multiplier 1 (or indeed is homotopic to any constant map). A key observation is that the datum σ, or more practically an equivalence class of σ ∈ C(X, Z 2 (T, T)) can be identified with the similar map in the parametrised deformation theory, and that the Landstad–Kasprzak dual deformation theory conveniently matches the duality in the definition of NCPT-bundles with the group V = T and the dual algebra B = C0 (X, K). The analysis in [3] starts with the case of X a point, where the algebra is shown to be the twisted C∗ group algebra of T defined by the multiplier σ, or, equivalently, the deformed algebra defined by σ. The same construction can be carried out in the case of general X using our parametrised deformation constructions, and this can then be twisted using an ordinary principal T -bundle. Now given a fibrewise smooth NCPT-bundle A∞ (X) the defining deformation σ can be removed by a further deformation by σ since then one has a total deformation σσ = 1, and a constant map 1 gives an ordinary principal torus bundle up to T -equivariant Morita equivalence over C0 (X). In other words one can recover the principal torus bundle q : Y → X in this way up to T -equivariant Morita equivalence over C0 (X) via an iterated parametrized strict deformation quantization. To summarize, we have the following main result, which follows from Theorem 3.1, §4, Example 6.2, and the observations above. Theorem 5.1. Given a fibrewise smooth NCPT-bundle A∞ (X), there is a defining deformation σ ∈ Cb (X, Z 2 (T, T)) and a principal torus bundle q : Y → X such that A∞ (X) is T -equivariant Morita equivalent over C0 (X), to the parametrised ∞ strict deformation quantization of Cfibre (Y ) (continuous functions on Y that are fibrewise smooth) with respect to σ, that is, ∞ A∞ (X) ∼ (Y )σ . = Cfibre

Conversely, by Example 6.2, the parametrised strict deformation quantization of ∞ ∞ Cfibre (Y ) is the noncommutative principal torus bundle Cfibre (Y )σ . 6. Fine Structure of Parametrised Strict Deformation Quantization We have seen in Theorem 5.1 that all fibrewise smooth NCPT-bundles are just parametrised strict deformation quantizations of ordinary principal torus bundles. We will use this to write out the fine structure of fibrewise smooth NCPT-bundles. Example 6.1. We begin by recalling the construction by Rieffel [15] realizing the smooth noncommutative torus as a deformation quantization of the smooth functions on a torus T = Rn /Zn of dimension equal to n. Recall that any translation invariant Poisson bracket on T is just  ∂a ∂b {a, b} = θij , ∂xi ∂xj for a, b ∈ C ∞ (T ), where (θij ) is a skew symmetric matrix. The action of T on itself is given by translation. The Fourier transform is an isomorphism between C ∞ (T ) and S(Tˆ), taking the pointwise product on C ∞ (T ) to the convolution product on S(Tˆ ) and taking differentiation with respect to a coordinate function to multiplication

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by the dual coordinate. In particular, the Fourier transform of the Poisson bracket gives rise to an operation on S(Tˆ ) denoted the same. For φ, ψ ∈ S(Tˆ), define  ψ(p1 )φ(p2 )γ(p1 , p2 ) {ψ, φ}(p) = −4π 2 p1 +p2 =p

where γ is the skew symmetric form on Tˆ defined by  θij p1,i p2,j . γ(p1 , p2 ) = For  ∈ R, define a skew bicharacter σ on Tˆ by σ (p1 , p2 ) = exp(−πγ(p1 , p2 )). Using this, define a new associative product  on S(Tˆ),  ψ(p1 )φ(p2 )σ (p1 , p2 ). (ψ  φ)(p) = p1 +p2 =p

This is precisely the smooth noncommutative torus A∞ σ . The norm || · || is defined to be the operator norm for the action of S(Tˆ) on 2 ˆ L (T ) given by  . Via the Fourier transform, carry this structure back to C ∞ (T ), to obtain the smooth noncommutative torus as a strict deformation quantization of C ∞ (T ), [15] with respect to the translation action of T . Example 6.2. We next generalize the above to the case of principal torus bundles q : Y → X of rank equal to n. Note that fibrewise smooth functions on Y decompose as a direct sum,   ∞ ∞ Cfibre (Y ) = Cfibre (X, Lα ) α∈Tˆ  φα φ= α∈Tˆ ∞ Cfibre (X, Lα )

∞ where is defined as the subspace of Cfibre (Y ) consisting of functions which transform under the character α ∈ Tˆ , and where Lα denotes the associated line bundle Y ×α C over X. That is, φα (yt) = α(t)φα (y), ∀ y ∈ Y, t ∈ T . The direct sum is completed in such a way that the function Tˆ  α → ||φα ||∞ ∈ R ∞ (Y ), it is easy to extend to this case, is in S(Tˆ). In this interpretation of Cfibre the explicit deformation quantization given in the previous example, which we ∞ now briefly outline. For φ, ψ ∈ Cfibre (Y ), define a new associative product  ∞ on Cfibre (Y ) as follows. For y ∈ Y , α, α1 , α2 ∈ Tˆ , let  (ψ  φ)(y, α) = ψ(y, α1 )φ(y, α2 )σ (q(y); α1 , α2 ), α1 α2 =α

using the notation ψ(y, α1 ) = ψα1 (y) etc., and where σ ∈ Cb (X, Z 2 (Tˆ, T)) is a continuous family of bicharacters of Tˆ such that σ0 = 1, which is part of the data that we start out with. We remark that one way to get such a σ is to choose a continuous family skew-symmetric forms on Tˆ , γ : X −→ Z 2 (Tˆ, R), and define σ = exp(−πγ). In the case of the principal torus bundle Y , we note that the 2 vert T Y , which can vertical tangent bundle of Y has a Poisson structure, i.e. γ ∈ be naturally interpreted as a continuous family of symplectic structures along the fibre, that is, γ is of the sort considered just previously. We denote the deformed ∞ algebra by Cfibre (Y ) , and using §3, we can realize it as a parametrised strict

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∞ deformation quantization of Cfibre (Y ). Since the construction is T -equivariant, ∞ ∞ Cfibre (Y ) has a T -action that is induced from the given T -action on Cfibre (Y ).

Example 6.3. We next consider the example which was one of the inspirations for Theorem 5.1. Although it is a special case of the previous example, and is probably also treated elsewhere, nevertheless we think that it is worthwhile to treat in our context. Consider a 3-dimensional torus, which we write as S 1 × T , where T is a two dimensional torus. Let {·, ·} denote the Poisson bracket on S 1 × T coming from T and trivial on S 1 . Then this Poisson bracket is invariant under the T action on S 1 × T , where T acts trivially on S 1 and via translation on itself. Here the fibres are T . As in the example above, we construct a strict deformation quantization of ∞ Cfibre (S 1 × T ). Taking the partial Fourier transform in the T -variables, we obtain ∞ (S 1 × T ) and Sfibre (S 1 × Tˆ). In the notation of the an isomorphism between Cfibre previous example, for φ, ψ ∈ Sfibre (S 1 × Tˆ), define  ψ(y, p1 )φ(y, p2 )σ (y; p1 , p2 ). (ψ  φ)(y, p) = p1 +p2 =p

∼ T is the family of bicharacters of Tˆ given where σ : S = R/Z −→ H (Tˆ, T) = by σ (y; p1 , p2 ) = exp(−πyγ(p1 , p2 )). 1

2

Here γ is defined as in Example 6.1. This gives us a family of smooth noncommutative tori, that is,  Sfibre (S 1 × Tˆ) = A∞ σ (y) y∈S 1

which in turn can be identified with (when  = 1) the fibrewise Schwartz subalgebra of the 3-dimensional integer Heisenberg group, HeisZ . That is, the norm closure of Sfibre (S 1 × Tˆ)=1 is isomorphic to C ∗ (HeisZ ). On the other hand, using the results of §3, we see that Sfibre (S 1 × Tˆ) is a ∞ parametrised strict deformation quantization of Cfibre (S 1 × T ). Example 6.4. Motivated by Theorem 5.1 and an example in [15], we define noncommutative non-principal torus bundles as follows. Let ρ : π1 (X) → Sp(2n, Z) be a representation of the fundamental group, T = R2n /Z2n be the torus and × T )/π1 (X), where q : Yρ → X be the non-principal torus bundle given by Yρ = (X we observe that the symplectic group is a subgroup of the automorphism group of −→ X via deck T and Γ = π1 (X) acts on T via ρ and on the universal cover r : X transformations. × T coming from T and trivial on Let {·, ·} denote the Poisson bracket on X × T , therefore X. Then this Poisson bracket is invariant under the T action on X descending to a Poisson bracket on X, denoted by the same symbol. As in the ∞ × T ), example above, we construct a strict deformation quantization of Cfibre (X × T that are smooth along which is the algebra of continuous functions on X the fibres. Taking the partial Fourier transform in the T -variables, we obtain an ∞ × T ) and Sfibre (X × Tˆ ). In the notation of the isomorphism between Cfibre (X ˆ previous example, for φ, ψ ∈ Sfibre (X × T ), define  ψ(y, p1 )φ(y, p2 )σ (r(y); p1 , p2 ). (ψ  φ)(y, p) = p1 +p2 =p

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where σ : X −→ Z 2 (Tˆ, T) is a continuous family of bicharacters of Tˆ, which is part of the data that we start out with. ∞ Then transporting this structure back to Cfibre (X×T ) gives a strict deformation quantization such that Γ = π1 (X) acts properly on it, cf. [15]. We denote the ∞ ∞ (X×T ) . The fixed point subalgebra Cfibre (X×T )Γ of the deformed algebra as Cfibre deformed algebra is then the desired parametrised strict deformation quantization, ∞ ×T )Γ = C ∞ (Yρ )σ , where we note that C ∞ (X ×T )Γ = C ∞ (Yρ ). This Cfibre (X   fibre fibre fibre is our definition of a noncommutative non-principal torus bundle. To summarize, it is determined by two pieces of data: • ρ ∈ Hom(π1 (X), Sp(2n, Z)); • σ ∈ C(X, Z 2 (Tˆ, T)), that is, a continuous family of bicharacters of Tˆ . Appendix A. Factors of Automorphy Appendix C to [1] introduced a method for lifting algebra bundles to a contractible universal cover and encoding information about the Dixmier–Douady class in a factor of automorphy  j. This also fits into a parametrised deformation picture, but with the further generalisation that the group Γ now acts on the parameter space X. The cocycle  j for a lifting can be reconstructed from the Dixmier–Douady class δ ∈ H 3 (X, Z) ∼ H 3 (Γ, Z), by first finding τ(k1 , k2 , x) (k1 , k2 ∈ Γ, x ∈ X) with d τ = δ, defining τ = exp(2πi τ ), and then finding  j(k, x) satisfying  j(k2 , x) = τ (k1 , k2 , x) j(k1 k2 , x), j(k1 , k2 x) which can be achieved by a modified τ -inducing construction, which gives  j in terms of τ . We know that τ is a C0 (X)-valued cocycle satisfying the cocycle condition τ (k1 k2 , k3 )αk−1 [τ (k1 , k2 )] = τ (k1 , k2 k3 )τ (k2 , k3 ) 3 where α just gives the translation action on C0 (X), and similarly suppressing the X-dependence in  j allows us to rewrite its cocycle condition as α−1 [ j(k1 )] j(k2 ) = τ (k1 , k2 ) j(k1 k2 ). k2

The cocycle condition on τ can also be written as αk3 [τ (k1 k2 , k3 )]τ (k1 , k2 ) = αk3 [τ (k1 , k2 k3 )]αk3 [τ (k2 , k3 )] so setting U (k1 ) : k → αk [τ (k1 , k)] we get U (k1 k2 )(k3 )τ (k1 , k2 ) = αk−1 [U (k1 )(k2 k3 )]U (k2 )(k3 ). 2 We can lift the automorphism αk2 to α k−1 [U (k1 )](k3 )] = αk−1 [U (k1 )(k2 k3 )] 2 2 and then

k−1 [U (k1 )]U (k2 ), U (k1 k2 )τ (k1 , k2 ) = α 2 the type of cocycle condition to be satisfied by  j. To compare these with the Landstad–Kasprzak construction we take Γ = V , j as a map from K ρ the left translation (Lk f )(x) = f (k−1 x). Now we think of  j(k, k−1 x), noting that the to unitary multipliers on C0 (X), and take U (k) : x →  j(k2 , x) = τ (k1 , k2 , x) j(k1 k2 , x) gives cocycle condition  j(k1 , k2 x) U (k1 ) ρ(k1 )[U (k2 )] = τ (k1 , k2 )U (k1 k2 ),

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precisely the condition arising in deformation (though the new ingredient is that K acts on the X argument of τ ). We note also that the predual algebra in the Landstad theory is the generalised fixed point algebra A = B ρ.  When this paper was completed, Marc Rieffel pointed out to us that [18] deals with an interesting bundle situation with varying deformation form, but where he was able to transform it in that special case into a constant-form situation. This is of course not possible to arrange in general. References [1] P. Bouwknegt, K. C. Hannabuss, and V. Mathai, C∗ -algebras in tensor categories, Clay Mathematics Proceedings. 12 (2009) 39 pages, (in press). [math.QA/0702802] ebres et g´ eometrie diff´ erentielle, C.R. Acad. Sci. Paris, Ser. A-B, 290, [2] A. Connes, C∗ -alg` (1980) no. 13, 599–604. [3] S. Echterhoff, R. Nest, and H. Oyono-Oyono, Principal non-commutative torus bundles, Proc. London Math. Soc. (3) 99, (2009) 1–31. [4] S. Echterhoff and D. P. Williams, Crossed products by C0 (X)-actions, J. Funct. Anal. 158 (1998) no. 1, 113–151. [5] K.C. Hannabuss, Representations of nilpotent locally compact groups, J. Funct. Anal. 34 (1979) no. 1, 146–165. [6] A. an Huef, I. Raeburn, and D.P. Williams, Functoriality of Rieffel’s generalised fixed point algebras for proper actions, [arXiv:0909.2860]. [7] S. Kaliszewski and J. Quigg, Categorical Landstad duality for actions, Indiana Math. J. 58 (2009) 415–441. [8] G. Kasparov, Equivariant K-theory and the Novikov conjecture, Invent. Math. 91, (1988) 147–201. [9] P. Kasprzak, Rieffel deformation via crossed products, J. Funct. Anal. 257 (2009) 1288–1332. [10] A. Kleppner, Multipliers on abelian groups, Math. Ann. 158 (1965) 11–34. [11] M. B. Landstad, Duality theory for covariant systems, Trans. Amer. Math. Soc. 248 (1979) 223–267. [12] M. B. Landstad, Quantization arising from abelian subgroups, Internat. J. Math. 5 (1994) 897–936. [13] V. Mathai and J. Rosenberg, T-duality for torus bundles via noncommutative topology, Commun. Math. Phys., 253 no. 3 (2005) 705–721. [hep-th/0401168] [14] V. Mathai and J. Rosenberg, T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group, Adv. Theor. Math. Phys., 10 no. 1 (2006) 123–158. [hep-th/0508084] [15] M. A. Rieffel, Deformation quantization for actions of Rd , Memoirs of the Amer. Math. Soc. 106 (1993), no. 506, 93 pp. [16] M. A. Rieffel, Quantization and C∗ -algebras, Contemporary Math. 167, (1994), 67–97. [17] M. A. Rieffel, Applications of strong Morita equivalence to transformation group C ∗ -algebras, Proceedings of Symposia in Pure Mathematics, 38 (1982) Part I, 299–310. [18] M. A. Rieffel, On the operator algebra for the space-time uncertainty relations, Operator algebras and quantum field theory (Rome, 1996), 374–382, Internat. Press, Cambridge, MA, 1997. (Keith Hannabuss) Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, and Balliol College, Oxford, OX1 3BJ, England E-mail address: [email protected] (Varghese Mathai) Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681762

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

A Survey of Noncommutative Yang-Mills Theory for Quantum Heisenberg Manifolds Sooran Kang Abstract. In this paper, we give a short overview of noncommutative YangMills theory developed by Connes and Rieffel and discuss Yang-Mills theory for quantum Heisenberg manifolds.

Introduction It has been over 50 years since the publication of the fundamental paper of [YM], when Yang and Mills introduced a new mathematical framework to describe the interactions among elementary particles. At the time, the importance of their idea was not fully recognized, because a Yang-Mills field is massless and, such a theory was incompatible to experiments. In the late 1960s, these problems were solved in [Hig], and now their idea is at the heart of theoretical physics including quantum field theory, string theory and the theory of gravitation. As mentioned elsewhere in this volume, the interplay between mathematics and physics has provided many interesting results since the birth of quantum physics. In fact, it was recognized soon by mathematicians that Yang-Mills non-abelian gauge theory is related to connections on fiber bundles, and the paper [WuY] providing a dictionary between the two different languages of physics and mathematics for the same subject was published by Wu and Yang shortly after. This recognition caused the rapid development of new realms of mathematics, especially in the study of three- and four-dimensional manifolds, and conversely, the existing mathematics provided new insights and rigorous mathematical formulations for this new area of physics. There has been a great deal of effort to extend Yang-Mills theory to various areas of mathematics in last twenty years, and some remarkable works can be found in [AtB], [CR], [Donal1], [SW], [Tau], [UY]. Historically, the motivation behind noncommutative geometry can be found in the development of quantum theory in the 1920s, when a noncommutative quantum mechanical system was constructed. However, the modern origins of noncommutative geometry first appeared in the algebraic version of differential geometry constructed by J. K. Koszul around 1960 in [Kosz], and the subject now known as noncommutative differential geometry was developed extensively by French mathematician Alain Connes in the early 1980s in [C1], [C2]. Objects in noncommutative 1991 Mathematics Subject Classification. Primary 46L87 ; Secondary 58B34. Key words and phrases. The Yang-Mills functional, quantum Heisenberg manifolds, Laplace’s equation, Morita equivalence. 1

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geometry are usually noncommutative deformations, sometimes called deformation quantizations, of commutative counterparts, and one studies the geometric theory of noncommutative algebraic structures, such as connections and curvatures of noncommutative spaces. A typical example of a noncommutative space is the noncommutative torus, which can be viewed as the deformation quantization of the function algebra of the ordinary two-torus. This example has been studied extensively and used as a test case for more complicated situations. Readers can find more about noncommutative tori in [Rf2], [Rf5], [Rf8], [RfSch], and some applications in [CR], [CDS], [KoSch], [KoSch2], [Ros]. As we find in the papers [C2], [C3], [CDS], [LLS], [Lnm], [SW], noncommutative geometry has naturally arisen in the development of recent quantum physics. In particular, the framework of noncommutative geometry can be found in the frameworks for open string theory and M(atrix) theory. More references can be found in [CDS], [KoSch] and [SW]. What should be noted here is that the core idea in these papers is based on Connes’ and Rieffel’s noncommutative Yang-Mills theory on the noncommutative torus. As mentioned in [SW], “the framework of noncommutative Yang-Mills theory seems very powerful since the T-duality acts within the noncommutative YangMills framework”. In other words, Morita equivalence of the noncommutative torus plays a key role in these theories. After the fundamental work of Connes and Rieffel in the mid1980s, no example of noncommutative Yang-Mills theory on any other different noncommutative manifold than the noncommutative torus within the same noncommutative framework has been completely understood to date. This may be because the proofs in [CR] depend highly on two specific properties of the noncommutative torus: a symmetry of the projective modules over the noncommutative torus, and a well-known relation between its generators. Using the same framework developed in [CR], the author in her Ph.D thesis [K1] has attempted to extend Yang-Mills theory to quantum Heisenberg manifolds, which are different type of noncommutative C ∗ -algebras first constructed by Marc Rieffel [Rf3]. The main theorem in this thesis what will appear in [K2] describes a certain family of connections on a projective module over the quantum Heisenberg manifold that give rise to critical points of the Yang-Mills functional on the quantum Heisenberg manifold, and that are also related to Laplace’s equation on quantum Heisenberg manifolds. Readers can find the full details in [K2]. Morita equivalence for quantum Heisenberg manifolds and construction of (finitely generated) projective modules over quantum Heisenberg manifolds have been studied by Abadie and her collaborators in [Ab1]–[Ab4] and [AEE]. Some other works related to quantum Heisenberg manifolds can be found in [Ch], [ChS], [ConDu1], [ConDu2], [Li], [W]. Also, Yang-Mills for commutative Heisenberg manifolds can be found in [Ur], and an explicit example of compatible connections on a quantum Heisenberg manifold is given in [L]. The main difference between the study of Yang-Mills theory for quantum Heisenberg manifolds and that of the noncommutative torus is the following. First of all, the projective modules over the quantum Heisenberg manifolds, denoted by c, c, Dμν , were constructed by realizing the algebra Dμν as generalized fixed point algebras of certain crossed product C ∗ -algebras in [Ab2], while projective modules over the noncommutative torus were constructed from a foliation of the ordinary

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two-torus. Also, in [K2] a particular Grassmannian connection is used to produce a compatible connection on the projective module, and the method of finding such a nontrivial connection is related to finding Rieffel projections in noncommutative tori, which is a different approach from that used by Connes and Rieffel in [CR] and [Rf7]. The last step in finding actual solutions to the Yang-Mills equation is related to solving an elliptic partial differential equation, which is also very different from the methods of [CR] and [Rf7]. On the other hand, the theory of Morita equivalence still plays an important role in our study, in particular in establishing a relationship between a particular family of critical points of the Yang-Mills functional on quantum Heisenberg manifolds and a set of solutions to Laplace’s equation on quantum Heisenberg manifolds. Recall that the Laplacian is the leading term for the coupled set of equations making up the Yang-Mills equation. This paper is organized as follows. In Section 1, we discuss noncommutative Yang-Mills theory as originally developed by Connes and Rieffel, based on their paper [CR]. In Section 2, we describe Rieffel’s quantum Heisenberg manifolds and Morita equivalence for quantum Heisenberg manifolds. In Section 3, we discuss the Yang-Mills functional and Laplace’s equation on quantum Heisenberg manifolds described in [K2]. 1. Connes’ and Rieffel’s Noncommutative Yang-Mills Theory To formulate Yang-Mills theory on a noncommutative manifold, we need a finitely generated projective module over a noncommutative counterpart of the manifold, usually a deformation quantization of the manifold. We also need a notion of compatible connection and curvature on the finitely generated projective module, which will provide geometric properties. (For brevity, we will use “projective” to mean “finitely generated projective” from now on). We then define the Yang-Mills functional on the set of compatible connections. This functional corresponds to the classical Yang-Mills functional, which can be also interpreted as the Energy functional in physics. We then analyze the nature of the set of critical points of the Yang-Mills functional, Y M . We are particularly interested in the critical points where Y M attains its minimum. We describe Connes’ and Rieffel’s noncommutative Yang-Mills theory introduced in [CR] as follows. Let A be a unital C ∗ -algebra. Let α be an action of Lie group G on A, and let g be the corresponding Lie algebra with basis {Z1 , . . . , Zn }. Then we can give a smooth structure on A using the Lie group action α, defined by A∞ = {a ∈ A | g → αg (a) is smooth in norm}. The infinitesimal form of α gives an action, δ, of the Lie algebra g of G, as a Lie algebra of derivations on A∞ . Remark 1.1. When we consider C ∞ (M ) as a commutative algebra corresponding to a manifold M , it is known that there is a natural isomorphism between the set of all vector fields on M and the set of all derivations of C ∞ (M ) as Lie algebras. Thus, for given noncommutative C ∗ -algebra A, we can consider the Lie algebra of derivations on A∞ as a proper analogue of vector field on M . Let Ξ be a (right) projective module over a unital C ∗ -algebra A. As shown in Lemma 1 of [C1], given a finitely generated projective A-module Ξ, there is a dense A∞ -submodule Ξ∞ ⊂ Ξ, such that Ξ∞ is finitely generated and projective over A∞ and Ξ is isomorphic to Ξ∞ ⊗A∞ A. Furthermore, Ξ∞ is unique up to isomorphism as an A∞ -module. We will denote Ξ∞ and A∞ by Ξ and A for notational simplicity from now on.

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Also, we can always equip Ξ with an A-valued positive definite inner product ·, ·A , called a Hermitian metric, such that ξ, η∗A = η, ξA , ξ, ηaA = ξ, ηA a, for ξ, η ∈ Ξ and a ∈ A. See the details of this construction for C ∗ -modules in [Rf1] and [Lnc]. Definition 1.2. [C1] Let Ξ, A and g be as above. A connection ∇ is a linear map from Ξ to Ξ ⊗ g∗ such that ∇X (ξa) = (∇X (ξ))a + ξ(δX (a)), for all X ∈ g, ξ ∈ Ξ and a ∈ A. We say that the connections are compatible with the Hermitian metric if δX (ξ, ηA ) = ∇X ξ, ηA + ξ, ∇X ηA . We denote the set of compatible connections by CC(Ξ). Note that the connection ∇ in the above definition is defined on Ξ that has a value in Ξ ⊗ g∗ . i.e. ∇(ξ) : g → Ξ so that ∇(ξ)(X) ∈ Ξ for ξ ∈ Ξ and X ∈ g. We then denote ∇(ξ)(X) by ∇X (ξ). According to Connes’ theory, we can always define a compatible connection on a projective module over A as follows. For a given unital C ∗ -algebra A and a projection Q ∈ A, QA is a projective right A-module in an obvious way. As described in [C1], we define a connection ∇0 on QA, called the “Grassmannian connection”, by ∇0X (ξ) = QδX (ξ) ∈ QA,

for all

ξ ∈ QA and

X ∈ g.

Obviously, this is a compatible connection with the canonical Hermitian metric on QA, such that ξ, η = ξ ∗ η for ξ, η ∈ QA. For given right A-module Ξ, let E = EndA (Ξ). Then the following facts are shown in [CR]. If ∇ and ∇ are any two connections, then ∇X − ∇X is an element of E, for each X ∈ g. If ∇ and ∇ are both compatible with the Hermitian metric, then ∇X − ∇X is a skew-symmetric element of E for each X ∈ g. Thus, once we have a compatible connection ∇, every other compatible connection ∇ is of the form ∇ + μ, where μ is a linear map from g into E s , the set of skew-symmetric element of E, such that μX ∗ = −μX for X ∈ g. The curvature of a connection ∇ is defined to be the alternating bilinear form Θ∇ on g, given by Θ∇ (X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] , for X,Y ∈ g. It is not hard to check that the values of Θ are in E for a given connection ∇, and the values of Θ are in E s if a connection ∇ is compatible with the Hermitian metric. For given A-valued inner product ·, ·A , we can define an E-valued inner product ·, ·E by ξ, ηE ζ = ξη, ζA , for ξ, η, ζ ∈ Ξ. So there is a natural bimodule structure (left E-right A) on Ξ. Now we assume that A has a faithful α-invariant trace, i.e. τ (δX (a)) = 0 for a ∈ A and X ∈ g. Then τ determines a faithful trace, τE , on E = EndA (Ξ), defined by τE (ξ, ηE ) = τ (η, ξA ).

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To define the Yang-Mills functional on CC(Ξ), we need a bilinear form1 on the space of alternating 2-forms with values in E. Let {Z1 , · · · , Zn } be a basis for g as stated before. We define a bilinear form {·, ·}E by  Φ(Zi ∧ Zj )Ψ(Zi ∧ Zj ), {Φ, Ψ}E = i 0 by Kj (A) = πj−1 (U∞ A). Thus there is a natural stabilization map πj (U A) −→ πj (U∞ A) ∼ = Kj+1 (A) 1Note that Cech ˇ and singular cohomology theories agree on finite complexes but not on all compact metric spaces. This was well-known in the 1940’s (Spanier mentions it in passing in a 1948 paper). The strong wedge of a countable number of copies of S 2 provides a dramatic example; see Barratt and Milnor. [2]

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FROM RATIONAL HOMOTOPY TO K-THEORY

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which we denote σ integral : π∗ (U A) −→ K∗+1 (A). At this point the functor Kj (A) is defined for j ≥ 0. Bott periodicity implies that K∗ (A) ∼ = K∗+2 (A) for ∗ > 0. So it is natural to regard K∗ (A) as a Z/2-graded theory (and we confess to having done so in the past.) However, for our present purpose it will be vital NOT to do so. That is, in this note we regard K-theory as defined for all non-negative integers. This allows us, for instance to distinguish between Morita-equivalent C ∗ algebras.2 Ideally we would like to compute the image of σ integral . However, this is probably as difficult as computing πj (U A), which is out of reach, as noted. Hence we have been focusing on the rationalization of this group which, while far simpler and hence carrying less information, has the advantage of being computable. Thus we focus on the rational stabilization map σ : π∗ (U A) ⊗ Q −→ K∗+1 (A) ⊗ Q. The first case to consider is A = Mn . Then we are looking at σ : π∗ (Un ) ⊗ Q −→ K∗+1 (Mn ) ⊗ Q. These are graded rational vector spaces which are abstractly isomorphic (after degree shift, of course) in degrees 1, . . . , 2n. Now π∗ (Un ) ⊗ Q ∼ = s1 , s3 , . . . , s2n−1  with |s2j−1 | = 2j − 1 and of course σ is linear. So σ is determined by its image on the basis vectors, and indeed σ(s2j−1 ) = 0 for each j. So there is an isomorphism ∼ =

σ : π2j−1 (Un ) ⊗ Q −→ K2j (Mn ) ⊗ Q for j = 1, . . . n and hence the range of σ is exactly   K2 (Mn ) ⊗ Q, K4 (Mn ) ⊗ Q, . . . , K2n (Mn ) ⊗ Q . In particular, if we regard the range of σ as an invariant of A with values in Z+ -graded K-theory, then this invariant distinguishes between Mn and Mk for n = k. Note that we are NOT saying that the classes σ(s2j−1 ) are actually in K∗ (Mn ). Consider the diagram π∗ (Un ) ⏐ ⏐ h 

σ

−−−−→ π∗ (Un ) ⊗ Q −−−−→ K∗+1 (Mn ) ⊗ Q ⏐ ⏐ h

H∗ (Un ; Z) −−−−→ H∗ (Un ; Q) 2J. F. Adams [1] does the same thing for a somewhat different reason. Having proved that

the Adams operations do not commute with the Bott periodicity maps in Corollary 5.3, he notes on page 619: “Owing to the state of affairs revealed by Corollary 5.3, we shall be most careful not to identify KC−n−2 (X, Y ) with KC−n (X, Y ). . . We therefore regard KC−n (X, Y ) as graded over Z, not over Z/2 . . . ”

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KLEIN, SCHOCHET, AND SMITH

where the vertical maps are the Hurewicz maps. Filling in three well-understood groups, this diagram becomes π∗ (Un ) ⏐ ⏐ h 

σ

−−−−→ s1 , s3 , . . . , s2n−1  −−−−→ K∗+1 (Mn ) ⊗ Q ⏐ ⏐ h

ΛZ (g1 , g3 , . . . g2n−1 ) −−−−→ ΛQ (g1 , g3 , . . . g2n−1 ) where ΛR denotes the exterior algebra over the ring R with given generators and h(sj ) = gj for each j. The map h is not onto the generators. 4. Second Example: X = S 3 In order to make Theorem A concrete, focus upon the special case where X = S 3 and contrast our results with those of Jonathan Rosenberg [5]. Theorem A asserts that the space of unitaries U Aζ is rationally equivalent to the space of functions F (S 3 , Un ). Actually, though, we know much more in this case. Any finite-dimensional P U (n)-bundle ζ has Dixmier-Douady invariant of finite order, and since H 3 (S 3 ; Z) ∼ = Z this implies that the invariant vanishes for ζ over S 3 . Thus ζ ∼ = End(V ) for some complex vector bundle V over S 3 . This bundle must also be trivial, since it is classified by an element of π3 (BUn ) = 0. Thus ζ must be a trivial bundle, and so U Aζ is homeomorphic to F (S 3 , Un ) even before rationalization. So let us examine this space carefully. Note first that its path components are interesting. Fix a base point x0 for S 3 . There is a standard fibration p

F• (S 3 , Un ) −→ F (S 3 , Un ) −→ Un where p(f ) = f (xo ) and F• denotes base point preserving maps. This fibration has a section (send a point u ∈ Un to the constant map S 3 → Un that takes every element of S 3 to u) and hence there are split short exact sequences in each degree p∗

0 → π∗ (F• (S 3 , Un )) −→ π∗ (F (S 3 , Un )) −→ π∗ (Un ) → 0. In particular, since Un is connected, π0 ( F (S 3 , Un ) ) ∼ = π0 (F• (S 3 , Un )) ∼ = π3 (Un ) ∼ = Z. The generator of π3 (Un ) ∼ = Z is given by the natural composition ∼ SU2 → U2 → Un S3 = where U2 is included in Un via u → u ⊕ 1. In higher degrees we obtain for each k the split short exact sequence p∗

0 → πk (F• (S 3 , Un )) −→ πk (F (S 3 , Un )) −→ πk (Un ) → 0. This helps us understand the result for Theorem A, which states (in this case) that π∗ ( F (S 3 , Un ) ) ⊗ Q ∼ = H ∗ (S 3 ; Q) ⊗ s1 , s3 , . . . , s2n−1 . Write

H ∗ (S 3 ; Q) =  1, x3  with x3 denoting the generator in dimension 3. Then π∗ (F (S 3 , Un )) ⊗ Q is spanned by two types of classes. There are the classes 1 ⊗ s2j−1 of degree 2j − 1 and the classes x3 ⊗ s2j−1 of degree 2j − 1 − 3 = 2j − 4. The short exact sequence p∗

0 → π∗ (F• (S 3 , Un )) ⊗ Q −→ π∗ (F (S 3 , Un )) ⊗ Q −→ π∗ (Un ) ⊗ Q → 0

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FROM RATIONAL HOMOTOPY TO K-THEORY

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becomes 0 → x3 ⊗s3 , . . . x3 ⊗s2n−1  −→ π∗ (F (S 3 , Un ))⊗Q −→ 1⊗s1 , . . . 1⊗s2n−1  → 0 Note that the class x3 ⊗ s1 is not present since it would have negative degree. Now, what happens when we map to K-theory? First, the fact that U Aζ is homeomorphic to F (S 3 , Un ) implies that K∗ (Aζ ) ∼ = K∗ (C(S 3 )) ∼ = K ∗ (S 3 ) ∼ =Z in every degree. The generator in even degree is simply the class of the trivial line bundle (i.e. the one dimensional trivial projection) and the generator in odd degree corresponds to the Bott generator in that degree. An easy naturality argument using the result of the previous section implies that the class 1 ⊗ s2j−1 maps to the class in K2j (Aζ ) ⊗ Q that corresponds to a multiple of the one-dimensional trivial projection in K0 (Aζ ) under the Bott map as is the case for A = Mn . The other classes are more interesting. The class x3 ⊗s2j−1 has degree 2j−4 and hence must map to K2j−3 (Aζ ). The first example is x3 ⊗s3 mapping to K1 (Aζ )⊗Q and the last is x3 ⊗ s2n−1 mapping to K2n−3 (Aζ ) ⊗ Q. To summarize: if X = S 3 then, independent of the (finite-dimensional) bundle ζ, the image of the stabilization map σ : π∗ (U Aζ ) ⊗ Q −→ K∗+1 (Aζ ) ⊗ Q has basis elements σ(1 ⊗ s2j−1 ) ∈ K2j (Aζ ) ⊗ Q

j = 1, . . . , n

and j = 3, . . . n. σ(x3 ⊗ s2j−1 ) ∈ K2j−3 (Aζ ) ⊗ Q Thus the image of the stabilization map σ consists of the groups   K0 (Aζ ) ⊗ Q, K1 (Aζ ) ⊗ Q, . . . , K2n−3 (Aζ ) ⊗ Q, K2n−2 (Aζ ) ⊗ Q, K2n (Aζ ) ⊗ Q and no others. Turning to the infinite-dimensional situation, the first thing to note is that Aζ is no longer unital; in fact it is stable. We add a unit to Aζ in the canonical fashion to obtain a short exact sequence 0 → Aζ −→ A+ ζ −→ C → 0 and then define

  1 U Aζ = Ker U (A+ ζ ) −→ U C = S .

Stability implies that

Kn (A) ∼ = πn−1 (U Aζ ). and we see that the homotopy groups are periodic. If the Dixmier-Douady invariant is trivial then Aζ ∼ = C(S 3 , K) and so U Aζ ∼ = 3 F (S , U K). Thus we see directly that ∼ πn (F (S 3 , U K)) πn (U Aζ ) = and an easy homotopy analysis shows this group to be Z for each n. This corresponds to K 0 (S 3 ) = K 1 (S 3 ) ∼ = Z. If the Dixmier-Douady class is s times the generator of H 3 (S 3 ; Z) then Rosenberg [5, 6] gives a spectral sequence argument to conclude that K0 (Aζ ) = 0 and K1 (Aζ ) ∼ = Z/s. In particular, if the Dixmier-Douady class is a generator then

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KLEIN, SCHOCHET, AND SMITH

K1 (Aζ ) = 0 and so Aζ is K-contractible. Thus if the Dixmier-Douady class of ζ is a generator then π∗ (U Aζ ) = 0, and for any non-trivial Dixmier-Douady class we have π∗ (Aζ ) ⊗ Q = 0. We hope to study Rosenberg’s result from the perspective of rational homotopy in subsequent work. 5. If One Grading is Good, then Two Gradings... We note that Theorem A gives a natural bigrading to π∗ (U Aζ ) ⊗ Q. In the X = S 3 example considered above, the classes 1 ⊗ s2j−1 have bidegree (0, 2j − 1) and total degree 2j − 1, and the classes x3 ⊗ s2j−1 have bidegree (−3, 2j − 1) and total degree −3 + (2j − 1) = 2j − 4. The bigrading has quite a bit of naturality associated with it. In the simplest case, with A = C(X) ⊗ Mn it is keeping track both of the cohomological degree and of the size of the matrix! This bidegree is of course completely lost when passing to Z/2-graded K-theory. For an elementary example, consider the case X = CP 2 . The rational (indeed, integral in this case) cohomology ring is a truncated polynomial algebra on a generator c ∈ H 2 (CP 2 ) with c3 = 0. Take A = F (CP 2 , M3 ). Then the classes c ⊗ s3 and c2 ⊗ s5 have different bidegrees, and hence are distinguished, but they have the same total degree and hence have the same degree when one passes to K-theory, even when K-theory is Z+ -graded! Now, take A = C(X) ⊗ Mn . One might reasonably ask for a calculation of [A, A], the homotopy classes of unital ∗-homomorphisms [A, A]. One source of such maps are the induced maps from functions f : X → X and so one might hope to determine [A, A] as some functor of [X, X]. (Determining [X, X] itself is extremely difficult even in fairly simple cases.) Another source of maps are the induced maps from ∗-homomorphisms Mn → Mn . These are known, of course. Non-trivial maps must be isomorphisms, every isomorphism is inner, and hence every such map is given by conjugation by a unitary, so we are back to P Un . The real problem is that there are other maps besides these two types that intertwine the two. There is a natural commuting diagram [A, A] ⏐ ⏐φ 

φK

−−−−→

EndZ/2 (K ∗ (X)) ⏐ ⏐ 



End∗∗ π∗ (U A) ⊗ Q −−−−→ EndZ/2 (K ∗ (X) ⊗ Q). It might seem at first glance that one would be better off using φK as an invariant rather than φ. This is illusory. The problem is that the map φK is only defined if we understand EndZ/2 (K ∗ (X)) to mean endomorphisms of Z/2-graded abelian groups since if A is non-commutative then there is no ring structure on K∗ (A). We can say something about the map φ. If the map h : A → A arises from a map f : X → X then of course φ(h) = f ∗ ⊗ 1. If h arises from conjugation by a unitary then φ(h) is just the identity, since Un is path-connected. We hope to compute φ(h) in some intertwined cases and expect it to be a helpful invariant. This is work in progress. References [1] J. F. Adams, Vector fields on spheres, Annals of Math. 75 (1962), 603 - 632. [2] M. G. Barratt and J. Milnor, An example of anomolous singular homology, Proc. Amer. Math. Soc. 13 (1962), 293–297.

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[3] G. Lupton, N. C. Phillips, C. L. Schochet, S. B. Smith, Banach algebras and rational homotopy theory, Trans. Amer. Math. Soc. 361 (2009), 267–295. [4] J. R. Klein, C. L. Schochet, and S. B. Smith, Continuous trace C ∗ -algebras, gauge groups and rationalization, J. Topology and Analysis 1 (2009), 261-288. [5] J. Rosenberg, Homological invariants of extensions of C ∗ -algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980), pp. 35–75, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, RI, 1982. [6] J. Rosenberg, Continuous - trace algebras from the bundle - theoretic point of view , J. Austral. Math. Soc. (Series A) 47 (1989), 368-381. unneth [7] C. Schochet, Topological methods for C ∗ -algebras. II. Geometric resolutions and the K¨ formula, Pacific J. Math. 98 (1982), no. 2, 443–458. Department of Mathematics, Wayne State University, Detroit MI 48202 E-mail address: [email protected] Department of Mathematics, Wayne State University, Detroit MI 48202 E-mail address: [email protected] Department of Mathematics, Saint Joseph’s University, Philadelphia PA 19131 E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681764

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Distances between Matrix Algebras that Converge to Coadjoint Orbits Marc A. Rieffel Abstract. For any sequence of matrix algebras that converges to a coadjoint orbit we give explicit formulas that show that the distances between the matrix algebras (viewed as quantum metric spaces) converges to 0. In the process we develop a general point of view that is likely to be useful in other related settings.

Introduction In earlier papers [6, 7, 9] I provided ways to give a precise meaning to statements in the literature of high-energy physics and string theory of the kind “Matrix algebras converge to the sphere”. I did this by equipping the matrix algebras with suitable “Lipschitz seminorms” that make the matrix algebras into “compact quantum metric spaces”, and then by defining convergence by means of a suitable “quantum Gromov-Hausdorff distance” between quantum metric spaces. By now a number of variations on this approach have been studied [1, 2, 3, 4, 5, 10]. When I then began to examine what consequences the convergence of quantum metric spaces had for the convergence of “vector bundles” (i.e. projective modules) over them [8], I found that it is very important that the Lipschitz seminorms satisfy a suitable Leibniz property. In [9] I showed that a very convenient source for seminorms that satisfy this Leibniz property consisted of normed bimodules, and in [9] I also constructed explicit normed bimodules that worked well for matrix algebras converging to coadjoint orbits. However, for our approach to work well, it should be the case that for a convergent sequence of matrix algebras the quantum Gromov-Hausdorff distances between the matrix algebras go to 0; but when I required that all of the seminorms satisfy the Leibniz property I did not see at first how to show this convergence directly. The purpose of the present paper is to give explicit normed bimodules and corresponding Leibniz Lipschitz seminorms that demonstrate this convergence to 0. In 2000 Mathematics Subject Classification. Primary 46L87; Secondary 53C23, 58B34, 81R15, 81R30. Key words and phrases. quantum metric space, Gromov–Hausdorff distance, Leibniz seminorm, coadjoint orbits, matrix algebras, coherent states, Berezin symbols. The research reported here was supported in part by National Science Foundation grant DMS-0753228. c 2010 American c Mathematical 0000 (copyright Society holder)

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MARC A. RIEFFEL

the process we develop a general point of view that is likely to be useful in other related situations. This point of view is motivated by the “nuclear distance” introduced and studied by Hanfeng Li [2, 4, 5], in which all of the bimodules are required to be C ∗ -algebras. I have so far not seen how to apply Hanfeng Li’s approach directly to obtain explicit normed bimodules for the matrix-algebra case. This is because Li’s nuclear distance requires implicitly that it always be the identity element of the C ∗ -algebra that is used to define the needed inner derivation, and I have not seen how to successfully arrange this for the situation discussed in this paper. But by trying just to arrange at least that all of the normed bimodules that I used be C ∗ -algebras, I was led to see the path to the explicit bimodules that I sought. The first section of this paper recalls the setting for matrix algebras converging to coadjoint orbits, reformulates the bimodules from [9] so that they are C ∗ algebras, and then uses these reformulated bimodules to construct candidates for C ∗ -bimodules between matrix algebras whose Leibniz Lipschitz seminorms might show that the distances go to 0. In Section 2 we place matters in a general framework, and obtain a basic theorem in this general framework. In Section 3 we prove that the candidate bimodules and corresponding Lipschitz seminorms of Section 1 do indeed show that the distances between the converging matrix algebras go to 0. An important step in the proof comes from the general theorem in Section 2. The full statement of the main theorem is given at the end of Section 3. 1. The Bimodules We recall the setting from [7, 9]. We let G be a compact connected semisimple Lie group, and we let g denotes the complexification of the Lie algebra of G. We choose a maximal torus in G, with corresponding Cartan subalgebra of g, its set of roots, and a choice of positive roots. We fix a specific irreducible unitary representation, (U, H), of G, and we choose a highest-weight vector, ξ, for (U, H) with ξ = 1. For any n ∈ Z≥1 we set ξ n = ξ ⊗n in H⊗n , and we let (U n , Hn ) be the restriction of U ⊗n to the U ⊗n -invariant subspace, Hn , of H⊗n that is generated by ξ n . Then (U n , Hn ) is an irreducible representation of G with highest-weight vector ξ n , and its highest weight is just n times the highest weight of (U, H). We denote the dimension of Hn by dn . We let B n = L(Hn ). The action of G on B n by conjugation by U n will be denoted simply by α. We assume that a continuous length function, , has been chosen for G, and we denote the corresponding C ∗ -metric on B n by LB n . It is defined by LB / eG } n (T ) = sup{αx (T ) − T /(x) : x ∈ for T ∈ B n . (The term “C ∗ -metric” is defined in definition 4.1 of [9].) We let P n denote the rank-one projection along ξ n . Then the α-stability subgroup, H, for P = P 1 will also be the α-stability subgroup for each P n . We let A = C(G/H), and we let LA be the C ∗ -metric on A for  and the left-translation action of G on G/H, defined much as is LB n. Roughly speaking, our goal is to obtain estimates on the distance between n B (B m , LB m ) and (B , Ln ) that show that the distance goes to 0 as m and n go to ∞. We want to do this in the setting of [9], where we insist that the Lipschitz seminorms involved satisfy a strong Leibniz property. We require this because of

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its importance for treating vector bundles (and projective modules), as shown in [8]. But in contrast to [9], our presentation here is influenced by Hanfeng Li’s definition of the “nuclear distance” between quantum metric spaces, although I have not seen how to use his nuclear distance directly. The effect of this influence is that we try to arrange that all of the bimodules that we consider are actually C ∗ -algebras. To motivate the construction of our bimodules, we first reformulate the corresponding constructions from [9] in terms of C ∗ -algebras. For any given n we form the C ∗ -algebra A ⊗ B n = C(G/H, B n ). There are canonical injections of A and B n into A ⊗ B n , and by means of these we view A ⊗ B n as an A-B n -bimodule. Let ωn ∈ C(G/H, B n ) be defined by ωn (x) = αx (P n ). We use the distinguished element ωn and the bimodule structure to define a seminorm, Nn , on A ⊕ B n by Nn (f, T ) = f ωn − ωn T . This seminorm is easily seen to be the same as the seminorm Nσ described by other means in proposition 7.2 of [9]. It is also easy to see that Nn satisfies the strong Leibniz property defined in definition 1.1 of [9], for the reasons discussed in example 2.3 of [9] if A ⊗ B n is viewed as an (A ⊗ B n )-bimodule in the evident way. For a suitable choice of the constant γ, as discussed in propositions 8.1 and 8.2 of [9], γ −1 Nn is a bridge, as defined in definition 5.1 of [6]. This implies that the ∗-seminorm Ln on A ⊕ B n defined by −1 (Nn (f, T ) ∨ Nn (f¯, T ∗ )) Ln (f, T ) = LA (f ) ∨ LB n (T ) ∨ γ

is a C ∗ -metric on A ⊕ B n (where ∨ means “maximum of”) that has the further property that its quotients on A and B n agree with LA and LB n on self-adjoint elements. (See notation 5.5 and definition 6.1 of [9].) This quotient condition on seminorms is exactly what we required in [6, 7, 9] in order to define distances between C ∗ -algebras such as A and B n . Specifically, for our situation, let S(A) denote the state space of A, and similarly for B n and A ⊕ B n . Then S(A) and S(B n ) are naturally viewed as subsets of S(A ⊕ B n ). Now Ln defines a metric, ρLn , on S(A ⊕ B n ) by ρLn (μ, ν) = sup{|μ(f, T ) − ν(f, T )| : Ln (f, T ) ≤ 1}. (By definition this supremum should be taken over just self-adjoint f and T , but by the comments made just before definition 2.1 of [6] it can equivalently be taken over all f and T because Ln is a ∗-seminorm. This fact is also used later for other ∗-seminorms.) The corresponding ordinary Hausdorff distance ρ

distHLn (S(A), S(B n)) gives, by definition, an upper bound for distq (A, B n ) as defined in definition 4.2 of [6] when we don’t require the strong Leibniz condition, and for prox(A, B n ) as defined in definition 5.6 of [9] when we do require the strong Leibniz condition. It is shown in theorem 4.3 of [6] that distq satisfies the triangle inequality. But prox probably does not satisfy the triangle inequality, basically because the quotient of a seminorm that satisfies the Leibniz condition need not satisfy the Leibniz condition.

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We always have distq (A, B) ≤ prox(A, B), so if we can show that prox(A, B) is “small” than it follows that distq (A, B) is “small” too. Since for our specific situation prox(A, B n ) converges to 0 as n goes to ∞, as seen in theorem 9.1 of [9], (and similarly for its matricial version, proxs , by theorem 14.1 of [9]), it is natural to expect that prox(B m , B n ) converges to 0 as m and n go to ∞. But because we can not invoke the triangle inequality, we need to give a direct proof of this fact. In the process of doing this we will construct a specific seminorm that gives quantitative estimates. Towards our goal we seek to construct a suitable B m -B n -bimodule. We can, of course, view the C ∗ -algebra A ⊗ B m as being the B m -A-bimodule B m ⊗ A, and then it is natural to form an “amalgamation” over A of these two C ∗ -algebras, to obtain the B m -B n -bimodule (B m ⊗ A) ⊗A (A ⊗ B n ) = B m ⊗ A ⊗ B n , which we can view as C(G/H, B m ⊗ B n ). Notice that this is again a C ∗ -algebra, and that we have natural injections of B m and B n into it. Inside this bimodule we choose a distinguished element, namely ωmn = ωm ⊗ ωn , viewed as defined by ωmn (x) = αx (P m ) ⊗ αx (P n ) = αx (P m ⊗ P n ). In terms of ωmn we define a seminorm, Nmn , on B m ⊕ B n by Nmn (S, T ) = Sωmn − ωmn T , where the norm is that of the C ∗ -algebra C(G/H, B m ⊗ B n ). We can now hope to find constants γ such that γ −1 Nmn is a bridge between B m and B n . In the next section we describe a more general setting within which to choose such bridge constants. 2. The Bridge Constants In this section we consider the following more general setting. We are given three compact C ∗ -metric spaces, (A, LA ), (B, LB ) and (C, LC ). We are also given unital C ∗ -algebras D and E together with injective unital homomorphisms of A and B into D, and of B and C into E. (Actually, we do not need the unital homomorphisms to be injective, but then we should provide notation for them, and that would clutter our calculations.) Thus we can consider D to be an AB-bimodule and E to be a B-C-bimodule. We assume further that we are given distinguished elements d0 and e0 of D and E respectively. For convenience we assume that d0  = 1 = e0 . We then define seminorms ND and NE on A ⊕ B and B ⊕ C by ND (a, b) = ad0 − d0 bD and similarly for NE . We assume that there are constants γD and γE such that −1 −1 γD ND and γE NE are bridges for (LA , LB ) and (LB , LC ) respectively. This means that when we form the ∗-seminorm −1 LAB (a, b) = LA (a) ∨ LB (b) ∨ γD (ND (a, b) ∨ ND (a∗ , b∗ ),

its quotients on A and B agree with LA and LB on self-adjoint elements, and similarly for LBC . Note that LAB and LBC are C ∗ -metrics by theorem 6.2 of [9]. Motivated by Hanfeng Li’s treatment of his nuclear distance [5], we consider any amalgamation, F , of D and E over B. This means that there are unital injections of D and E into F whose compositions with the injections of B into D

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and E coincide. We denote the images of d0 and e0 in F again by d0 and e0 , and we set f0 = d0 e0 . Unfortunately in this generality it could happen that f0 = 0. (In Hanfeng Li’s definition of his nuclear distance this problem does not occur since his distinguished elements are, implicitly, the identity elements.) Theorem 2.1. Let notation be as above, and assume that f0 = 0. View F as an A-C-bimodule in the evident way, and define a seminorm, NF , on A ⊕ C by NF (a, c) = af0 − f0 cF . Then for any γ ≥ γD + γE the seminorm γ −1 NF is a bridge for (LA , LC ). Proof. It is clear that γ −1 NF (1A , 0C ) = 0 since f0 = 0, and that γ −1 NF is norm-continuous. Thus the first two conditions of definition 5.1 of [6] are satisfied. We must verify the third, final, condition. To simplify notation, we identify A, B, C, D and E with their images in F . For any a ∈ A, b ∈ B and c ∈ C we have NF (a, c) = af0 − f0 cF ≤ ad0 e0 − d0 be0 F + d0 be0 − d0 e0 cF ≤ ad0 − d0 bD e0 E + d0 D be0 − e0 cE = ND (a, b) + NE (b, c). −1 ND is a Now let a ∈ A with a = a∗ be given, and let ε > 0 be given. Since γD ∗ bridge for (LA , LB ), there is by definition a b ∈ B with b = b such that −1 LB (b) ∨ γD ND (a, b) ≤ LA (a) + ε. −1 Then since γE NE is a bridge for (LB , LC ), there is a c ∈ C with c∗ = c such that −1 LC (c) ∨ γD ND (b, c) ≤ LB (b) + ε.

Consequently LC (c) ≤ LB (b) + ε ≤ LA (a) + 2ε, and, from the earlier calculation, NF (a, c) ≤ ND (a, b) + NE (b, c) ≤ γD (LA (a) + ε) + γE (LB (b) + ε) ≤ (γD + γE )LA (a) + ε(γD + 2γE ). The situation is basically symmetric between A and C, so one can make a similar calculation but starting with a c ∈ C to obtain a b ∈ B and then an a ∈ A satisfying the corresponding inequalities. This shows that (γD + γE )−1 NF is indeed a bridge. Then also γ −1 NF will be a bridge for any γ ≥ γD + γE .  However, I have so far not seen any good general conditions that yield estimates showing that if the corresponding seminorm ∗ LAB = LA ∨ LB ∨ γ −1 (ND ∨ ND )

brings (A, LA ) and (B, LB ) close together, and similarly for LBC , then LAC using (γD + γE )−1 NF brings (A, LA ) and (C, LC ) close together, in the sense that ρL distH AC (S(A), S(C)) is small. In Hanfeng Li’s nuclear distance, in which the distinguished elements are all, implicitly, the identity elements, this aspect works much better. And since the nuclear distance satisfies the triangle inequality, it is clear that distnu (B m , B n ) converges to 0 as m and n go to ∞. But so far I find the nuclear distance to be more elusive, as I discuss briefly in section 6 of [9], though it is certainly attractive. I do not yet see how to obtain for the nuclear distance the kind of quantitative estimates that we will obtain here for prox.

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3. The Proof and Statement of the Main Theorem For the context of Section 1 the role of F of Section 2 is played by C(G/H, B m ⊗ B ), while the roles of d0 and e0 are played by ωm and ωn , with f0 being ωmn . Let A B γm be defined as in proposition 8.1 of [9] but for P = P m , and let γm be defined as m A B in proposition 8.2 of [9] but for P = P . Let γm = max{γm , γm }. All that we need to know here about γm is that propositions 8.1 and 8.2 of [9] tell us that, for Nm as −1 defined in Section 1 above, γm Nm is a bridge for (LA , LB m ), and that propositions 10.1 and 12.1 of [9] tell us that γm converges to 0 as m goes to ∞. From Theorem 2.1 above and from the identifications made above, it follows immediately that for B any γ with γ ≥ γm + γn the seminorm γ −1 Nmn is a bridge for (LB m , Ln ). m n We now investigate how close S(B ) and S(B ) are in the metric from the corresponding seminorm Lmn on B m ⊕B n . Given μ ∈ S(B m ), we want a systematic way to find a ν ∈ S(B n ) that is “relatively close” to μ. For this purpose we use the Berezin symbols σ n and σ ˘ n that we used in [7, 9]. We recall that σ n is the completely positive unital map from B n to A defined by σTn (x) = tr(αx (P n )T ), while σ ˘ n is the completely positive unital map from A to B n defined by  f (x)αx (P n )dx, σ ˘fn = dn n

G/H

where we recall that dn is the dimension of Hn , and the G-invariant measure on G/H gives G/H measure 1. Then σ ˘ m ◦ σ n will be a completely positive unital map n m from B to B , whose transpose will map S(B m ) into S(B n ), for any m and n. For any T ∈ B n we have  αx (P m ) tr(αx (P n )T )dx. σ ˘ m (σTn ) = dm G/H

Let Nmn be the seminorm on B m ⊕ B n determined by ωmn , so that Nmn (S, T ) = Sωmn − ωmn T  = sup{(S ⊗ In )αx (P m ⊗ P n ) − αx (P m ⊗ P n )(Im ⊗ T ) : x ∈ G/H}. Then Lmn is defined on B m ⊕ B n by B −1 Lmn (S, T ) = LB (Nmn (S, T ) ∨ Nmn (S ∗ , T ∗ )) m (S) ∨ Ln (T ) ∨ γ

for some γ ≥ γm + γn . Let μ ∈ S(B m ) be given, and as state ν ∈ S(B n ) potentially close to μ we choose ν to be defined by ν(T ) = μ(˘ σ m (σTn )). We then want an upper bound on ρLmn (μ, ν). Now ρLmn (μ, ν) = sup{|μ(S) − ν(T )| : Lmn (S, T ) ≤ 1}, and ˘ m (σTn )||. |μ(S) − ν(T )| = |μ(S) − μ(˘ σ m (σTn ))| ≤ ||S − σ So we need to understand what the condition Lmn (S, T ) ≤ 1 implies for ||S − σ ˘ m (σTn )||. This seems difficult to do directly, so we use a little gambit that we have used before, e.g. shortly before notation 8.4 of [9], namely ||S − σ ˘ m (σTn )|| ≤ ||S − σ ˘ m (σSm )|| + ||˘ σ m (σSm ) − σ ˘ m (σTn )|| B B ≤ δm Lm (S) + σSm − σTn ∞ ,

where for the last inequality we have used theorem 11.5 of [9], which includes the B definition of δm . (We remark that theorem 11.5 of [9] is the same as theorem 6.1 of [7], but [9] gives a simpler proof of this theorem.) Note that Lmn (S, T ) ≤ 1

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m n implies that LB m (S) ≤ 1. Thus we see that it is σS − σT ∞ that we need to control. In preparation for this we establish some additional notation in order to put the situation into a comfortable setting. Notice that B m ⊗ B n = L(Hm ⊗ Hn ). Furthermore ξ m ⊗ ξ n is a highest-weight vector in (U m ⊗ U n , Hm ⊗ Hn ), and its weight is just the sum of the highest weights of (U m ⊗ Hm ) and (U n ⊗ Hn ), which is just the highest weight of (U, H) multiplied by m + n. Thus ξ m ⊗ ξ n is just the highest-weight vector for a copy of (U m+n , Hm+n ) inside Hm ⊗ Hn . To simplify notation we now just set ξ m+n = ξ m ⊗ ξ n , and view Hm+n as being the G-invariant subspace of Hm ⊗ Hn generated by ξ m+n . Then the rank-1 projection P m+n on ξ m+n is exactly P m ⊗ P n . We let Πmn denote the projection from Hm ⊗ Hn onto Hm+n . Our notation will not distinguish between viewing the domain of P m+n as being Hm ⊗ Hn or as being Hm+n , and we will use below the fact that αx (P m+n ) = αx (P m+n )Πmn for any x ∈ G.

Lemma 3.1. For any S ∈ B m and T ∈ B n we have m+n σSm − σTn = σR

where R = Πmn (S ⊗ In − Im ⊗ T )Πmn , viewed as an element of B m+n . Proof. For any x ∈ G we have σSm (x) − σTn (x) = trm (αx (P m )S) − trn (T αx (P n )) = (trm ⊗ trn )(αx (P m ⊗ P n )(S ⊗ In − Im ⊗ T )αx (P m ⊗ P n )) = trm+n (αx (P m+n )Πmn (S ⊗ In − Im ⊗ T )Πmn ) m+n = σR (x).

 Notice now that for R defined as just above, because the rank of P m+n is 1, we have for any x ∈ G m+n |σR (x)| = | trm+n (αx (P m+n )Πmn (S ⊗ In − Im ⊗ T )Πmn )|

= αx (P m+n )(S ⊗ In − Im ⊗ T )αx (P m+n ) ≤ αx (P m+n )(S ⊗ In ) − (Im ⊗ T )αx (P m+n ), and consequently

m+n σR  ≤ Nmn (S ∗ , T ∗ ). But if Lmn (S, T ) ≤ 1, then Nmn (S ∗ , T ∗ ) ≤ γm + γn if we have taken γ = γm + γn . Thus we find that B |μ(S) − ν(T )| ≤ δm + γm + γn . Since the situation is symmetric in m and n, we conclude that ρ

B B A B , δn } + max{γm , γm } + max{γnA , γnB }. distHLmn (S(B m ), S(B n )) ≤ max{δm

As mentioned in part above, it is shown in proposition 10.1, theorem 11.5, and A B B proposition 12.1 of [9] that, respectively, γm , δm , and γm all converge to 0 as m goes to ∞. We thus obtain the main theorem of this paper: Theorem 3.2. With notation as above, for all m and n we have B B A B prox(B m , B n ) ≤ max{δm , δn } + max{γm , γm } + max{γnA , γnB },

and in particular, prox(B m , B n ) converges to 0 as m and n go to ∞.

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One can also obtain matricial versions of this theorem along the lines discussed in section 14 of [9]. References [1] David Kerr, Matricial quantum Gromov-Hausdorff distance, J. Funct. Anal. 205 (2003), no. 1, 132–167. arXiv:math.OA/0207282. MR 2020211(2004m:46153) [2] David Kerr and Hanfeng Li, On Gromov-Hausdorff convergence for operator metric spaces, J. Operator Theory 62 (2009), no. 1, 83–109, arXiv:math.OA/0411157. [3] Hanfeng Li, Order-unit quantum Gromov-Hausdorff distance, J. Funct. Anal. 231 (2006), no. 2, 312–360, arXiv:math.OA/0312001. MR 2195335 (2006k:46119) , C*-algebraic quantum Gromov-Hausdorff distance, arXiv:math.OA/0312003. [4] , Metric aspects of noncommutative homogeneous spaces, J. Funct. Anal. 257 (2009), [5] no. 7, 2325–2350, arXiv:0810.4694. [6] Marc A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), no. 796, 1–65, arXiv:math.OA/0011063. MR 2055927 , Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance, [7] Mem. Amer. Math. Soc. 168 (2004), no. 796, 67–91, arXiv:math.OA/0108005. MR 2055928 , Vector bundles and Gromov-Hausdorff distance, J. K-Theory, published online 2009, [8] arXiv:math.MG/0608266. , Leibniz seminorms for “Matrix algebras converge to the sphere”, arXiv:0707.3229. [9] [10] Wei Wu, Quantized Gromov-Hausdorff distance, J. Funct. Anal. 238 (2006), no. 1, 58–98, arXiv:math.OA/0503344. MR 2234123 Department of Mathematics, University of California, Berkeley, CA 94720-3840 E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681765

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Geometric and Topological Structures Related to M-branes Hisham Sati Abstract. We consider the topological and geometric structures associated with cohomological and homological objects in M-theory. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying spacetime to admit a String structure and a Fivebrane structure, respectively. For the former, we study how the fields in M-theory are associated with the above structures, with homotopy algebras, with twisted cohomology, and with generalized cohomology. We also explain how the corresponding charges should take values in Topological Modular forms. We survey background material and related results in the process.

Contents 1. Introduction and Setting 2. The M-theory C-Field and String Structures 3. The M-theory Dual C-Field and Fivebrane Structures 4. The Gauge Algebra of Supergravity in 6k − 1 Dimensions 5. Duality-Symmetric Twists 6. M-brane Charges and Twisted Topological Modular Forms References

181 188 200 209 216 220 231

1. Introduction and Setting String theory is concerned with a worldsheet Σ2 , usually a Riemann surface, a target spacetime M , usually of dimension ten, and the space of maps φ : Σ2 → M between them. The study of the field theory on Σ2 is the subject of two-dimensional conformal field theory (CFT). The study of the maps is the sigma (σ-) model and the study of the target space is the target theory where low energy limits, i.e. field and supergravity theories, can be taken. The target space theory involves fields called the Ramond-Ramond (RR) or the Neveu-Schwarz (NS) fields. These are differential forms or cohomology classes which 2010 Mathematics Subject Classification. Primary 53C08, 55R65; Secondary 81T50, 55N20, 11F23. Key words and phrases. String structures, Fivebrane structures, n-bundles, differential cohomology, K-theory, topological modular forms, generalized cohomology theories, anomalies, dualities. 1

c 2010 American Mathematical Society

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can be paired with homology cycles, the branes, namely D-branes and NS-branes respectively. These extended objects carry charges, generalizing those carried by electrons. The RR fields are classified by K-theory [MM] [W6] and are twisted by the NS fields leading to a twisted K-theory classification [W6] (see also [BM]), in the sense of [Ro] [BCMMS]. Such a description has been refined to more generalized cohomology theories, most notably elliptic cohomology. This was approached from cancellation of anomalies in type IIA string theory [KS1], studying the compatibility of generalized cohomology twists with S-duality in type IIB string theory [KS2], and the study of modularity in the actions of type IIB string theory and F-theory [KS3] [S4]. The study of sigma models involves loop spaces as follows. The definition of spinors require lifting the classifying map of the tangent bundle from the special orthogonal group SO(n) to the spinor group Spin(n), which corresponds to killing the first homotopy group π1 (SO(n)) of SO(n). In string theory a further step is needed, namely lifting the spinor group to the String group String(n) by killing 1 π3 (SO(n)), giving rise to String structures. Existence of such structures is a condition for the vanishing condition of the anomaly of a string in the context of the index theory of Dirac operators on loop space [W3] [Ki]. The String structure is regarded as a lift of an LSpin(n)-bundle over the free loop space LX through  the Kac-Moody central extension LSpin(n)-bundle [Ki] [CP] [PW] [Mc]. This lift can also be interpreted as a lift of the original Spin(n)-bundle down on target space X to a principal bundle for the topological group String(n) [ST1]. This is, in fact, the realization of the nerve of a smooth categorified group, the String 2-group [BCSS] [H]. The above classification of String-bundles coincided with that of 2bundles with structure 2-group the String 2-group [BS] [BBK] [BSt]. In addition to the above infinite-dimensional models, now there is a finite-dimensional model for the String 2-group [S-P]. The elliptic genus is a loop space generalization  of the A-genus as the index of the Dirac-Ramond operator [SW] [PSW] [W2] [AKMW]. The Green-Schwarz [GS] anomaly can be computed as essentially the elliptic genus [LNSW]. Mathematically, the connection between elliptic genera and loop spaces has been studied, notably in [A] and [Liu]. The String structure , required by modularity, provides an orientation [AHS] [AHR] for TMF, the theory of Topological Modular Forms [Ho] [Go]. At the level of conformal field theory, which is the quantum field theory on the worldsheet of the string, one has fields that are pulled back from the spacetime theory via the sigma model map, in addition to other fields. In two-dimensional supersymmetric quantum field theory the partition function, which is an integral modular function, is argued in [ST1] [ST2] to be an element in TMF. Another geometric description of elliptic cohomology via CFT is given in [HK2], which builds on Segal’s definition of CFT and on vertex operator algebras. A variant that has features of both [ST1] and [HK2] is proposed in [GHK] using a newlyintroduced notion of infinite loop spaces. We emphasize two main points, central to the theme of this paper, concerning the above mathematical structures: 1This might perhaps more correctly be called “cokilling” since it corresponds to the Whitehead tower rather than to the Postnikov tower.

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(1) First, generally, that various structures appearing in this part of theoretical physics are much deeper (and thereby richer and more interesting) than the sketchy physics literature about them indicates. (2) Second, more specifically, that the above mathematical structures appearing in string theory are beginning to appear, even in a perhaps richer form, within the study of another theory, namely M-theory. M-theory (cf. [W5] [To3]) is a conjectured theory in eleven dimensions that unifies all five ten-dimensional superstring theories. The theory is best understood through these string theories and also via its classical low energy limit, eleven-dimensional supergravity theory [CJS]. Thus one strategy in studying the theory is to take eleven-dimensional supergravity and perform semi-classical quantization. Due to quantum effects the process is only selectively reliable. Among the reliable terms are the topological terms, i.e. the terms that are not sensitive to the metric. Metricdependent quantities might undergo drastic changes due to quantum gravitational effects. One way of keeping the metric requires taking some large volume limit, making sure that the scale is larger than the critical scale at which Riemannian geometry can no longer provide a good description. As the theory is supersymmetrc, it will at least have a fermion (in this case, a section of the tensor product of the tangent bundle and the spin bundle), and since it involves gravity, it will also contain a metric, or graviton. There is also an a priori metric-independent field, called the C-field. This is a higher-degree analog of a connection whose field strength – the analog of a curvature – is denoted by G4 . The fields (aside from the metric and fermions) in string theory and M-theory are differential forms at the rational level, i.e. at the level of description of the corresponding supergravity theories. Gauge invariance leads to a description of the fields in terms of de Rham cohomology. Quantum mechanically, these generically become integral-valued and hence one needs to go beyond de Rham cohomology to integral cohomology. Dually, these fields can be described via homology cycles that admit extra geometric structures, such as Spin structures and vector bundles. This dual homological picture is captured by the notion of branes, namely D-branes in string theory and M-branes in M-theory. The fields and the branes are of specified dimensions, determined by the corresponding theory. In particular, D-branes have odd (even) spacetime dimensions, and hence even (odd) spatial dimensions, for type IIA (type IIB) string theory. For M-branes, spatial dimensions two for the M2-brane [BST] and five for the M5-brane [Gu] occur. The five string theories in ten dimensions are related through a web of dualities (see e.g. [Sch2] for a survey). The first kind of duality is called T-duality ( “T” for Target space), which relates two different theories on torus bundles, where the first theory with fiber a torus is related to a second theory with a fiber the dual torus. An example of this is T-duality between type IIA and type IIB string theories. The second kind of dualities is S-duality, or strong-weak coupling duality, which relates a theory at a high value of some parameter (strong coupling) to another theory at a low value of the same parameter (weak coupling). This generalizes the usual electromagnetic duality between electric and magnetic fields in four dimensions. Whenever we discuss dualities in this paper we will focus mostly on S-duality. A duality within the same theory is called self-duality. An example of this is the selfduality among the RR fields in type IIA/B string theory. At the rational level this is

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simply a manifestation of Hodge duality. A delicate discussion of such matters can be found in [Fr2] [FMS]. Another important example of this is S-(self)duality in type IIB string theory. Some subtleties on the relation of this duality to generalized cohomology are discussed in [BEJMS] [KS2]. The above dualities can be most directly seen at the level of fields. By Poincar´e duality, such dualities also manifest themselves at the level of branes. Branes of even (odd) spatial dimensions in type IIA (type IIB) are dual to odd (even) branes in type IIB (IIA) string theory. Furthermore, within the same type II theory, D-p-branes are dual to D-(6 − p)-branes, p even or odd for type IIA or type IIB, respectively. This is a homological manifestation of the self-duality of the RR fields. There is a similar duality in eleven dimensions which relates the C-field C3 to its dual C6 , which, at the rational level, is Hodge duality between the corresponding fields strengths G4 = ∗G7 . This duality on the fields also has a homological interpretation as a duality between the M2-brane and the M5-brane. String theories in ten dimensions can be obtained from eleven-dimensional Mtheory via dimensional reduction and/or duality transformations. M-theory on the total space of a circle bundle gives rise to type IIA string theory on the base space. By pulling back the fields along the section of the circle bundle π, assumed trivial, one gets a D2-brane from an M2-brane [To2] and a NS5-brane from an M5-brane. On the other hand, upon integration over the fiber of π, the M2-branes give rise to strings [DHIS] and the M5-brane give rise to D4-branes [To2]. The branes of type IIB string theory can also be obtained from those of M-theory on a torus bundle [Sch1]. Similar relations hold at the level of fields: Integrating G4 in eleven dimensions over the circle gives H3 = π∗ (G4 ), the field strength of the NS B-field. On the other hand, pulling back G4 along a section s, again assuming the circle bundle is trivial, gives a degree four field F4 = s∗ G4 , which is one component of the total RR field. An invariant description of this is given in [FS] and [MSa]. Branes carry charges– a notion that can be made mathematically precise– that can be viewed either as classes of bundles in generalized cohomology or as their images in rational or integral cohomology under a (normalized) version of the Chern character map. A working mathematical definition of D-branes and their charges can be found in [BMRS]: A D-brane in ten-dimensional spacetime X is a triple (W, E, ι), where ι : W → X is a closed, embedded submanifold and E ∈ Vect(W ) is a complex vector bundle over W . The submanifold W is called the worldvolume and E the Chan-Paton bundle of the D-brane. The charges of the D-branes [Po] can be classified, in the absence of the NS fields, by K-theory of spacetime [MM], namely by K 0 (X) for type IIB [W6] and by K 1 (X) for type IIA [Ho]. The fields are also classified by K-theory of spacetime but with the roles of K 0 and K 1 interchanged [MW] [FH]. In the presence of the NS B-field, or its field strength H3 , the relevant K-theory is twisted K-theory, as was shown in [W6] [FW] [Ka] by analysis of worldsheet anomalies for the case the NS field [H3 ] ∈ H 3 (X, Z) is a torsion class, and in [BM] for the nontorsion case. Twisted K-theory has been studied for some time [DK] [Ro]. More geometric flavors were given in [BCMMS]. Recently, the theory was fully developed by Atiyah and Segal [AS1] [AS2]. The identification of twisted D-brane charges with elements in twisted K-theory requires a push-forward map and a Thom isomorphism in the latter, both of which are established in [CW].

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The study of the M-brane charges and their relation to (generalized) cohomology is one of the main goals of this paper. Given the relation between generalized cohomology and string theory on one hand, and between string theory and M-theory on the other, it is natural to ask whether elliptic cohomology or T M F have to do with M-theory directly. Suggestions along these lines have been given in [KS1] [S1] [S2] [S3]. In particular in [KS1] it was proposed that the M-branes should be described by TMF in the sense that the elliptic refinement of the partition function originates from interactions of M2-branes and M5-branes. Furthermore, in [S3] it was observed that the M-theory field strength G4 , rationally, can be viewed as part of a twist of the de Rham complex and suggested that the lift to generalized cohomology would be related to a twisted version T M F . A twisted differential is one of the form d+α∧ acting on differential forms, where d is the de Rham differential and α is a differential form. Twisted rational cohomology is then the kernel modulo the image of d + α∧. When α is a 3-form then one gets the image of twisted K-theory under the twisted Chern character (cf. [AS2] [BCMMS] [MSt]). The field strength G4 on an eleven-manifold Y 11 satisfies the shifted quantization condition [W7] (1.1)

1 [G4 ] + λ(Y 11 ) = a ∈ H 4 (Y 11 ; Z) , 2

with λ(Y 11 ) = 12 p1 (Y 11 ), where p1 (Y 11 ) is the first Pontrjagin class of the tangent bundle T Y 11 of Y 11 and a is the degree four class that characterizes an E8 bundle in M-theory [DMW]. There is a one-to-one correspondence between H 4 (M, Z) and isomorphism classes of principal E8 bundles on M , when the dimension of M is less than or equal to 15, which is the case for Y 11 in M-theory. This follows from homotopy type of E8 being of the form (3, 15, · · · ). Up to the 14th-skeleton E8 is homotopy equivalent to the Eilenberg-MacLane space K(Z, 3) so that up to the 15th-skeleton the classifying space BE8 is ∼ K(Z, 4). For the homotopy classes of maps [M, E8 ] = [M, K(Z, 3)] = H 3 (M, Z) if dimM ≤ 14, and similarly {Equivalence classes of E8 bundles on M } = [M, BE8 ] = [M, K(Z, 4)] = H 4 (M, Z) if dimM ≤ 15. Therefore, corresponding to an element a ∈ H 4 (M, Z) we have an E8 principal bundle P (a) → M with p1 (P (a)) = a. In [Wa] the notion of a twisted String structure was defined, where the twist is given by a degree four cocycle. This degree four generalization of the twisted Spinc structure in degree three was anticipated there to be related to the flux quantization condition (1.1). This was made explicit and precise in [SSS3], where also the Green-Schwarz anomaly was shown to be more precisely the obstruction to having a refined twisted String structure. The study the dual NS- and M-theory fields of degree eight leads to even higher structures. The first appears in a dual form of the Green-Schwarz anomaly cancellation. The second satisfies a condition analogous to (1.1) in degree eight, as shown in [DFM]. In [SSS1] Fivebrane structures were introduced and the corresponding differential geometric structures, i.e. the higher bundles, are constructed. They are systematically studied in [SSS2] and shown to emerge within the description of the dual of the Green-Schwarz anomaly, involving the Hodge dual of H3 in ten dimensions, as well as in the description of the dual of the M-theory C-field in eleven

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dimensions. Fivebrane structures can be twisted in the same way that String structures can. Twisted Fivebrane structures are defined and studied in [SSS3], and their obstructions can be matched with the dual of the Green-Schwarz anomaly. In fact, both twisted String and Fivebrane structures are refined in [SSS3] to the differential case, using a generalization of the discussion in [HS]. The notion of (twisted) String and, to some extent, Fivebrane structures can in fact be described in various ways, via: 1. Principal and associated bundles. 2. Gerbes and differential characters. ˇ 3. Cech cohomology and Deligne cohomology. 4. Loop bundles. 5. 2-bundles and 6-bundles and their 2-algebras and 6-algebras, respectively. Another purpose of this paper is to provide the generalized cohomology aspect. The appearance of λ in (1.1) and the subsequent interpretation in terms of twisted String structures suggests a relation to a theory that admits that structure as an orientation. A Spin manifold M has a characteristic class λ such that 2λ = p1 (M ). The paper [AHS] shows that M admits a T M F orientation if λ = 0. More precisely, a String structure on a Spin manifold M is a choice of trivialization of λ, and in [AHS] it is shown that a String structure determines an orientation of M in T M F cohomology. In this paper we argue that the shifted quantization condition (1.1) provides a twist for T M F and hence that (1.1) defines a twist for T M F . More precisely, the cohomology class of G4 is 12 λ − a, which we view as a twist of the T M F orientation by the degree four class a of the E8 bundle. Since E8 ∼ K(Z, 4) up to dimension 14 then a a priori can be any class in K(Z, 4). However, fixing an E8 bundle completely fixes a. Conjectures for using a twisted form of T M F and elliptic cohomology – and hence that such structures should exist– to describe the fields in M-theory go back to [S2] [S3]. Given the interpretation of G4 and its dual in terms of twisted String and Fivebrane structures and the proposed connection to twisted T M F , it is natural to consider the corresponding homological objects. The charge of the M5-brane is, rationally, the value of the integral of G4 over the unit sphere in the normal bundle in the eleven-dimensional manifold. We interpret the charge of the M5-brane in full and not just rationally, in a Riemann-Roch setting, as the direct image of an element in twisted T M F on the M-brane. The interpretation of the charges as elements of twisted T M F uses some recent work [ABG] on push-forward and Thom isomorphism in T M F . This is a higher degree generalization of the case of D-branes, where the H-field defines a twist for the Spinc structure and the charge is defined using the push-forward and Thom isomorphism in twisted K-theory. Given the topological structures defined by G4 and its dual, it is natural to ask for geometric models for the corresponding potentials, i.e. the C-field C3 and its dual. There is the E8 model of the C-field [DMW] [DFM], mentioned above, which is essentially a Chern-Simons form, or more precisely a shifted differential character, where the shift on the E8 bundle class a is given by the factor 12 λ. In an alternative model in terms of 2-gerbes [AJ], picking a connection A on the E8 principal bundle P (a) gives a Deligne class, the Chern-Simons 2-gerbe CS(a), with a its characteristic class. The interpretation we advocate in terms of twisted String

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structures allows for other interpretations of the C-field, where the term 12 λ is now the ‘main’ term, since it is responsible for defining the String structure, and the term a coming from the E8 bundle is merely a twist for that structure. This gives the tangent bundle-related term a more prominent role. In fact, this model can be described at the differential twisted cohomology level [SSS3]. In this paper we provide an identification of the C-field as the String class, i.e. as the class that provides the trivialization of the String structure. We also do the same for the dual of the C-field which we identify with the Fivebrane class, the class that provides the trivialization for the Fivebrane structure. We also give an alternative description using differential characters. The appearance of higher chromatic phenomena in [KS1] [KS2] [KS3] in relation to string theory, and the appearance of Fivebrane structures in degree eight in string theory and M-theory naturally leads to the question of whether higher degree twists exist in this context. Indeed, in [S6] it was shown that a degree seven twist occurs in heterotic string theory, manifested via a differential of the form d + H7 ∧. The lift of this twisted rational cohomology to generalized cohomology suggests the appearance of the second generator v2 at the prime 2 in theories descending from complex cobordism M U . This generalizes the situation in degree three, where the Bott generator u = v1 in K-theory appears via d+v1 H3 ∧. In this paper we consider the fields in M-theory as part of a twist in de Rham cohomology, extending and refining the discussion in [S1] [S2] [S3]. In addition we consider duality-symmetric twists, i.e. twisted differentials whose twists are uniform degree combinations of the H-field and its dual as well as G4 and its dual. The second case leads to an interesting appearance of the M-theory gauge algebra, which in turn leads to the super-tranlation algebra. We also provide an L∞ -algebra description of this gauge algebra.

String Theory

M-Theory

sigma model φ : Σ2 → X 10

sigma model Φ : M 3 → Y 11

ψ ∈ Γ(SΣ2 ⊗ φ∗ T X 10 ), B2 1-gerbe

ψ ∈ Γ(SM 3 ⊗ N (M 3 → Y 11 ), C3 2-gerbe

D-brane ⊃ ∂Σ2

M5-brane ⊃ ∂M 3

Freed-Witten condition W3 + [H3 ] = 0

Witten Flux quantization

1 λ 2

+ [G4 ] = a

Table 1. Extended objects in string theory and in M-theory. This paper is written in an expository style, even though it is mainly about original research. In fact, there are three types of material: • Survey of known results, with some new perspectives a well as providing some generalization. • New research established here. • New research announced and outlined here and to be more fully developed in the future. This material is mostly based on discussions with Matthew Ando, Chris Douglas, Corbett Redden, Jim Stasheff, and Urs Schreiber.

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We hope that the expository style makes it more self-contained and accessible to mathematicians interested in this area of interaction between physics on one hand and geometry and topology on the other. 2. The M-theory C-Field and String Structures Consider an eleven-dimensional Spin manifold Y 11 with metric g. Corresponding to the tangent bundle T Y 11 with structure group SO(11) there is the Spin bundle SY 11 with structure group the Spin group Spin(11), the double cover of SO(11). Let ω be the Spin connection on SY 11 with curvature R. Using the metric we can identify the cotangent bundle T ∗ Y 11 with the tangent bundle. The field content of the theory is the metric g, a spinor one-form ψ , i.e. a section ∈ Γ(SY 11 ⊗ T ∗ Y 11 ), and a degree three form C3 . 2.1. The Quantization Condition and the E8 model for the C-field. The quantization condition on G4 on Y 11 is given in equation (1.1). This can be obtained either from the partition function of the membrane or from the reduction to the heterotic E8 × E8 theory on the boundary [W7]. We will reproduce this result using the membrane partition function in a fashion that is essentially the same as appears in [W7] but with details retained. Since the Spin cobordism 3 groups ΩSpin 4k+3 are zero, we can extend both the membrane worldvolume M and the target spacetime Y 11 as Spin manifolds to bounding manifolds X 4 and Z 12 , respectively. In fact we can also extend E8 bundles on M 3 and Y 11 to E8 bundles on X 4 and Z 12 , respectively, since M Spini (K(Z, 4)) = 0 for i = 3, 11 [Stg] where BE8 has the homotopy of K(Z, 4) in our range of dimensions. The effective action involves two factors:  (1) The ‘topological term’ exp i M 3 C3 ,  (2) The fermion term exp i M 3 ψDψ, where the integrand in the exponential is the pairing in spinor space of the spinor ψ with the spinor Dψ, where D is the Dirac operator.  The first factor will simply give exp i X 4 G4 . Now consider the second factor. Corresponding to the map φ : X 4 → Z 12 we have the index of the Dirac operator for spinors that are sections of S(X 4 ) ⊗ φ∗ T Z 12 given via the index theorem by the degree two expression   4 ∗ 12  IndexD = A(X ) ∧ ch(φ T Z ) 

X4



X

4

= X4

(2)

 1 p1 (T X 4 ) rank(T Z 12 ) + ch2 (φ∗ T Z 12 ) 24 1 1 p1 (φ∗ T Z 12 ) − p1 (X 4 ) . 2 2 1−

= (2.1)





On the other hand we have a split of the restriction of the tangent bundle T Z 12 to X 4 as T Z 12 = T X 4 ⊕ N X 4 with N X 4 the normal bundle of X 4 in Z 12 . Taking the characteristic class λ of both sides we get that the index is equal to λ(N X 4 ). The effective action involves the square root of the index so that the contribution from the second factor in the effective action is exp 2πi 12 λ(N X 4 ). This gives the result in [W7] provided we assume that N (M 3 → Y 11 ) ∼ = N (X 4 → Z 12 ).

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Proposition 2.1. The M-theory field G4 satisfies the quantization condition (1.1). Remarks 2.2. The condition (1.1) has the following consequences [W7]: (1) When λ is divisible by two then G4 cannot be set to zero. (2) When λ2 is integral then the tadpole anomaly of [SVW] vanishes. (3) The relation is parity-invariant, i.e. requiring G4 − λ2 to be integral is equivalent to G4 + λ2 being integral. The E8 model of C-field. The C-field at the level of supergravity will be just a real-valued three-form C3 ∈ Ω3 (Y 11 ; R). The field strength is G4 = dC3 ∈ Ω4 (Y 11 ). This is invariant under gauge transformations C3 → C3 + dφ2 , where φ2 is a twoform. Factoring out by the gauge transformations amounts to declaring the fields to be in cohomology. Upon quantization, several features become important: integrality via holonomy, the presence of torsion, and possible appearance of anomalies. Taking these into account, a model for the C-field was obtained in [W7] and further developed in [DMW] [DFM] [Mo]. Let P be a principal E8 bundle over Y 11 with the characteristic class a pulled back from H 4 (BE8 ; Z). Let A be a connection on P with curvature two-form F . The C-field in this model is given by C = CS3 (A) − 12 CS(ω) + c, where CS3 (A) is the Chern-Simons invariant for the connection A, CS(ω) is the ChernSimons invariant of the connection ω on the Spin bundle, and c is the harmonic representative of the C-field which dominates at long distance approximation, i.e. in the supergravity regime. The Chern-Simons forms and the Pontrjagin forms are related as dCS3 (A) = TrF ∧ F , dCS3 (ω) = TrR ∧ R, so that the field strength of the C-field is given by 1 (2.2) G4 = TrF ∧ F − TrR ∧ R + dc . 2

The cohomology classes are [TrF ∧ F ]DR = aR , [TrR ∧ R]DR = 12 p1 (T Y 11 ) R . Globally, the C-field can be described as the pair [DFM] (2.3)

(A, c) ∈ EP (Y 11 ) := A(P (a)) × Ω3 (Y 11 ),

where A(P (a)) is the space of smooth connections on the bundle P with class a. 2.2. Twisted String structures. Definition 2.3. An n-dimensional manifold X admits a String structure if the classifying map X → BO(n) of the tangent bundle T X lifts to the classifying space BString := BO 8. fˆ

(2.4) M

f

BO(n) 8 :  / BO(n) .

Remarks 2.4. (1) The obstruction to lifting a Spin structure on X to a String structure on X is the fractional first Pontrjagin class 12 p1 (T X). (2) The set of lifts, i.e. the set of String structures for a fixed Spin structure is a torsor for a quotient of the third integral cohomology group H 3 (X; Z).

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Definition 2.5 ([Wa] [SSS3]). An α-twisted String structure on a brane ι : M → X with Spin structure classifying map f : M → BO(n) 4 is a cocycle α : X → K(Z, 4) and a map c : BO(n) 4 → K(Z, 4) such that there is a homotopy η between BO(n) 4 and X, M ι

 X

f

ssss ssssss s s s sssss η u} ssss α

/ BO(n) 4 c

.

 / K(Z, 4)

The homotopy η is the coboundary that relates the cocycle cf to the cocycle αι. Hence on cohomology classes it says that the fractional Pontrjagin class of M does not necessarily vanish, but is equal to the class ι∗ [α]. The twisted String case was originally considered in [Wa]. The definition is refined to structures dubbed F m that account for obstructions that are fractions of the ones for the String structures. For example, the fractional class 14 p1 shows up in (1.1). As an application, which was originally the motivation: Theorem 2.6 ([SSS3]). (1) The Green-Schwarz anomaly cancellation condition defines a twisted String structure pulled back from BO(10) 4 = BSpin(10). The twist α in this case is given by (minus) the degree four class of the E8 × E8 bundle. (2) The anomaly cancellation condition in heterotic M-theory and the flux quantization condition in M-theory each define a twisted String structure pulled back from F 4 = BO 41 p1 . The twist α in this case is given by [G4 ] minus the class of the E8 bundle. The division of λ by 2 require some refinement of the structure [SSS3] as mentioned in Remarks 6.11. The relation to orientation in generalized cohomology is discussed in the study of the membrane partition function in section 2.4. Note that when we consider M-theory with a boundary ∂Y 11 , where essentially the heterotic string theory is defined, [G4 ] would be zero when restricted to ∂Y 11 . In this case the flux quantization condition defines a a-twisted String structure 1 11 ) = a|∂Y 11 = 0 on that boundary. This is discussed further in section 6.3. 2 p1 (∂Y 2.3. The C-field as a String class. Let ·, · be a suitably normalized Adinvariant metric on the Lie algebra spin(11) of Spin(11). Then the 4-dimensional Chern-Weil form R ∧ R ∈ Ω4 (Y 11 ) on S(Y 11 ) is one-half the first Pontrjagin 1 class (restoring normalization) 12 p1 (S, ω) = − 16π 2 Tr(R ∧ R). We will later assume the following condition on the Pontrjagin class 12 p1 (Y 11 ) = 0 in H 4 (Y 11 ; R). This means that the bundle S(Y 11 ) admits a String structure. A choice of String structure is given by a particular cohomology class S ∈ H 3 S(Y 11 ); Z . This element restricts to the fiber as the standard generator of H 3 (Spin(11); Z) ∼ = Z. In terms of the curvature R of the connection ω on S(Y 11 ), the condition = 0 ∈ H 4 (Y 11 ; R) means that [TrR ∧ R] = 0 ∈ H 4 (Y 11 ; R). The Chern-Simons 3-form of the connection ω on the principal bundle S(Y 11 ) is the right-invariant form CS3 (ω) := ω ∧ R − 16 ω ∧ [ω, ω] ∈ Ω3 (S), whose pull-back to 1 11 )) 2 p1 (S(Y

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the fiber via the inclusion map i : Spin(11) → S is the real cohomology class − 16 ω∧ [ω, ω] ∈ H 3 (Spin(11); R) associated to the real image of the standard generator of H 3 (Spin(11); Z) ∼ = Z. The Chern-Simons form CS3 (ω) provides a trivialization for the zero cohomology class above: dCS3 (ω) = TrR ∧ R ∈ Ω4 (S(Y 11 )). Now we consider the geometry on the total space of the Spin bundle. Corresponding to a choice of Riemannian metric g on Y 11 and a connection ω on S(Y 11 ) gives rise to a metric gS on the total space S(Y 11 ). Under the decomposition T S(Y 11 ) ∼ = π ∗ (T Y 11 ⊕ spin(11)) of the tangent space into orthogonal vertical and horizontal subspaces, the metric decomposes as gS := π ∗ (g ⊕ gSpin(11) ), where gSpin(11) is the metric on the fiber. In relating the fields on the base to classes on the total space, one is forced to use the adiabatic limit, introduced in [W1], of the metric on the total space. In particular, as is constructed in [R1], the String structure S on S(Y 11 ) is related to a form on the base in this fashion. This way there is a one-parameter family of

metrics gδ = π ∗ δ12 g ⊕ gSpin(11) on the bundle S(Y 11 ) with parameter δ. This is reminiscent of a Kaluza-Klein ansatz frequently used in supergravity. Now consider the adiabatic limit δ → 0 of gδ . Note that metrics in the adiabatic limit have been used in this form e.g. in [MSa] in the reduction from M-theory in eleven dimensions to type IIA string theory in ten dimensions. Relative Chern-Simons form on Y 11 . Before considering the C-field we will need the following. Consider an E8 bundle P over Y 11 with connection A ∈ Ω1 (P ; e8 ) and curvature π ∗ Ω = dA + 12 [A, A] ∈ Ω2 (P ; e8 ), where Ω ∈ Ω2 (Y 11 ; adP ), a two-form with values in adP . The space AP of connections on P is an affine space modeled on Ω1 (Y 11 ; adP ). Using [Fr1] we can define a relative Chern-Simons invariant on the base. Given two connections A0 and A1 in AP , the straight line path At = (1 − t)A0 + tA1 , 0 ≤ t ≤ 1, determines a connection A on the bundle / [0, 1] × Y 11 . The relative Chern-Simons form is then [0, 1] × P  CS3 (A1 , A0 ) := −

(2.5)

TrF 2 (A) ∈ Ω3 (Y 11 ). [0,1]

Using Stokes’ theorem  dCS3 (A1 , A0 )

= −d  = −

TrF 2 (A) [0,1]

 2

dTrF (A) + (−1)

(11−4)

[0,1]

(2.6)

TrF 2 (A) ∂[0,1]

= 0 + TrF 2 (A1 ) − TrF 2 (A0 ) .

Now we are ready to consider the C-field. Invariance of the C-field. The C-field is invariant under the following trans  formations [DFM]:

A = A + α and C3 = C3 − CS3 (A, A + α) + Λ3 , where α ∈ Ω1 adS(Y 11 ) and Λ3 is a closed 3-form on Y 11 . The Chern-Simons invariant takes values in R/Z so that it is not defined as a differential form unless exponentiated. The relative Chern-Simons invariant is defined as in (2.5).

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If we include the Spin bundle S(Y 11 ) then we also have an invariance of the connection ω and a corresponding shift in the relative Chern-Simons form of ω. Thus we have Proposition 2.7. The C-field is invariant under the following transformations (1) ω  = ω + β, (2) A = A + α, (3) C3 = C3 − CS3 (A, A + α) + 12 CS3 (ω, ω + β) + Λ3 , where α ∈ Ω1 (adP ), β ∈ Ω1 (adS), and Λ3 ∈ Ω3 (Y 11 ) is a closed differential form on Y 11 . Note that in terms of differential characters, Λ3 will be integral as in [DFM]. Harmonic part of the C-field. The C-field has a classical harmonic part, which we now characterize. The Bianchi identity and equation of motion for the C-field in M-theory are dG4 1 d ∗ G4 3p

(2.7)

=

0 1 G 4 ∧ G 4 − I8 , (2.8) = 2

1 p2 − ( 12 p1 )2 , a polynomial where I8 is the one-loop term [VW] [DLM] I8 = 48 in the Pontrjagin classes pi of Y 11 , ∗ is the Hodge duality operation in eleven dimensions, and p is the scale in the theory called the Planck constant. The classical (or low energy) limit given by eleven-dimensional supergravity, is obtained by taking p → 0 and is dominated by the metric-dependent term. The other limit is the high energy limit probing M-theory and is dominated by the topological, i.e. metric-independent terms.



Let Δ3g : Ω3 (Y 11 ), g −→ Ω3 (Y 11 ), g be the Hodge Laplacian on 3-forms on ∗ ∗ ∗ the base Y 11 with respect to the metric g given by Δ3g = d d + d d, where d is the adjoint operator to the de Rham differential operator d. Assuming [G4 ] = 0 in H 4 (Y 11 ; R) so that G4 = dC3 , then applying the Hodge operator on (2.8) gives ∗

Proposition 2.8. In the Lorentz gauge, d C3 = 0, we have (1) Δ3g C3 = ∗je , where je is the electric current associated with the membrane given by  1 3 G 4 ∧ G 4 − I8 . (2.9) je = p 2 (2) C3 is harmonic if p → 0 and/or there are no membranes. The space of harmonic 3-forms on Y 11 is Hg3 (Y 11 ) := kerΔ3g ⊂ Ω3 (Y 11 ). 11 We would to consider 3-forms

harmonic

on the Spin bundle S(Y ). Let 3 like 11 3 3 11 Δgδ : Ω (S(Y )), gδ → Ω (S(Y )), gδ be the Hodge Laplacian for 3-forms on S(Y 11 ) with respect to the metric gδ . The harmonic forms, which are in kerΔ3gδ , on the Spin bundle can be calculated in the adiabatic limit δ → 0. The expression for kerΔ30 := limδ→0 kerΔ3gδ was calculated in [R1], using the spectral sequence of [MaM], further developed in [D] [Fo]. Applying the results of [R1] to our case gives

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Proposition 2.9. (1) When 12 p1 (Y 11 ) = 0 ∈ H 4 (Y 11 ; R), i.e. [TrR ∧ R] = 0 ∈ H 4 (Y 11 ; R) then kerΔ30 = π ∗ Hg3 (Y 11 ) ⊂ Ω3 (S(Y 11 )). (2) When 12 p1 (Y 11 ) = 0 then kerΔ30 = R [CS3 (ω) − π ∗ h] ⊕ π ∗ Hg3 (Y 11 ) ⊂ Ω3 (S(Y 11 )) ,

(2.10)

where h ∈ Ω3 (Y 11 ) is the unique form such that dh = TrR ∧ R, h ∈ ∗ d Ω4 (Y 11 ) . We see that for the C-field in M-theory we have Proposition 2.10. (1) When Y 11 is a Spin manifold such that 1 11 ) = 0, the little c-field is a harmonic form both on Y 11 and on 2 p1 (Y 11 S(Y ). (2) When Y 11 is a String manifold, i.e. with 12 p1 (Y 11 ) = 0 in cohomology, so that [G4 ] = a, then there is a gauge in which the 3-form part of the C-field is that defining a String class, as in the above discussion. Remark 2.11. The combination of forms appearing in proposition 2.9 are exactly the ones also appearing in heterotic string theory. Indeed, the ChaplineManton coupling is a statement about the String class. The String class from the String condition on the target Y 11 . From (2.2), we see that when G4 = TrF ∧ F then 12 TrR ∧ R = dc. At the level of cohomology this means that 12 p1 (S, ω) = 0, i.e. that our space admits a String structure S. Let us form the combination CS3 (ω) − π ∗ c ∈ Ω3 (S). Consider a choice of String structure S ∈ H 3 (S(Y 11 ); Z). From (2.10), using the results in [R1], the adiabatic limit of the harmonic representative of S is given by (2.11)

[S]0 := limδ→0 [S]gδ = CS3 (ω) − π ∗ c3 ∈ Ω3 (S),

where the form c3 ∈ Ω3 (Y 11 ) has the properties: (2.12)

dc3

(2.13)

d c3

∗

1 p1 (S, ω), 2 = 0 (Lorentz condition) . =

Observation 2.12. Under these assumptions, the C-field can be identified with a String class. Remark 2.13. Consider a change of the String structure S. If the String structure is changed by ξ ∈ H 3 (Y 11 ; Z) then the cohomology class of the String structure changes, in the adiabatic limit, as (2.14)

[S + π ∗ ξ]0 = [S]0 + π ∗ [ξ]g .

Since ξ is a degree three cohomology class, the field strength G4 does not see the change in String structure. However, at the level of the exponentiated C-field, i.e. at the level of holonomy or partition function of the membrane, there will be an effect. See the discussion leading to observation 2.16. This will be formalized and considered in more detail in a future work.

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2.4. The String class from the membrane. Consider the embedding of the membrane Σ3 → Y 11 with normal bundle N . The fields on the membrane worldvolume include a metric h, the pullback of the C-field and a spinor ψ ∈ Γ(S(Y 11 )|Σ3 ). Restricting S(Y 11 ) to Σ3 gives the splitting (2.15)

S(Y 11 )|Σ3 = S(Σ3 ) ⊗ S − (N ) ⊕ S(Σ3 ) ⊗ S + (N ),

where kappa symmetry– a spinorial gauge symmetry– requires the fermions to be sections of the first factor [BST]. Taking the membrane as an elementary    object, the exponentiated action will contain a factor exp i Σ3 (C3 + i−3 vol(h) . This is p one part of the partition function, with another being the spinor part given by the Pfaffian of the Dirac operator. Neither of the factors in    (2.16) ZM 2 = Pfaff(DS(Σ3 )⊗S − (N ) ) exp i (C3 + i−3 vol(h) p Σ3

3 are separately well-defined,  but the product is [W7]. Taking Σ to be the boundary 4 of a 4-manifold B we get B 4 G4 in place of the first factor in the exponent in (2.16). The partition function is independent of the choice of bounding manifold B 4 .

The quantization condition (1.1) for the C-field in M-theory was derived in [W7] by studying the partition function of the membrane of worldvolume M 3 embedded in spacetime Y 11 . There, the manifold M 3 was assumed to be Spin. Here we notice that M 3 already admits a String structure because 12 p1 (M 3 ) = 0, by dimension reasons. Since this is automatic, one might wonder what is gained by assuming this extra structure. We will proceed with justifying this. The idea is that while M 3 always admits a String structure, we can have more than one String structure. We have the following diagram (2.17)

/ BString s9 ψ sss s s ss  ss σ / BSpin M3

/∗

K(Z, 3) O

λ

 / K(Z, 4) .

Choosing a String structure ψ is equivalent to trivializing λ ◦ σ. If we fix one String structure ψ then any other is classified by maps from M 3 to K(Z, 3), which is H 3 (M 3 ; Z). If we take K(Z, 3) = BK(Z, 2) then we can say that the set of String structures on M 3 is a torsor over the group of equivalence classes of gerbes on M 3 . From one given (equivalence class of) String structure we obtain for each (equivalence class of a) gerbe another (equivalence class of a) String structure. Notice that in the non-decomposible part of the C-field c3 + h3 , h3 is the curvature of the gerbe. It is closed as a differential form. We can see that there is a gerbe on the membrane worldvolume by taking the membrane to be of open topology and having a boundary on the fivebrane. There is then a gerbe connection C3 − db2 , where b2 is the chiral 2-form on the M5-brane worldvolume (see e.g. [AJ]). In this case we have the exponentiated action     (2.18) exp i b2 + i −3 vol(h) . p ∂Σ3

Σ3

(See also Remark 2.13 and the discussion around equation (2.25)).

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Now we consider the bounding 4-dimensional space B 4 , ∂B 4 = M 3 . Starting with a Spin M 3 , there are two cases to consider, according to whether B 4 is Spin or String. We will make use of an approach due to David Lipsky and to Corbett Redden. Let us start with the Spin case. Including B 4 in diagram (2.17) we get

(2.19)

/ BString s9 ψ sss s s ss  ss σ / BSpin M 3 _ ss9 τ sss s ss  sss B4

/∗

K(Z, 3) O

 / K(Z, 4)

λ

.

Let CM 3 be the cone on M 3 . The fact that

(2.20)

M3

ψ

/ BString

/∗

 BSpin

 / K(Z, 4)

commutes up to homotopy means precisely that there is a strictly commuting diagram /∗

M3

(2.21)

M3

i0

ψ

 BString  BSpin



.

i1

/ M3 × I 99 99 99 99 99 99 99 99 9  / K(Z, 4)

Moreover, the cone is precisely the pushout

(2.22)

M3

/∗

 M3 × I

 / CM 3

,

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through which, hence, diagram (2.21) factors (2.23)

/∗

M3 

i1

 / M3 × I / CM 3 99 99 99 99 99 Cψ 99 99 99 9  / K(Z, 4)

i0

M3

.

ψ

 BString  BSpin

Here Cψ is the extension of ψ from M 3 to the cone on M 3 . Therefore, we find that the following diagram λ◦σ

/ CM 3

M 3 _

(2.24)

 B4

/B

 4

Cψ

'/

∗

ρ

M3

CM 3

λ(τ,ψ)

 / K(Z, 4) 6

λ◦τ

commutes. Note that the fact that ψ extends from M 3 to the cone of M 3 , as indicated, is crucially another incarnation of the fact that ψ is homotopic to the map through the point. The map ρ is equivalent to a String structure, and the map λ(τ, ψ) is the relative String class. Let [B 4 , M 3 ] be the relative fundamental class and ,  the pairing between cohomology and homology. For this pairing we will study integrality and (in)dependence on the choice of B 4 or structures on B 4 . The long exact sequence for relative cohomology is (2.25)

· · · → H 3 (M 3 ) −→ H 4 (B 4 , M 3 ) −→ H 4 (B 4 ) −→ H 4 (M 3 ) · · · αω3 → λ(τ, ψ) + ∂(αω3 ) −→ λ(τ ) ,

where ω3 is the volume form on M 3 and α is a real number. The sequence then is explained as follows. Given a choice of initial String structure on M 3 , any other choice will be given by the difference with multiples of the volume form ω3 . Note that the only parameter which is varying is α ∈ R. Then, let B 4 be another bounding manifold with corresponding Spin structure τ . Then, by the index theorem,   λ(τ, ψ) λ(τ  , ψ) − =x∈Z, (2.26) 24 24 B4 B 4 

so that



 λ(τ, ψ) −

(2.27) B4

B 4

λ(τ  , ψ) = 24x, x ∈ Z .

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Now taking e2πi of both sides gives that the expression is an integer. So this together with what we explained above also gives that integral does not depend on the choice of B 4 or structures on B 4 , and we have the following Proposition 2.14. In the case when M 3 is String and B 4 is Spin, the relative pairing λ(τ, ψ) , [B 4 , M 3 ] = B 4 λ(τ, ψ) is well-defined mod 24.  Now consider the case when B 4 also admits a String structure, so that B 4 λ = 0. In this case, the question simply reduces to a statement in cobordism of String  8 ∼ = 3-manifolds Ω3 −→Z/24 defined by (M 3 , ψ) −→ B 4 λ(τ, ψ). Therefore Proposition 2.15. In the case when both M 3 and B 4 are String manifolds, the relative pairing λ(τ, ψ) , [B 4 , M 3 ] = B 4 λ(τ, ψ) is well-defined mod 24. Dependence of the membrane partition function on the String structure. Taking (2.16) into account and the fact mentioned above (in the proof of proposition 2.14 that changing the String structure of the membrane amounts to changing its volume, we have for membrane worldvolumes with String structure Observation 2.16. The membrane partition function depends on the choice of String structure on the membrane worldvolume. Remarks 2.17. (1) There are nonperturbative effects, namely instantons, resulting from membranes wrapping 3-cycles in spacetime. See for example [HM]. (2) In the case of string theory, the partition function depends crucially on the Spin structure of the string worldsheet. Modular invariance requires summing over all such structures [SW]. The observations we made above then suggest that the membrane theory would require a careful consideration of dependence on the String structure, and possibly summing over such structures. We hope to address this important issue elsewhere. The framing in Chern-Simons theory. We can look at the dependence of the membrane partition function on the String structure through the dependence of Chern-Simons theory on the choice of framing. The partition function of ChernSimons theory on M 3 depends on [W4]: M 3 , the structure group G, the ChernSimons coupling k, and a choice of framing f of the manifold. In particular, the semiclassical partition function, while independent of the metric, it does depend on the choice of framing, and different framings generally give different values for the partition function. However, there are transformations that map the value corresponding to one framing to the value corresponding to another. A framing of M 3 is a homotopy class of a trivialization of the tangent bundle T M 3 . Given a framing f : M 3 → T M 3 of M 3 the gravitational Chern-Simons term can be defined as  1 (2.28) IM 3 (g, f ) = f ∗ CS(ω) , 4π M 3 where g is the metric on M 3 , ω is the Levi-Civita connection on M 3 , and the integrand is the pullback via f of the Chern-Simons form on T M 3 .

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In dimension three, there is an isomorphism between the String cobordism and the framed cobordism group Ωfr group ΩString 3 . Thus, the study of Chern3 Simons theory with a String structure is then equivalent to the study of ChernSimons theory with a framing. Therefore, it is natural to consider a String structure in Chern-Simons theory, and hence on the membrane worldvolume, since the latter is essentially described by Chern-Simons theory. We argue that not only is a String structure allowable, but is in fact desirable. This is because such a structure explains the framing anomaly in a very natural way. Under the transformation IM 3 → IM 3 + 2πs, where s is the change

in framing, the d partition function transforms as [W4] ZM 3 → ZM 3 ·exp 2πis · 24 , for d a constant related to the level of the theory. This factor of 24, making the partition function essentially a 24th root of unity, is reflection of the fact that both the String- and the framed cobordism group are isomorphic in dimension three to Z/24. Any Lie group G has two canonical String structures defined by the left invariant framing fL and the right invariant framing fR of the tangent bundle T G. For example taking M 3 to be G = SU (2), there are three framings: a left framing and a right framing (related by orientation reversal) and a trivial framing given by taking S 3 = ∂D4 to be the boundary of the 4-disk. The invariants associated to these framings are the images of ΩString under the σ-map (the String orientation [AHS]) in tmf −3 ∼ = Z/24. This 3 map depends on the String structure in an analogous way that its more classical  cousin, the Atiyah α-invariant, refining the A-genus from Z to KO, depends on the Spin structure. From the isomorphisms π3 S 0 → π3 M String → π3 tmf [Ho] these values are as follows −→ tmf −3 1 [SU (2), fL ] −→ − 24 1 [SU (2), fR ] −→ 24   SU (2), ∂D4 −→ 0 . ΩString 3

(2.29)

The transformation of the partition function can then be soon more transparently using String cobordism. Thus, we get more confirmation to observation 2.16 and, in fact, we can also add Observation 2.18. The dependence on framing of Chern-Simons theory (and hence also for the membrane partition function) can be seen more naturally within String cobordism. We thus conjecture that the membrane partition function takes values in (twisted) tmf . Note that within Spin cobordism there would be no nontrivial expressions in dimension three. This is because ΩSpin = 0 and also the target for the Atiyah α3 invariant, KO3 (pt), is also zero. Furthermore any generalization of α, for example to the Ochanine genus with target KO3 (pt)[[q]] would also be trivial. Further relation to generalized cohomology. There is further a connection of the membrane partition function to generalized cohomology from another angle as follows. The Stiefel-Whitney class w4 is the mod 2 reduction of λ = 12 p1 . This implies that λ is even if and only if w4 = 0. The latter is in fact an orientation condition in real Morava E-theory EO(2) [KS1] (worked out there for a different but

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related purpose). Therefore, in order to remove the ambiguity in the quantization, we require EO(2)-orientation. Proposition 2.19. The membrane partition function is well-defined in EO(2)theory. In particular, G4 is an integral class when the underling spacetime is EO(2)oriented. Remarks 2.20. (1) This means that eleven-dimensional spacetime backgrounds in M-theory with no fluxes should be EO(2)-oriented. (2) Note that EO(2) is closely related to the theory EO2 of Hopkins and Miller, which in turn is closely related to T M F . (3) The M-theoretic partition function via E8 gauge theory of [DMW] is considered for the String case in [S7]. The case when the Spin bundle is trivial. Consider the case when S(Y 11 ) is trivial as a principal bundle. This then means that there is a global section. Such a section s : Y 11 → S(Y 11 ) gives an isomorphism of principal bundles s

S(Y 11 ) ∼ = Y 11 × Spin(11) .

(2.30)

unneth formula gives an Using the fact that H i (Spin(11); Z) = 0 for i = 1, 2, the K¨ induced isomorphism on integral cohomology H 3 (S; Z) S

(2.31)

∼ = ←→

H 3 (Y 11 ; Z) ⊕ H 3 (Spin(11); Z) (0 , 1Spin ) .

The String structure is then determined by s and is an element S ∈ H 3 (S; Z) which corresponds to the pullback of the generator 1Spin ∈ H 3 (Spin(11); Z). Note that the differential d acting on p∗ CS3 (ω) is 12 p1 (ω), which is the same as dc. This means that [p∗ CS3 (ω) − c] ∈ H 3 (Y 11 ; R). Consider the membrane worldvolume M 3 , taken as a 3-cycle X ∈ Map(M 3 , Y 11 ). Then p(X) ⊂ Y 11 × {pt} ⊂ S under the isomorphism (2.30) induced by the global section s. The triviality of the bundle implies that any class ξ ∈ H 3 (S; Z) is zero when evaluated on 3-cycles Σ3 in Y 11 ⊂ S, ξ, [s(Σ3 )], where [s(Σ3 )] is the fundamental class of the 3-cycle s(Σ3 ) in S(Y 11 ). Then, in real cohomology    ∗ [ξ]0 = (CS3 (ω) − π c) = (s∗ CS3 (ω) − c) . (2.32) 0= s(Σ3 )

s(Σ3 )

Σ3 ∗

This holds for an arbitrary 3-cycle Σ so that [s CS3 (ω) − c] = 0 ∈ H 3 (Y 11 ; R). ∗ ∗ Now if d (s∗ CS3 (ω)) = 0, the since d c = 0, then s∗ CS3 (ω) − c is harmonic, ∗ and hence zero, so s CS3 (ω) = c. Therefore, in this case, following the general construction [R1] [R2], s∗ CS3 (ω) and c are equal to the coexact and harmonic components, respectively. 3

The C-field as a Chern-Simons 2-gerbe. In [CJMSW] a Chern-Simons bundle 2-gerbe is constructed, realizing differential geometrically the Cheeger-Simons invariant [CS]. This is done by introducing a lifting to the level of bundle gerbes of the transgression map from H 4 (BG; Z) to H 3 (G; Z). A similar construction is given in [AJ]. Both groups of interest, Spin(n) and E8 are simply-connected, a fact that removes some subtleties from the discussion.

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For any integral cohomology class in H 3 (Spin(n); Z), there is a unique stable equivalence class of bundle gerbes [Mu, MuS] whose Dixmier-Douady class is the given degree three integral cohomology class. Geometrically H 4 (BSpin(n); Z) can be regarded as stable equivalence classes of bundle 2-gerbes over BSpin(n), whose induced bundle gerbe over Spin(n) has a certain multiplicative structure. More precisely, given a bundle gerbe G over Spin(n), G is multiplicative if and only if its Dixmier-Douady class is transgressive, i.e., in the image of the transgression map τ : H 4 (BSpin(n); Z) → H 3 (Spin(n); Z) [CJMSW]. Consider a principal Spin(n)-bundle P with connection A on a manifold M . For the Chern-Simons gauge theory canonically defined by a class in H 4 (BSpin(n); Z), there is a Chern-Simons bundle 2-gerbe Q(P, A) associated with the P which is defined to be the pullback of the universal Chern-Simons bundle 2-gerbe by the classifying map f of (P, A) [CJMSW]. With the canonical isomorphism between the Deligne cohomology and CheegerSimons cohomology, the Chern-Simons bundle 2-gerbe Q(P, A) is equivalent [CJMSW] in Deligne cohomology to the Cheeger-Simons invariant S(P, A) ∈ ˇ 3 (M, U (1)), which is the differential character that can be associated with each H principal G-bundle P with connection A [CS]. A bundle gerbe version of the discussion of the invariance of the C-field in section 2.3 is provided by the following Proposition 2.21 ([AJ] (also [CJMSW])). A Chern-Simons 2-gerbe is contained in the data of the C-field. The metric torsion part of C-field. Connections with torsion come in various classes. Especially interesting is the case when the torsion is totally antisymmetric. In this case the new connection is metric and geodesic-preserving, and the Killing vector fields coincide with the Riemannian Killing vector fields. Connections with torsion arise in in eleven-dimensional supergravity [E] and in heterotic string theory and type I [Str]. In the first case an ansatz is taken such that the little c-field is proportional to the torsion tensor T ∈ Ω3 (Y 11 ), most prominently when Y 11 is an S 7 bundle over 4-dimensional anti-de Sitter space AdS4 , in which case the torsion is parallelizing – see [DNP]. In the second case, the H-field in heterotic string theory acts as torsion, which is important for compactification to lower dimensions [Str]. 3. The M-theory Dual C-Field and Fivebrane Structures 3.1. The E8 model for the dual of the C-field. In [DFM] the electric charge associated with the C field is studied. From the nonlinear equation of motion (2.8) of the C-field, the induced electric charge that is given by the cohomology class 1 (3.1) [ G2 − I8 ]DR ∈ H 8 (Y, R) . 2 ˇ (and also ΘY (a)), In [DFM] the integral lift of (3.1) is studied and denoted ΘY (C) ˇ where C = (A, c). A tubular neighbourhood of the M5-brane worldvolume V in Y is diffeomorphic to the total space of the normal bundle N → V . Let X = Sr (N ) be the 10π dimensional sphere bundle of radius r, so that the fibers of X →V are 4-spheres.

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An 11-dimensional manifold Yr with boundary X is then constructed by removing from Y the disc bundle Dr (N ) of radius r. Yr = Y − Dr (N ), the bulk manifold, is the complement of the tubular neighbourhood Dr (N )). There are two path integrals or wavefunctions:  (1) The bulk C field path integral Ψbulk (CˇX ) ∼ exp[G∧∗G]Φ(CˇYr ) where the integral is over all equivalence classes of CˇYr fields that on the boundary assume the fixed value CˇX . This wavefunction is a section of a line bundle L on the space of CˇX fields. (2) The M5-brane partition function ΨM 5 (CˇV ), which depends on the Cˇ field on an infinitesimally small (r → 0) tubular neighbourhood of the M5brane. In general, Ψbulk ΨM 5 is not gauge invariant and therefore it is a section of a line bundle. However, one can consider a new partition function ΨM 5 that is obtained from multiple M5-branes stacked on top of one another instead of just a single M5-brane. This stack gives rise to a twisted gerbe on V as follows [AJ]. In order for Ψbulk ΨM 5 to be well defined, the twisted gerbe has to satisfy [CS(π∗ (ΘX ))] − [ϑijkl , 0, 0, 0] = [DH ] + [1, 0, 0, CV ] ,

(3.2)

where, CS(π∗ (ΘX )) is the Chern-Simons 2-gerbe associated with π∗ (ΘX ) and a choice of connection on the E8 bundle with first Pontryagin class π∗ (ΘX ) (all other 2-gerbes differ by a global 3-form), while [ϑijkl , 0, 0, 0] is the 2-gerbe class associated with the torsion class θ on V , β(ϑ) = θ, and [1, 0, 0, CV ] is the trivial Deligne class associated with the global 3-form CV . In particular (3.2) implies π∗ (ΘX ) − θ = ξDH ,

(3.3)

where the RHS is the characteristic class of the lifting 2-gerbe. 3.2. Twisted Fivebrane structures. Fivebrane structures are obtained by lifting String structures as follows. Definition 3.1 ([SSS1][SSS2]). An n-dimensional manifold X has a Fivebrane structure if the classifying map X → BO(n) of the tangent bundle T X lifts to the classifying space BFivebrane := BO 9.

fˆ

(3.4) M

f

BO(n) 9 :  / BO(n) .

Theorem 3.2 ([SSS2]). (1) The obstruction to lifting a String structure on X to a Fivebrane structure on X is the fractional second Pontrjagin class 16 p2 (T X). (2) The set of lifts, i.e. the set of Fivebrane structures for a fixed String structure in the real case, or the set of BU 9 structures for a fixed BU 7 structure in the complex case, is a torsor for a quotient of the seventh integral cohomology group H 7 (X; Z).

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Twisted Fivebrane structures. Twisted cohomology refers to a cohomology which is defined in terms of a twisted differential on ordinary differenial forms. The twisted de Rham complex Ω• (X, dH2i+1 ) means the usual de Rham complex but with the differential d replaced by dH2i+1 := d + H2i+1 ∧, which squares to zero by virtue of the Bianchi identity for H2i+1 , provided that H2i+1 is closed. The form on which this twisted differential acts  would involve components that are 2i formm degrees apart, e.g. of the form F = n=0 Fk+2in . The main case considered in [S6], and which corresponds to heterotic string theory, corresponds to i = 3, k = 2, and m = 1. There it was observed that, at the rational level, the (abelianized) field equation and Bianchi identity in heterotic string theory can be combined into an equation given by a degree seven twisted cohomology. This is discussed further in section 5. In [S6] it was also proposed that such a differential should correspond to a twist of what was there called a higher String structure and later in [SSS2] was defined, studied in detail and given the name Fivebrane structure. Definition 3.3 ( [SSS3]). A β-twisted Fivebrane structure on a brane ι : M → X with String structure classifying map f : M → BO(n) 8 is a cocycle β : X → K(Z, 8) and a map c : BO(n) 8 → K(Z, 8) such that there is a homotopy η between BO(n) 8 and X, M ι

 X

f

ssss ssssss s s s sssss η u} ssss β

/ BO(n) 8 c

.

 / K(Z, 8)

1 The fractional class 48 p2 show up in physics. As an application, which was originally the motivation:

Theorem 3.4 ([SSS3]). (1) The dual formula for the Green-Schwarz anomaly cancellation condition on a String 10-manifold M is the obstruction to defining a twisted Fivebrane structure, with the twist given by ch4 (E), where E is the gauge bundle with structure group E8 × E8 . (2) The integral class in M-theory dual to G4 defines an obstruction to twisted Fivebrane structure, which is the obstruction to having a well-defined partition function for the M-fivebrane. The precise description actually requires some refinement of the Fivebrane 1 structure to account for appearance of fractional classes such as 48 p2 rather than 1 9 p . This is called F -structure in [SSS3]. Also, a corresponding fivebrane Lie 6 2 6-algebra is defined in [SSS3], where the description of the twist in terms of L∞ algebras is also given. Higher bundles: The following applications are of interest (see [SSS1]). 1. Chern-Simons 3-forms arise as local connection data on 3-bundles with connection which arise as the obstruction to lifts of ordinary bundles to the corresponding String 2-bundles and are shown to be governed by the String Lie 2-algebra. For g an ordinary semisimple Lie algebra and μ its canonical 3-cocycle, the obstruction to lifting a g-bundle to a String 2-bundle is a Chern-Simons 3-bundle with characteristic class the Pontrjagin class of the original bundle.

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2. The formalism immediately allows the generalization of this situation to higher degrees. Indeed, certain 7-dimensional generalizations of Chern-Simons 3-bundles obstruct the lift of ordinary bundles to certain 6-bundles governed by the Fivebrane Lie 6-algebra. The latter correspond what was defined in [SSS1] as the Fivebrane structure, for which the degree seven NS field H7 plays the role that the degree three dual NS field H3 plays for the n = 2 case. Using the 7-cocycle on so(n), lifts through extensions by a Lie 6-algebra, defined as the Fivebrane Lie 6-algebra, is obtained. Accordingly, Fivebrane structures on String structures are indeed obstructed by the second Pontrjagin class. 3.3. Harmonic part of the dual of the C-field. Consider the equation of motion (2.8). Set C7 := ∗G4 , so that we get (3.5)

dC7 = je ,

where je is the electric current given in (2.9). Let



(3.6) Δ7g : Ω7 (Y 11 ), g −→ Ω7 (Y 11 ), g ∗

be the Hodge Laplacian on 7-forms on Y 11 with respect to the metric g. Taking d of both sides of equation (3.5) we get ∗

Proposition 3.5. In the Lorentz gauge, d C7 = 0, we have (1) Δ7g C7 = ∗jm , where jm is the fivebrane magnetic current, related to the electric current je (2.9) by (3.7)

jm = d(∗je ) . (2) C7 is harmonic if p → 0 and/or there are no fivebranes.

This is the degree seven analog of proposition 2.8. The geometry of the de Rham representatives of the Fivebrane class for the dual C-field is considered in [RS]. 3.4. The integral lift. The quantization law (1.1) on the C-field leads to an integral lift of the electric charge, the left hand side of the equation of motion (2.8), as mandated by Dirac quantization. The lift is [DFM]   1 1 1 [G8 ] = a− λ a − λ + I8 2 2 2 1 8 . a(a − λ) + 30A (3.8) = 2 Proposition 3.6. Properties of [G8 ]. (1) [G8 ] and [G4 ] obey the multiplicative structure on KSpin. (2) G8 defines an obstruction to having a (twisted) Fivebrane structure. The first part is proved in [S7] and the second part in [SSS3]. Let us expand a bit on the first point. The quadratic refinement defined in [DFM] is encoded in the multiplicative structure in the K-theory for Spin bundles. Starting with a real unoriented bundle ξ, the condition w1 (ξ) = 0 turns ξ into an oriented bundle, and the condition w2 (ξ) = 0 further makes ξ a Spin bundle. Obviously then, a real O-bundle becomes a Spin bundle when W = 0, and so the kernel of W is the  reduced group KSpin(X). Thus W fits into the exact sequence [LD] (3.9)

  0 −→ KSpin(X) = ker W −→ KO(X) −→H 1 (X; Z2 ) × H 2 (X; Z2 ). W

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Among the properties of this class proved in [DFM] is that it is a quadratic refinement of the cup product of two degree four classes a1 and a2 (3.10)

Θ(a1 + a2 ) + Θ(0) = Θ(a1 ) + Θ(a2 ) + a1 ∪ a2 .

We would like to look at this from the point of view of the structure on the product of the cohomology groups H 4 ( · ; Z)×H 8 ( · ; Z). For this we consider the two classes a and Θ(a) as a pair (a, Θ(a)) in H 4 ( · ; Z) × H 8 ( · ; Z). Then the linearity of the addition of the degree four classes a and the quadratic refinement property (3.10) of Θ(a) can both be written in one expression in the product H 4 ( · ; Z) × H 8 ( · ; Z), which makes use of the ring structure, namely (3.11)

(a1 , Θ(a1 )) + (a2 , Θ(a2 )) = (a1 + a2 , Θ(a1 ) + Θ(a2 ) + a1 ∪ a2 ) .

The second entry on the RHS is just Θ(a1 + a2 ) − Θ(0), and so it encodes the property (3.10). We can define the shifted class Θ0 (a) as the difference Θ(a) − Θ(0), so that (3.11) is replaced by





(3.12) a1 , Θ0 (a1 ) + a2 , Θ0 (a2 ) = a1 + a2 , Θ0 (a1 + a2 ) , corresponding to the special case (3.13)

Θ0 (a1 + a2 ) = Θ0 (a1 ) + Θ0 (a2 ) + a1 ∪ a2 .

This is then just a realization of the multiplication law on H 4 ( · ; Z) × H 8 ( · ; Z) which, for (a, b) in the product group, is (3.14)

(a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 + a1 ∪ a2 ).

Note that in order to get this law we had to use the modified eight-class Θ0 (a), or 8 . alternatively discard Θ(0) = 30A We now make the connection to Spin K-theory. Similarly to the case of other kinds of bundles, e.g. complex or real, one can get a Grothendieck group of isomorphism classes of Spin bundles up to equivalence. The reduced KSpin group  of a topological space can be defined as KSpin(X) = [X, BSpin]. For the case of BSpin, we will be interested in relating Spin K-theory to cohomology of degrees 4 and 8. Such a homomorphism of abelian groups (3.15)

 QX : KSpin(X) → H 4 (X; Z) × H 8 (X; Z)

 where Q1 and Q2 are the is defined by [LD] QX (Q1 (ξ), Q2 (ξ)) for ξ ∈ KSpin(X),  Spin characteristic classes of [Th]. For two bundles ξ and γ in KSpin(X), and for k ≤ 3,  (3.16) Qk (ξ ⊕ γ) = Qi (ξ) ∪ Qj (γ). i+j=k

Remark 3.7. The above multiplicative structure is a Z-analog (or 4k-analog) of the Z2 -structure in the case of KO-theory. Given a topological space X, let  KO(X) be the reduced KO group for X and let (3.17)

 W : KO(X) −→ H 1 (X; Z2 ) × H 2 (X; Z2 )

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be the map W (ξ) = (w1 (ξ), w2 (ξ)), where wi (ξ) denotes the i-th Stiefel-Whitney  class of ξ ∈ KO(X). There is a group structure on H 1 (X; Z2 ) × H 2 (X; Z2 ) making W a homomorphism, i.e. a map that preserves the group structure. 3.5. Invariance of the dual C-field. From (3.8) we see that we can write the dual C-field at the level of differential forms as 1 (3.18) G8 = G4 ∧ G4 − I8 + dc7 . 2 3.5.1. Case I: Trivial cohomology, no one-loop term. In this case we have 12 p1 = 0 = 16 p2 . From (3.18) we have dC7 = 12 G4 ∧ G4 + dc7 . When C3 = CS3 (A) we get C7 = 12 CS3 (A) ∧ G4 . 3.5.2. Case II: 12 p1 = 0. The invariance of G8 will include the invariance of the three terms in (3.18), hence invariance of G4 , of the Pontrjagin classes of S(Y 11 ), and of the differential form C7 ∈ Ω7 (Y 11 ). Therefore, we have Proposition 3.8. The dual field G8 is invariant under the following transformations: (1) The invariances of the C-field from proposition 2.7, (2) C7 → C7 + Λ7 , (3) CS7 (ω) → CS7 (ω) + λ7 , where Λ7 and λ7 are closed differential forms in Ω7 (Y 11 ). As in the case for the C-field, the expressions above when recast in terms of differential characters will result in requiring Λ7 and λ7 to be closed integral forms, i.e. to be in Ω7Z (Y 11 ). The Fivebrane class. Remark 3.9. The Fivebrane class can be discussed in a manner that is very similar to that of the String class. The change of Fivebrane structure F can be seen from the M-theory fivebrane, similarly to the way the String structure S is seen from the membrane. Here we assume that spacetime is ten-dimensional. We relate the dual Hfield H7 to the degree seven Chern-Simons form CS7 above via a class C7 that we introduce. Consider the principal bundle (3.19)

String(n) → STRING(M ) → M,

where STRING(M ) is the total space of the String(n) bundle on our ten-dimensional spacetime M corresponding to heterotic string theory. Recall that this is one of the two bundles in that theory, namely the one obtained from the lift of the tangent bundle (non-gauge one). We can build a class C7 out of the Chern-Simons 7-form CS7 (A) on STRING(X) for the given connection 1-form A on the gauge bundle as C7 = CS7 (A) − π ∗ H7 ,  with dCS7 = π ∗ (p2 (M )) and such that Σ7 C7 = 1 where Σ7 is a fundamental 7-cycle in the fiber. Similarly to the Spin case [ASi], we have

(3.20)

Proposition 3.10. If X is 6-connected, C7 represents a generator of H 7 (STRING(M ); Z).

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Proof. C7 is closed since dC7

= =

dCS7 (A) − dπ ∗ H7 π ∗ (p2 (M )) − π ∗ dH7

= π ∗ (p2 (M ) − dH7 ) = 0.

(3.21)

Consider the homotopy exact sequence (3.22) · · · → πi (String(n)) → πi (STRING(M )) → πi (M ) → πi−1 (String(n)) → · · · corresponding to the bundle (3.19). Assuming πi (M ) = 0 for i ≤ 6, the sequence gives that π7 (STRING(M )) ∼ = π7 (String(n)) = Z. Therefore, C7 represents a generator of H 7 (STRING(M ), Z).  Remark 3.11. If M is ten-dimensional and is 6-connected then it is topologically the ten-dimensional sphere. This follows from Poincar´e duality and the Poincar´e conjecture in ten dimensions. The first gives that the cohomology groups in degrees 8 and 9 are the same as those in degrees 2 and 1, respectively, and hence are zero by 6-connectedness. M , having cohomology groups Z in degrees 0 and 10, is a homology ten-sphere. However, by the Poincar´e conjecture, which is a theorem in ten dimensions, M must be the sphere S 10 itself. 3.6. Higher differential characters. We consider the space Map(Z, M ), with Z of dimension six. The space Map(M, STRING(M )) is a bundle over Map(Z, M ) with structure ‘group’ Map(M, String(10)). Let (3.23)

ev  : Z × Map(Z, STRING(M )) → STRING(M )

be the evaluation map. Proposition 3.12. There exists a Cheeger-Simons differential 6-character B6 with dB6 = C7 and such that ev  ∗ B6 exponentiates to a differential 0-character on Map(Z, STRING(M )) with values in U (1). Proof. This is analogous to [ASi] where the degree two case is established. Our case corresponds to replacing B2 with B6 and the spin condition with the  0 : Z → P0 ∈ STRING(M ), let γ be a path String condition. For a trivial map Φ   from Φ0 to Φ ∈ Map(Z, STRING(M )) so that γ maps the interval [0, 1] times Z to  This path exists since πi (STRING(M )) = 0 for i ≤ 6. STRING(M ) with γ(1) = Φ. The function ev  ∗ B6 will be given by       (3.24) exp 2πi C7 = exp 2πi α1 

γ([0,1]×Z)

γ([0,1])

∗

with α1 is the one-form Z ev  C7 . The function is independent of the path γ: if γ1 is another path then γ1−1 γ is a map of S 1 × Z → P0 and    (3.25) C7 = C7 − C7 ∈ Z, (γ1−1 γ)(S 1 ×Z)

γ(S 1 ×Z)

γ1 (S 1 ×Z)

i.e. α1 represents an integral 1-cocycle. Since C7 represents an element of H 7 (STRING(M ); Z), there a Cheeger-Simons differential 6-character B6  exists with dB6 = C7 . Then Z ev  ∗ B6 exponentiates to a differential 0-character on

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Map(Z, STRING(M )) with values in S 1 .

Proposition 3.13. The curvature of the 0-character ev  ∗ B6 is the one-form α1 , which can be interpreted as a flat connection on a circle bundle over Map(Z, STRING(M )). Remarks 3.14. 1. We consider the effect of gauge transformation on H7 . Changing H7 to H7 + dβ6 leads to a shift in B6 as B6 + β6 . 2. The topological term  (3.26) I= dx1 ∧ · · · ∧ dx6 Bμ1 ···μ6 ∂1 X μ1 · · · ∂6 X μ6 Z

 can be interpreted as log Z ev  ∗ B6 , in analogy with the case of the string. 3. Similar results can be extrapolated to eleven dimensions. 3.7. Mapping Space Description. As we recalled in the introduction, String structures on a space M is related to Spin structure on the loop space LM . Is there a corresponding statement about Fivebrane structures? We do not fully answer this question here but we do give possible scenarios. In the case of String structure, the string class on the loop space LM is obtained by pull-back of the second Chern character, ch2 (E), of a bundle E on M to give a bundle LE on LM via the evaluation map ev : S 1 × LM → M,   so that the String class is S 1 ev ∗ ch2 (E), where S 1 : H ∗ (S 1 ×LX; C) → H ∗−1 (LM ; C) is the integration along S 1 .

(3.27)

Now the idea is to generalize this to the case of the Fivebrane structure. We see two directions for doing so: (1) Replace the second Chern character ch2 by a higher degree Chern character chq , q > 2, and keeping the same evaluation map (3.27). This will yield higher degree analogs of the String class, but still on the loop space LM . (2) In addition, replace the circle S 1 by a higher-dimensional space Y so that the loop space LM = [S 1 , M ] is replaced by a higher degree generalization Map[Y, M ], the space of maps from Y to M . We start with the first. Here, for a vector bundle E over M , one has the higher degree analogs of the String class as  q q+1 (3.28) C (LE) = −(2πi) q! ev ∗ chq+1 (E), S1

which gives a class of degree 2(q + 1) − 1 = 2q − 1 on LM . Such a generalization of the usual String structure has been defined in [As]. Kuribayashi in [Ku] finds fairly special conditions under which it is still true that C p (LE) = 0 if and only if chp+1 (E) = 0, namely when H ∗ (M ; R) is a tensor product of truncated polynomial and exterior algebras. This generalizes McLaughlin’s result [Mc] in the case p = 1 for the usual String structure, where 12 p1 (P ) = 0 implies a String structure on a bundle P on M only when π2 (M ) = 0.

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An example of the higher classes (3.28) is the first term in the anomaly polynomial   1 1 1 2 (3.29) dH7 = 2π ch2 (A) − p1 (ω)ch2 (A) + p1 (ω) − p2 (ω) , 48 64 48 where A and ω are the connections on the gauge bundle V and the tangent bundle T M , respectively. Note that we can use p = 3 in (3.28) to get a degree seven 1 1 3 class upon integration over the circle S 1 ev ∗ ch4 (E), which gives (2π) 3 4! C (LV ) + decomposables + non − gauge factors. In this case the fivebrane class can be described as follows. We have, for n ≥ 5, the isomorphism  (3.30) ◦ ev ∗ : H 8 (BString(n); Z) → H 7 (LBString(n); Z) . S1

Since H (BString(n); Z) = 0 and H 7 (String(n); Z) = Z, we have that the image in (3.30) is Z. In terms of the space itself, the evaluation map and integration over the circle give  (3.31) ev ∗ : H 8 (M ; Z) → H 7 (LM ; Z). 7

S1

Next we consider the second case. In addition, replace the circle S 1 by a higherdimensional space Y so that the loop space LM = [S 1 , M ] is replaced by a higher degree generalization Map[Y, M ]. Then what replaces the evaluation map (3.27)  is ev : Y × Map[Y, M ] → M and (3.28) would then become Y ev ∗ chp+1 (E). The result will be a class on Map[Y, M ] of degree 2p − dimY . Obviously, when p = 1 and dimY = 1, we get back the String case. We will discuss further aspects of the general case in section 3.8. There are two special cases of interest, the first when Y is a torus and the second when Y is a sphere. Let the dimension of Y be m and that of X be n. Then the two cases are • Y = T m : This gives Map[T m , M ] = Lm M , the higher iterated loop spaces of M , i.e. Lm M = LL · · · LX n (m times). This is the iterated loop space of M which is obtained by looping on X m times. The bundle replacing the loop bundle of the String case will be a bundle with structure group the toroidal group Map[T m , G] = Lm G. • Y m = S m : In this case the space to consider is Map[S m ; M ], corresponding to the homotopy groups πm (M ) of M . Such spaces, at least for low m, have been studied in connection to gauge theory in physics in [Mi]. In the physical situation under consideration, Y can be taken to be the spatial part W 5 of the fivebrane worldvolume in spacetime of dimension ten for the heterotic fivebrane and dimension eleven for the M5-brane. Then, for p = 3, the integration over the worldvolume yields a degree three class on the fifth loop space L5 X 10 (likewise for eleven dimensions). The homotopy groups of the two groups, G and Lm G are related by πn (Lm G) = πn+m (G) ⊕ πn (G), so that, in particular, π3 (L5 G) = π8 (G) ⊕ π3 (G). For example, 7-connectedness of G would be the same as 2-connectedness of L5 G.

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3.8. Determinantal 6-gerbes. We can look at the relation to higher gerbes by looking at the structure on the worldvolume itself. Consider the case when the worldvolume has five compact dimensions, i.e. time is considered to be R. The Fivebrane structure can be described in terms of a 5-gerbe in general. In the special case of an index 5-gerbe, this can be interpreted as the degree five part of the Family’s index theorem for Dirac operators [Lo] [BKTV]. The construction of the index 1-gerbe is given in [Lo] where also general features of the higher gerbes are given. The construction for those is given in [BKTV] which we follow below. The degree two component of the families index theorem is given by the first Chern class, or the curvature, of a determinant line bundle. There is an obstruction to realizing the component of the families index theorem of degree higher than two as a curvature of some geometric object. This, however, is automatic if we choose our spacetime X to be 5-connected. so X is homeomorphic to S 10 , and the explicit construction is given in [Lo] in this case. The corresponding Deligne classes form a countable sets corresponding to different trivializations of the bundle on the five-skeleton of the triangulation of X and are labeled by 2 index5−2j (X; Z) = H 1 (X; Z) ⊕ H 3 (X; Z). j=1 H Assume Y m to be a compact oriented C ∞ -manifold of dimension m over which we have a smooth complex vector bundle E. Then the space of sections ΓE of the bundle is expected to give rise to a determinental 5-gerbe DetΓ(E), generalizing the determinant line bundle [BKTV]. Thus, in the case of the fivebrane, if we take the spatial part then the currently perceived wisdom leads us to a determinantal 5-gerbe and if we take the even-dimensional spacetime then we are led to a determinantal 6-gerbe. We now consider a family of fivebranes by considering the map to spacetime q : W 5 → X 10 of relative dimension 5 and a C ∞ bundle E on W 5 . Then the ∞∗ 7 10 characteristic class of the 5-gerbe would be a class in H 6 (X 10 , CX , Z). 10 ) = H (X The class of the determinantal 6-gerbe in complex cohomology should be thought of as a 6-fold delooping of the usual first Chern (determinantal) class. Still following [BKTV], the Real Riemann-Roch formula gives    (3.32) C1 (q∗ E) = ch(E) ∧ Td(T W 5 ) 12 ∈ H 7 (X 10 , C), W5

where we are integrating the degree twelve part of the index formula over the five-dimensional spatial part of the fivebrane worldvolume to get a degree seven class. This involves the Dolbeault operator ∂ over a complex envelope of W 5 . In  the the smooth category, we just replace the Todd class Td in (3.32) with the A, roof-genus of T W 5 . 4. The Gauge Algebra of Supergravity in 6k − 1 Dimensions Five-dimensional SO(2) supergravity on a five-dimensional Spin manifold X is the theory obtained by coupling pure supergravity in five dimensions to an SO(2) vector multiplet [CN][C]. The former is made of a metric g on X and a RaritaSchwinger field ψ, which is a section of the spin bundle SX coupled to the tangent bundle T X, ψ ∈ Γ(SX ⊗T X). The latter contains an SO(2)-valued, hence abelian, one-form C1 with curvature two-form G2 = dC1 . The Lagrangian L(5) will have

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• a bosonic part L(5),bos for the bosonic fields (g, G2 ), • a fermionic part L(5),ferm for the fermionic field Ψ, • and an interaction part L(5),int for the terms that are mixed in the bosonic and fermionic fields. In L(5) = L(5),bos + L(5),ferm + L(5),int we will consider only the bosonic part, given by the five-form 1 1 (4.1) L(5),bos = R ∗1l − G2 ∧ ∗G2 − G2 ∧ G2 ∧ C1 , 2 6 where ∗ is the Hodge duality operator on differential forms in five dimensions, and R is the scalar curvature of the metric g of X. Eleven-dimensional supergravity [CJS] has some common features with fivedimensional supergravity [C] [CN], described above. The bosonic field content is the same, except that the potential C3 , replacing C1 , is now of degree three so that the corresponding field strength G4 is of degree four. The Hodge dual in eleven dimensions to G4 is G7 . The bosonic part of the Lagrangian is given by the eleven-form 1 1 (4.2) L(11),bos = R ∗1l − G4 ∧ ∗G4 − G4 ∧ G4 ∧ C3 . 2 6 From here on we treat both theories at the same time. We thus take X to be a (6k − 1)-dimensional Spin manifold on which we define a supergravity with ChernSimons term built out of the potential C2k−1 , with corresponding field strength G2k . The value k = 1 corresponds to the five-dimensional case and the value k = 2 to the eleven-dimensional case. The equations of motion are obtained from the Lagrangian via the variational δL(6k−1),bos principle. The variation δC = 0 for C2k−1 gives the corresponding equation 2k−1 of motion 1 (4.3) d ∗ G2k + G2k ∧ G2k = 0 . 2 We also have the Bianchi identity (4.4)

dG2k = 0 .

The second order equation (4.3) can be written in a first order form, by first

writing d ∗G2k + 12 C2k−1 ∧ G2k = 0 so that 1 ∗G2k = G4k−1 := dC4k−2 − C2k−1 ∧ G2k , 2 where C4k−2 is the potential of G4k−1 , the Hodge dual field strength to G2k in 6k − 1 dimensions.  The action Sbos = X dvol(X)Lbos , and hence the equations of motion, are invariant under the abelian gauge transformation δC2k−1 = dλ2k−2 , where λ2k−2 is a (2k−2)-form. We can alternatively write the gauge parameter as Λ2k−1 = dλ2k−2 . In fact, the first order equation (4.5) is invariant under the infinitesimal gauge transformations 1 δC2k−1 = Λ2k−1 , δC4k−2 = Λ4k−2 − Λ2k−1 ∧ C2k−1 , (4.6) 2 (4.5)

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where Λ4k−2 is the (4k − 2)-form gauge parameter satisfying dΛ4k−2 = 0. Applying two successive gauge transformations with different parameters Λi and Λi , i = 2k − 1, 4k − 2, for the first and second one, respectively, and forming the the commutators gives   δΛ2k−1 , δΛ2k−1 = δΛ4k−2 ,   δΛ2k−1 , δΛ4k−2 = 0,   (4.7) = 0, δΛ4k−2 , δΛ4k−2 with the new parameter Λ4k−2 = Λ2k−1 ∧ Λ2k−1 . Note that the transformations are nonlinear, and this can be tracked back to the presence of the Chern-Simons form in the Lagrangian (4.1). We now introduce generators v2k−1 and v4k−2 for the Λ2k−1 and Λ4k−2 gauge transformations, respectively. On the generators, from the commutation relations (4.7), we get the graded Lie algebra {v2k−1 , v2k−1 }

= −v4k−2 , [v2k−1 , v4k−2 ] = 0 , [v4k−2 , v4k−2 ] = 0 ,

(4.8)

Note that we can use a graded commutator, which unifies a commutator and an anticommutator, so that the above algebra (4.8) becomes [v2k−1 , v2k−1 ] = −v4k−2 , [v2k−1 , v4k−2 ] = 0 , [v4k−2 , v4k−2 ] = 0 ,

(4.9)

where it is now understood that we are using graded commutators. The generators satisfy the following properties (1) The generators v2k−1 and v4k−2 are constant: dv2k−1 = 0 = dv4k−2 . (2) The grading on the generators v2k−1 and v4k−2 follow that of the potentials A2k−1 and A4k−2 , respectively. Hence, v2k−1 is odd and v4k−2 is even. Thus, d(v2k−1 α) = −v2k−1 dα and d(v4k−2 α) = v4k−2 dα, for any α. We will think of these “generators” vi as elements of a graded Lie algebra, where we will write C2k−1 ⊗ v2k−1 , etc. instead of just C2k−1 for the fields (see the discussion around equation (4.26)). The field strengths can be combined into a total uniform degree field strength G by writing V = eC2k−1 ⊗ v2k−1 eC4k−2 ⊗ v4k−2 ,

(4.10) so that G (4.11)

1 dC2k−1 ⊗ v2k−1 + (dC4k−2 − C2k−1 ∧ dC2k−1 ) ⊗ v4k−2 2 = G2k ⊗ v2k−1 + G4k−1 ⊗ v4k−2 = G2k ⊗ v2k−1 + ∗G2k ⊗ v4k−2 .

=

Note the analogy with usual (i.e. not higher-graded) nonabelian gauge theory. V is the analog of g and G = dVV −1 is the analog of dgg −1 .

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The equation of motion for C2k−1 (= Bianchi identity for C4k−2 ) and the Bianchi identity for C2k−1 are obtained together from dG − G ∧ G = −dV ∧ dV −1 − dVV −1 ∧ dVV −1 = 0 .

(4.12)

Indeed, using the commutators (4.9), we have G ∧G

= (G2k ⊗ v2k−1 ) ∧ (G2k ⊗ v2k−1 ) + (G2k ⊗ v2k−1 ) ∧ (∗G2k ⊗ v4k−2 )

+(∗G2k ⊗ v4k−2 ) ∧ (G2k ⊗ v4k−2 ) 1 [G2k ⊗ v2k−1 , G2k ⊗ v2k−1 ] + [G2k ⊗ v2k−1 , ∗G2k ⊗ v4k−2 ] = 2 1 G2k ∧ G2k ⊗ [v2k−1 , v2k−1 ] − G2k ∧ ∗G2k ⊗ [v2k−1 , v4k−2 ] = 2 1 = − G2k ∧ G2k ⊗ v4k−2 . (4.13) 2 Hence, (4.12) follows from the equation of motion (4.3) and the Bianchi identity (4.4), which indeed correspond, respectively, to the coefficient of v2k−1 and v4k−2 in the expression for dG. The case k = 2, corresponding to eleven-dimensional supergravity, was derived in [CJLP]. 4.1. Models for the M-Theory Gauge Algebra. In the previous section we have seen that G4 and its dual can be written in terms of the total uniform degree field strength G, the generators in which satisfy an algebra. It is natural to ask about the nature of the generators and the graded structure in which they result. In this section we provide a description in terms of homotopy (or highercategorical) Lie algebras: L∞ -algebras based on the constructions in [SSS1], and superalgebras corresponding to (1|1) supertranslations. 4.2. The gauge algebra as an L∞ -algebra. One connection to L∞ -algebras is the appearance of higher form abelian Chern-Simons theory. Recall that for g any semisimple Lie algebra and μ = ·, [·, ·] the canonical 3-cocycle on it, we call gμ the corresponding (skeletal version of the) stringμ (g) Lie 2-algebra. Similarly, for g any semisimple Lie algebra and μ7 the canonical 7-cocycle on it, we call gμ the corresponding (skeletal version of the) Fivebrane Lie 6-algebra [SSS1] [SSS3]. Reminder on L∞ -algebra valued diferential forms. Recall from [SSS1] that for g any L∞ -algebra with CE(g) its Chevalley-Eilenberg differential graded commutative algebra (DGCA), the space of GCA-morphisms CE(g) → Ω(X) is isomorphic to the degree zero elements in the graded vector space Ω• (X) ⊗ g, where g is in negative degree. Flat L∞ -algebra valued forms can be realized as graded tensor products A ∈ Ω• (Y ) ⊗ g of forms with L∞ -algebra elements with the special property that • A is of total degree 0 , • A satisfies a flatness constraint of the form (4.14)

dA + [A ∧ A] + [A ∧ A ∧ A] + · · · = 0 ,

where d and ∧ are the operations in the deRham complex and where [·, · · · , ·] are the n-ary brackets in the L∞ -algebra. It is usually more convenient to shift g by one into non-positive degree (hence with the usual Lie 1-algebra part in degree 0) and accordingly take A to be of total degree 1.

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Recall one of the central constructions of [SSS1] involving the Weil algebra W(g). For g any L∞ -algebra with degree (n + 1)-cocycle μ that is in transgression with an invariant polynomial P  (4.15) 0O PO o P dCE(g)

_ μo

dW(g)

_  cs

CE(g) o o

? _ W(g)basic

W(g) o

we can form the String-like extension Lie n-algebra gμ and the corresponding ChernSimons Lie (n + 1)-algebra csP (g) with the property that W(gμ )  CE(csP (g)). In [SSS1] this construction was of interest for the case that μ was a nontrivial cocycle on a semisimple Lie 1-algebra. Another interesting case in which the construction works is when an invariant polynomial P suspends to 0, i.e. if it is in transgression with the 0-cocycle μ = 0. Notice that in particular all decomposable invariant polynomials P = P1 ∧ P2 , for P1 and P2 nontrivial and with transgression elements csi , dW(g) csi = Pi , suspend to 0, since for them we can choose the Chern-Simons element cs = cs1 ∧P2 , which vanishes in CE(g) because P2 does, by definition:  (4.16) P1 ∧ P2 . P1 ∧O P2 o 0O dCE(g)

_ 0o

CE(g) o o

dW(g)

_  cs1 ∧ P2

? _ W (g)basic

W(g) o

Higher abelian Chern-Simons forms. A very simple but useful example are the decomposable invariant polynomials on shifted u(1) in an even number of shifts: b2k−2 u(1), for k any positive integer. In this case (4.17)

CE(b2k−2 u(1)) = (

•  ( c ), d = 0)  2k−1

and (4.18)

W(b2k−2 u(1)) = (

•  ( c ⊕ g ), dc = g, dg = 0) .   2k−1

2k

The invariant polynomials are all the wedge powers g, , g ∧ g, g ∧ g ∧ g of the single indecomposable one P := g (4.19)

inv(b2k−2 u(1)) = (

•  ( P  ), dP = 0) .  2k

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Notice that in this case the CE-algebra of the “String-like extension” b2k−2 u(1)μ=0 is that of b2k−2 u(1) ⊕ b4k−1 u(1): CE(b2k−2 u(1)μ=0 ) = CE(b2k−2 u(1) ⊕ b4k−3 u(1)) .

(4.20)

Abelian Chern-Simons L∞ -algebras. Let k ∈ N be a positive integer. Then the Lie (2k − 2)-algebra b2k−2 u(1) has a decomposable degree 4k invariant polynomial P4k which is the product of two copies of the standard degree 2k-polynomial. The corresponding Chern-Simons Lie (4k − 1)-algebra csP4k (b2(k−1) u(1)) is given by the Chevalley-Eilenberg algebra of the form •   (4.21) CE(csP4k (b2k−2) u(1))) = ( c2k−1 , g2k , c4k−2 , g4k−1 ), d where dc2k−1 dc4k−2 dg2k dg4k−1

(4.22)

= g2k , = c2k−1 ∧ g2k + g4k−1 , = 0, = g2k ∧ g2k .

This has a canonical morphism onto 2k−2

(4.23)

CE(b

u(1) ⊕ b

4k−3

u(1)) =

• 

 ( c2k−1 , c4k−2 ), d = 0

with respect to which we can form the invariant or basic polynomials (4.24) CE(b2k−2 u(1) ⊕ b4k−3 u(1)) o This is the DGCA (4.25)

i∗

CE(csP4k (b2k−2) u(1))) o

? _ basic(i∗ ) .

⎛

⎞ •  ⎜ ⎟ basic(i∗ ) = ⎝ ( g2k  ⊕ g4k−1 ), (dg2k = 0 , dg4k−1 = g2k ∧ g2k )⎠ .       2k

4k−1

We consider the L∞ -algebra sa that admits the above as its Chevalley-Eilenberg algebra. This sa is a graded Lie algebra with generators v3 and v6 in degree 3 and 6, respectively, and with the graded Lie brackets being [v3 , v3 ] = v6 [v3 , v6 ] = 0 [v6 , v6 ] = 0 .

(4.26)

Flat differential form data with values in this L∞ -algebra is given by a degree 1-element A = G4 ⊗ v3 + G7 ⊗ v6 ∈ Ω• (X) ⊗ g, where • G2k a closed 2k-form; • G4k−1 is a 4k − 1-form satisfying dG4k−1 = G2k ∧ G2k .

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In total we have that a Cartan-Ehresmann connection with respect to i∗ is given by differential form data as follows: (4.27) Ω•vert (Y ) o OO

Avert

CE(b2k−2 u(1) ⊕ b4k−3 u(1)) OO i∗

Ω• (Y ) o O

(A,FA )

? Ω• (X) o

CE(csP4k (b2k−2) u(1))) O

? basic(i∗ )

G2k = dC2k−1 G4k−1 = dC4k−2 + C2k−1 ∧ G2k

dG2k = 0 dG4k−1 = G2k ∧ G2k

This can be regarded as a certain Cartan-Ehresmann connection for the product of a line (2k − 1)-bundle and a line 4(k − 2)-bundle The situation for 11-dimensional supergravity. The local gauge connection data of 11-dimensional supergravity is given by a 3-form C3 with curvature 4-form G4 = dC3 which can be captured in a duality-symmetric manner by regarding C3 as the data giving a flat connection with values in the abelian Chern-Simons Lie 6-algebra obtained by setting k = 2, subject to a self-duality constraint: A = G4 ⊗ v3 + (∗G4 ) ⊗ v6 . The flatness condition satisfied by this is then equivalent to the equations of motion for G4 # " dG4 = 0 (4.28) (dA + [A ∧ A] = 0) ⇔ d ∗ G4 = − 12 G4 ∧ G4 Therefore, Theorem 4.1. The C-field and its dual in M-theory define an L∞ -algebra as their gauge algebra. 4.3. The gauge algebra as a Superalgebra. We next interpret the Mtheory gauge algebra in another novel way. The commutator of v3 with v6 and that of v3 and v6 are zero. Furthermore, the commutator of two v3 ’s gives v6 , it is natural to suspect that each one of the two generators belongs to a different subspace in some grading. Indeed, these are the even and odd gradings, and we have Proposition 4.2. The generators v3 and v6 form a Lie superalgebra of translations in (1|1) dimensions. ∂ ∂ Remark 4.3. This is analogous to the generator ∂θ + θ ∂x , where x is an even coordinate and θ is an odd coordinate. Thus we see an analog of the supersymmetric quantum mechanical relation {Q, Q} = H, where Q is the supercharge and H is the Hamiltonian of the system.

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.

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5. Duality-Symmetric Twists In the twisted cohomology setting one can form uniform degree expressions for both the fields, e.g. the cohomology classes, and the twisted differential. In this section we consider twists of degrees higher than the familiar three. Also, given the discussion of uniform degree fields in the previous section, it is natural to use these for more exotic twists. 5.1. Degree seven twists. The NS H-field in type II string theory serves as the twist in the K-theoretic classification of the RR fields. This involves unifying the fields of all degrees into one total RR field. We investigate, based on [S6], the case of heterotic string theory where there are E8 × E8 or Spin(32)/Z2 gauge fields in addition to the H-field. a. Rationally: Considering the gauge field and its dual as a unified field, the equations of motion at the rational level contain a twisted differential with a novel degree seven twist. Consider the case where the Yang-Mills group G, is broken down to an abelian subgroup, thus making the curvature F2 be simply dA. The manifolds M 10 are chosen such that this breaking via Wilson lines (= line holonomy)  is possible. The result of the variation of the action S = H3 ∧∗H3 + F2 ∧∗F2 with respect to A gives (d − H7 ∧)F = 0, where F = F2 + ∗F2 is defined as the combined curvature, and H7 is equal to ∗H3 at the rational level, in analogy with the RR fields. In this analysis, following [S6], we used the “Chapline-Manton coupling” H3 = CS3 (A), where CS3 (A) is the Chern-Simons three-form for the connection A, whose curvature is F2 . This gives a twisted differential dH7 = d − H7 ∧ which in nilpotent, i.e. squares to zero, d2H7 = 0, since H7 is closed. (i) Let vn denote the nth generator of the complex oriented cobordism ring. Consider the case n = 2 and let R = R[[v2 , v2−1 ]] be a graded ring. The generator vn has dimension 2pn − 2, so that at the prime p = 2, v2 has dimension 6. Let dH7 = d − v2−1 H7 be the twisted de Rham differential of uniform degree one. Denote by ΩidH (M 10 ; R) the space of dH7 -closed differential forms of total degree i on 7 M 10 . The total curvature F is an element of degree two, i = 2, in the above space of forms. The equation dH7 F = 0 defining the complex is just the Bianchi identity and the equation of motion of the separate fields. Another possibility is to use the −1 combination F = u−1 1 F2 + u2 F8 , where u1 has degree two and u2 has degree eight. (ii) The second step is to ask whether the argument at the level of rational cohomology generalizes to some rational generalized cohomology theory. A twisting of complex K-theory over M is a principal BU⊗ -bundle over M . From BU⊗ ≡ K(Z, 2) × BSU⊗ , the twisting is a pair τ = (δ, χ) consisting of a determinantal twisting δ, which is a K(Z, 2)-bundle over M and a higher twisting χ, which is a BSU⊗ -torsor. Twistings are classified, up to isomorphism, by a pair of classes [δ] ∈ H 3 (M, Z) and [χ] in the generalized cohomology group H 1 (X, BSU⊗ ). The former is twisted K-theory, where the twist is given by the Dixmier-Douady (DD) class. The twistings$of the rational K-theory of X are classified, up to isomorphism, by [Te] the group n>1 H 2n+1 (M ; Q). This shows that, in addition to the usual H 3 Q-twisting, one can in principle have twistings from H 5 Q and H 7 Q etc. It is thus possible that a degree seven twist comes from complex K-theory. However, it is not obvious how to isolate just the degree seven part from the tower of all n > 3 odd-dimensional twists.

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b. Integrally: The above generalizes the usual degree three twist that lifts to twisted K-theory and raises the natural question of whether at the integral level the abelianized gauge fields belong to a generalized cohomology theory. (i) The appearance of the higher degree generator connects nicely with the discussion in [KS1][KS2] [KS3] on generalized cohomology in type II (and to some extent type I) string theories. Further the appearance of the w4 = 0 condition in [KS1], interpreted in [S4] in the context of F-theory, which is a condition in heterotic string theory, is another hint for the relevance of generalized cohomology in the heterotic theory. Given the appearance of elliptic cohomology through W7 = 0 and the H7 -twist, then the condition W7 + [H7 ] = 0 is expected to make an appearance, which would give rise to some notion of twisted structure in a similar way that the analogous condition W3 + [H3 ] = 0 of [FW] amounts to a twisted Spinc structure. This structure that we seek would be related to a twisted String structure, but is not quite the same but is implied by it, since the String orientation condition implies W7 = 0 via the action of the operation Sq 3 . Further, we expect the modified condition to correspond to a differential d7 in the AHSS of twisted generalized theories, possibly Morava K-theory and elliptic cohomology, since the ‘untwisted’ differential is the first nontrivial differential there, in analogy to the ‘twist’ for d3 generated by [H3 ] in the K-theory AHSS. Conjecture 5.1. The cohomology class W7 +[H7 ] corresponds to a differential in twisted Morava K-theory and twisted Morava E-theory, where [H7 ] acts as the twist. This is a generalization of the statement that W3 + [H3 ] corresponds to the differential d = Sq 3 + [H3 ]∪ in twisted K-theory. Of course the construction of such twisted generalized theories is not yet established. Nevertheless we note the following. (ii) In general, twists of a cohomology theory E are classified by BGL1 (E), i.e. the twisted forms of E ∗ (X) correspond to homotopy classes of maps [X, BGL1 (E)]. An equivalent way of saying this, which the more familiar one in the context of K-theory, is that the twists are classified by BAut(E), where Aut(E) is the automorphism group of E. The homotopy groups of BGL1 (E) are given as units of the ring E 0 (pt) in degree 1, and as E k−1 (pt) in degree k > 1. For K-theory, K 0 (pt) = Z gives π0 BGL1 (K) = Z/2, K 2 (pt) = Z gives π3 BGL1 (K) = Z, which is detected by a map K(Z, 3) → BGL1 (K) giving the standard degree twist. In addition, there is π7 BGL1 (K) = Z. (iii) Another possibility is the following. Twists of T M F are classified by BGL1 (T M F ). This may have nontrivial homotopy in degree 7 coming from the homotopy in degree 6 of the connective theory tmf . 2 5.2. Duality-symmetric twists in ten-dimensional string theory. In  type II string theory one encounters the Ramond-Ramond (RR) fields F = i Fi u−i , where u is the Bott generator and i is the degree of the RR field, which is even for type IIA and odd for type IIB. This satisfies the equation dH3 F = 0, where dH3 2The homotopy groups of the spectrum tmf and those of its periodic version T M F are   related as π∗ (T M F ) = π∗ (tmf ) (Δ24 )−1 , where Δ is the discriminant of elliptic curves.

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is the twisted differential, whose uniform degree expression is dH3 = d + u−1 H3 ∧. Here H3 is the Neveu-Schwarz (NS) 3-form. This is explained very well in [Fr2]. As we saw in section 5.1, in [S6] a degree seven twist was uncovered in heterotic and type I string theory, where the twist is given by the dual H7 of the usual NS H-field H3 in ten dimensions. The differential is of the form dH7 = d + v −1 H7 ∧. In this theory, for instance for v = u3 , one can form the uniform total degree one field strength [Fr2] (5.1)

H = u−1 H3 + u−3 H7 ,

with corresponding potentials, or B-fields, of total degree zero B = u−1 B2 +u−3 B6 . Given that the total field strengths are built of more than one component, we can ask whether the corresponding differential of uniform degree might be built out of a twist that has more than one component. Consider a candidate twisted de Rham differential with an expression of the form (5.2)

dH = d + u−1 H3 ∧ +u−3 H7 ∧ .

The square is d2H contains the terms that are zero because dH3 and dH7 are differentials. In addition, there is the cross-term u−4 (H3 ∧ H7 + H7 ∧ H3 ), which is zero by antisymmetry of the wedge product. Of course, another way to immediately see this is to write dH as dH3 + u−3 H7 ∧ or as dH7 + u−1 H3 ∧. Thus one can build a twisted graded de Rham complex out of such a differential. Remarks. 1. In fact, one can build a differential by adding to dH all expressions of the form u−i H2i+1 ∧, i.e. (5.3)

dH = d +

∞ 

u−i H2i+1 ∧ .

i=0

2. As differential forms, the u are constant, i.e. du = 0. We can conceive of two modifications of this: First consider the generators appearing in front of the H2i+1 to be independent. For example, in [S6], instead of (5.1), we used the expression 3 (5.4)

−1 −1 H = v(1) H3 + v(2) H7 ,

where v(1) is still the Bott generator and v(2) is the generalization of that generator, i.e. identified as coming from a complex-oriented generalized cohomology theory at the prime p = 2. In this case it still holds that v(1) and v(2) are constants as differential forms. From the above discussion the following immediately follows. Proposition 5.2. There is a twisted graded de Rham complex with differential  −1 d+ ∞ i=1 v(i) H2i+1 ∧ , provided the differential forms H2i+1 are closed. The coefficients v(i) are constant as differential forms and can be taken to be either dependent or independent. 3In section 4 we used v to indicate a generator of degree i, so to make a distinction we i are using the notation v(i) to indicate a generator of level i in complex-oriented generalized cohomology. We understand that the first notation is more standard for the second notion, but since this is the only occurrence of the higher Bott generators, then we hope it will not cause a confusion.

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One can associate analytic torsion [MWu] to this type of twisted de Rham complex and a spectral sequence for the corresponding twisted cohomology [LLW]. 5.3. Duality-symmetric twists in eleven-dimensional M-theory. In Mtheory the situation is much more interesting. In this case we will see that we can have twists of even degree and the generators are not independent in the sense that they satisfy relations. Some aspects of this discussion have been observed in [S1] [S2] [S3]. In the low energy limit of M-theory, in addition to the metric and the gravitino, there is the C-field C3 with field strength G4 . We can build a differential with G4 a twist as follows. The square of the expression dG4 = d + v3 G4 ∧ is 4 d2G4 = d2 + d(v3 G4 ∧) + v3 G4 ∧ d + v3 G4 ∧ v3 G4 ∧ .

(5.5)

On the right hand-side of (5.5), the first term is always zero since the bare d is the de Rham differential. For the second term we need to decide whether v3 is even or odd as a differential form. Since G4 is even we see that we have to choose v3 to be odd in order to cancel the third term. In addition, for the left-over from the second term to be zero, G4 has to be closed. The last term has no other term against which to cancel, so it has to be zero by itself. We need v3 to be idempotent. This can be achieved either by the fact that the form degree is odd or by the stronger condition that it squares to zero, i.e. that it is a Grassmann variable. The above discussion generalizes in an obvious way to the case when the coefficient has degree 2i − 1 and the field has degree 2i. Therefore we have Proposition 5.3. The de Rham complex can be twisted by a differential of the form d + v2i−1 G2i ∧ provided that G2i is closed and v2i−1 is Grassmann algebravalued. In M-theory one can consider the field dual to the C-field. This is a field strength G7 , which at the rational level is Hodge dual to G4 . We can use G7 to twist the de Rham differential in the same way that H7 did. Furthermore, in the same way as in (5.4) one could form a duality-symmetric uniform degree field strength G = v3−1 G4 + v6−1 G7 . This expression can now be used to twist the de Rham differential, leading to dG = d + G∧ = d + v3−1 G4 ∧ +v6−1 G7 ∧ .

(5.6)

The conditions for (5.6) to be a differential are given in the following. Proposition 5.4. The de Rham complex can be twisted by the differential dG provided that either (1) dG7 = 0 and v32 = 0, or (2) {v3 , v3 } = v6 and dG7 = − 12 G4 ∧ G4 . Furthermore, this differential on G is equivalent to the equations of motion and the Bianchi identity of the C-field. 4In this section we have suppressed the tensor product between generators and fields for ease of notation.

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Remarks 5.5. 1. The first case would hold when there is no Chern-Simons term in the M-theory action. 2. The second case arises in M-theory and realizes the equation of motion for the C-field. While this is what appears in M-theory, mathematically we can have combination of even and odd fields of any degrees −1 −1 d + v2m−1 G2n + v2m G2m+1 .

(5.7)

3. The generators in proposition 5.4 have appeared in [CJLP] in the context of the M-theory gauge algebra (which is generalized in section 4). What we have done above is relate them to twisted cohomology. 6. M-brane Charges and Twisted Topological Modular Forms 6.1. Evidence for TMF. 6.1.1. Construction of anomaly-free partition functions. The E-theoretic partition function in type IIA. The K-theoretic partition function encounters an anomaly [DMW], given by the seventh integral Stiefel-Whitney class W7 , whose cancellation [KS1] is the orientation condition in elliptic cohomology for Spin manifolds, and in second integral  Morava K-theory at p = 2 K(2) (cf. [Mor]) for oriented manifolds. The above class W7 is the result of applying the Steenrod square operation Sq 3 on w4 , the fourth Z/2 Stiefel-Whitney class or, equivalently, the result of applying the Bockstein operation β = Sq 1 on the degree six class Sq 2 w4 , W7 = Sq 3 (w4 ) = βSq 2 (w4 ), by the Adem relation Sq 3 = Sq 1 Sq 2 . Theorem 6.1 ([KS1]). (1) A 10-manifold X is orientable with respect ˜ to K(2) iff W7 (X) = 0. ˜ (2) The M-theory partition function is anomaly-free when constructed on K(2)orientable spaces. Similar results hold also for Morava E(2)-theory. In [KS1] an elliptic refinement of the mod 2 index j is obtained. Assuming that X is orientable with respect to a real version EO(2) of E(2)-theory, there is an EO(2)-orientation class [X]EO(2) ∈ EO(2)10 (X). Now for x ∈ E 0 (X), the class xx lifts canonically to EO(2)0 (X), so −1 3 v(2) ] j(x) = xx, [X]EO(2)  ∈ EO(2)10 , the right hand side being EO(2)10 = Z/2[v(1) by [HK1]. The assumption on EO(2)-orientation is made precise: Theorem 6.2 ([KS1]). (1) A spin manifold X is orientable with respect to EO(2) if and only if it satisfies w4 (X) = 0, where w4 is the fourth Stiefel-Whitney class. (2) When w4 = 0, this uncovers another anomaly to the existence of an elliptic cohomology partition function. Remark 6.3. The class w4 is the mod 2 reduction of the integral class λ = 12 p1 , so that the vanishing of λ implies the vanishing of w4 , which in turn implies the vanishing of W7 . Therefore, the String orientation condition λ = 0 is a necessary condition for the cancellation of the DMW anomaly. There is a “character map” E → K[[q]][q −1 ] where q is a parameter of dimension 0, with K[[q]] a product of infinitely many copies of K, and the notation [q −1 ] signifies that q is inverted. The map is determined by what happens on coefficients [AHS].

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Theorem 6.4 ([KS1]). The refined partition function is a one-parameter fam−1 3 v(2) is of dimension zero and serves ily of theta functions. At the prime p = 2, v(1) as the expansion parameter q. The E-theoretic partition function in type IIB. The construction of the partition function in the IIB case is analogous to that of type IIA. Instead of K 1 (X), one starts with E 1 (X) where E is a complex-oriented elliptic cohomology. The construction proceeds precisely analogously as in the K 1 (X) case. However, the discussion of the phase is delicate. First there is the pairing in E 1 (X): E 1 (X) ⊗ E 1 (X) → E 2 (X) → E −8 = E 0 where the second map is capping with the fundamental orientation class in E10 (X). To construct a θ-function, a quadratic structure is needed, which amounts to considering real elliptic cohomology: A product of an x ∈ E 1 (X) with itself can be given a real structure, which gives rise to an element of ω(x) ∈ ER1+α X, which when capped with the fundamental orientation class in ER10 (X) gives an element in E α−9 . This is a Z/2-vector space 3n−1 2−n −4 2 v(2) σ a , n ≥ 1[HK1]. Therefore, generated by the classes v(1) Theorem 6.5 ([KS2]). There is a quadratic structure depending on one free parameter, which leads to a precise IIB analog of the θ-function constructed for IIA using real elliptic cohomology in [KS1]. TMF and the type IIB fields. A particularly convenient combination of the two degree-three fields in type IIB string theory is G3 = F3 − τ H3 , where τ is the parameter on the upper half plane. This is a field with modular weight −1 since it transforms as G3 = G3 · (cτ + d)−1 under τ  = (aτ + b)/(cτ + d). In tmf , a class of modular weight k appears in tmf 2k (X 10 ). Therefore, Proposition 6.6 ([KS3]). The fields of type IIB string theory as elements in ˜ 3 ∈ tmf −2 X 10 . tmf, satisfy G This points to the 12-dimensional picture: suppose, in the simplest possible physical scenario [V] that V 12 = X 10 × E where E is an elliptic curve, then G3 × μ ∈ tmf 0 (V 12 ) where μ ∈ tmf 2 (E) is the generator given by orientation. It is consistent that the class ends up in dimension 0 and no odd number shows up. Modular classes of weight 0, however, must be in dimension 0. The mathematical interpretation of τ appears only when we apply the forgetful map E k (X) → K k (X)[[q 1/24 ]][q −1/24 ] with q = exp(2πiτ ). In fact, it is necessary to generalize to an elliptic cohomology theory E which is in general modular only with respect to some subgroup Γ ⊂ SL(2, Z). For forms with such modularity, fractional powers of q are needed: in the case of complex-oriented cohomology, one encounters q 1/24 . The map E → K[[q 1/24 ]][q −1/24 ] whose induced map on coefficients (homotopy groups) makes the k-th homotopy group modular of weight k/2 ˜ 0 (S k ) = E −k (pt) would is not the correct normalization to use because then E have modular weight −k/2 and not 0. The correct normalization is given by composition with Adams operations (or alternatively with Ando operations [A]) ψ η : K[[q 1/24 ]][q −1/24 ] → K[[q 1/24 ]][q −1/24 ], where η is the Dedekind function (Δ1/24 where Δ is the discriminant form), which is a unit in K[[q 1/24 ]][q −1/24 ]. For general k, multiplication by η k is needed. More on elliptic curves and F-theory. Roughly speaking, there is an elliptic cohomology theory for every elliptic curve. There is no universal elliptic curve over

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a commutative ring, so on what basis should one ‘favor’ one elliptic curve over the other, and hence one elliptic cohomology over the other? If one considers all elliptic curves at once then the corresponding generalized cohomology theory is tmf , which is then, in a sense, the universal ‘elliptic’ cohomology theory. However, the price to pay is that tmf is not an elliptic cohomology theory and also not even-periodic. For more on this see e.g. [Lu]. In our identification of the F-theory elliptic curve with the elliptic curve in elliptic cohomology (after suitable reduction of coefficients), we hence, along the lines of [S4], expect elliptic curves with a fixed modulus in Ftheory to correspond to elliptic cohomology while ones with a modulus parameter varying in the base of the elliptic fibration, i.e. families, to correspond to tmf . It is then tempting to propose that tmf sees all possible compactifications of F-theory on an elliptic curve, i.e. all admissible elliptic fibrations. S-duality and twisted K-theory are not compatible. Type IIB string theory has a duality symmetry, S-duality, which is analogous to electric/magnetic duality in gauge theory. In the presence of H3 , the description of the RR fields of type IIB using twisted K-theory is not immediately compatible with S-duality. The origin of this is that the RR fields are considered as elements of K-theory while the S-dual field H3 is taken to be a cohomology class, leading to the breaking of the symmetry. Furthermore, type IIB string theory has a five-form in place of the four-form in type IIA and in M-theory. In [DMW] the apparent puzzle about the incompatibility of twisted K-theory and S-duality is raised. A definite statement is proved in [KS2]. The condition for anomaly cancellation for F3 is (Sq 3 + H3 ) ∪ F3 = 0, which is not invariant under the full SL(2, Z) group. The direct SL(2, Z)-invariant extension of the above equation is [DMW] F3 ∪ H3 + βSq 2 (F3 + H3 ) = 0. One immediate question is that of justification (and interpretation) of the nonlinear term βSq 2 H3 = H3 ∪ H3 . The point is to exhibit this as a differential, or obstruction for the cohomological pair (H3 , F3 ) to lift to the theory. The usual requirement that twisted K-theory be a module over K-theory, which forces the ‘structure group’ of the bundle of K-theories in question to be the multiplicative infinite loop space GL1 (K) of K-theory, violates the condition. One can then ask for some further generalized twisting, where, for a particular H3 , the choice of allowable F3 ’s would not form a vector space, i.e. whether one could consider a form of twisted K-theory which is not a module cohomology theory over ordinary K-theory. In [KS2] this is shown not to exist if the twisting space is K(Z, 3). The classifying space, i.e. a topological space B such that the affine-twisted K 1 -group would be classified by homotopy classes of maps X → B, cannot exist because the ‘group cohomology’ H 2 (CP ∞ , BU ) vanishes. The above equation cannot occur as first Postnikov invariant, so Theorem 6.7 ([KS2]). K(Z, 3)-twisted K-theory is not compatible with Sduality in type IIB string theory. 4. Re-interpreting the twist: Due to the above, one has to seek a solution in the realm of higher generalized cohomology theories. However, a question arises whether or not to leave the twist. The twist introduces an intrinsic non-commutativity which seems to prevent further delooping of the theory into a modular second cohomology group, giving an indication for a an untwisted generalized cohomology theory.

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There is a map K(Z, 3) → T M F coming from the String orientation. A twist of K-theory of the form X → K(Z, 3) then gives rise to an element of T M F (X) by composition X → K(Z, 3) → T M F , in fact defining “elliptic line bundles” (cf. [D]). Let us explain this. The classifying space BGL1 K for elementary twistings of complex K-theory splits, as an infinite loop space, as a product of two factors A × B. The first factor is a K(Z, 3) bundle over K(Z/2, 1) which splits as a space but has nontrivial infinite loop structure classified by Sq 3 ∈ H 3 (H(Z/2); Z). There is a natural infinite loop map B → T M F from B to the representing space for topological modular forms, and so by projecting through B a map BGL1 K → T M F . In particular an elementary twisting of K-theory for X determines a T M F class on X [D]. The geometric interpretation of these T M F classes is simplified if restricted to those classes coming from twistings involving only the K(Z, 3) factor of B. Such a twisting is determined by a map X → K(Z, 3) or equivalently by a BS 1 bundle on X. This bundle can be thought of as a stack 5 locally isomorphic to the sheaf of line bundles on X and hence as a 1-dimensional 2-vector bundle on X. In this sense the T M F classes coming from K-theory twistings can be viewed as 1-dimensional elliptic elements and twisted K-theory as K-theory with coefficients in this “elliptic line bundle” [D]. Therefore, what looks like twisting to the eyes of K-theory, untwists and becomes merely a multiplication by a suitable element in T M F or any suitable form of elliptic cohomology. Thus, Observation 6.8 ([KS2]). If both F3 and H3 are viewed as elements of elliptic cohomology, i.e. symmetrically, and the twisting is replaced by multiplication then the S-duality puzzle is solved. 6.2. Review of D-brane charges and twisted K-theory. The analogy with the more familiar case of the NS field H3 is as follows. The cohomology class [H3 ] appears in the definition of a twisted Spinc -structure (6.1)

W3 + [H3 ] = 0,

a condition for consistent wrapping of D-branes around cycles in ten-dimensional spacetime [FW], where W3 is the third integral Stiefel-Whitney class of the normal bundle, the vanishing of which allows a Spinc -structure. In the presence of the NS B-field, or its field strength H3 , the relevant K-theory is twisted K-theory, as was shown in [W6] [FW] [Ka] by analysis of worldsheet anomalies for the case the NS field [H3 ] ∈ H 3 (X, Z) is a torsion class, and in [BM] for the nontorsion case. Twisted K-theory has been studied for some time [DK] [Ro]. More geometric flavors were given in [BCMMS]. Recently, the theory was fully developed by Atiyah and Segal [AS1] [AS2]. It is a further result that [H3 ] acts as a determinantal K(Z, 2)-twist for complex K-theory. The left hand side of the expression (6.1) is in fact the first differential d3 in the Atiyah-Hirzebruch spectral sequence for twisted K-theory – see [BCMMS] [AS1] [AS2]. Then, a twisted D-brane in a B-field (X, H3 ) is a triple (W, E, ι), where ι : W → X is a closed, embedded oriented submanifold with ι∗ H = W3 (W), and E ∈ K 0 (W) (see [BMRS]). 5These are yet another incarnation of the line bundle gerbes mentioned earlier, for instance in the paragraphs just before proposition 2.21. This stack incarnation is precisely the “gerbe” in the original sense of the word (a locally non-empty and transitive stack).

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A D-brane wrapping a homology cycle is inconsistent if it suffers from an anomaly, and is sometimes inconsistent if the homology cycle cannot be represented by any nonsingular submanifold. This is detected by the first Milnor primitive cohomology operation Q1 = βP31 where β is the Bockstein and P31 is the Steenrod power operation, both at the prime p = 3. In contrast, the twisted Spinc condition is at p = 2. In fact, we have Observation 6.9 ([ES2]). The twisted Spinc condition is not sufficient. ∗ D-brane charges are classified by the twisted K-group KH (X). A rigorous formulation of such D-brane charges requires a Thom isomorphism and a pushforward map. Indeed the Thom isomorphism and push-forward in twisted K-theory are established in [CW]: Corresponding to the map ι : W → X there is (1) Push-forward map: ι! : Kι•∗ σ+W3 (ι) (W) → Kσ• (X). (2) Thom isomorphism: K • (W) ∼ = Kι•∗ σ+W3 (ι) (W).

With the use of the Riemann-Roch formula and index theorem in twisted K-theory it is now established that the RR charges in the presence of an H-field are indeed classified by twisted K-theory [CW2]. Now applying the push-forward map for ι in twisted K-theory, one can associate a canonical element in KH (X), the desired D-brane charge of the underlying Dbrane [CW] (6.2)

∗ ι! : K(W) ∼ (X), = Kι∗ H+W3 (W) (W) −→ KH

Hence, Definition 6.10. For any D-brane wrapping W determined by an element E ∈ K(W), the charge is (6.3)

∗ (X). ι! (E) ∈ KH

6.3. The M-brane charges and twisted TMF. In the case of M-theory, the object carrying charges with respect to G4 is the M5-brane and we will study conditions for consistent wrapping of such branes on cycles in eleven-dimensional spacetime. Hence, the candidate object to carry charges with respect to TMF is the M-theory fivebrane. In this section we will provide a point of view on the interpretation of Witten’s quantization condition (1.1) for G4 , which will give the context within which we describe M-brane charges in the following section. In the case of the string, the target spacetime is assumed to be Spin, i.e. w2 (X 10 ) = 0. Then this also implies that X 10 is certainly Spinc . Then the requirement that the brane’s worldvolume be Spinc is equivalent to requiring the normal bundle to the D-brane to be Spinc . On the other hand, for the M-theory case, Witten’s flux quantization is obtained from the embedding of the membrane in spacetime Y 11 . Taking Y 11 be Spinc will not be enough this time. We will see below how the twisted String condition [Wa] [SSS3] in the three bundles: tangent bundle to the worldvolume, the normal bundle, and the tangent bundle of the target will be related. Remarks 6.11. The Interpretation of Witten’s quantization.

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1. The equation (1.1) makes sense only if λ is divisible by two. It means that it is not [G4 /2π] but [G4 /π] that is well defined as an integral cohomology class and that this class is congruent to λ modulo two [W7]. 2. A weaker condition than (1.1) can be obtained by multiplying by two, provided there is no 2-torsion, (6.4)

2[G4 ] + λ = 2a.

Condition (6.4) gives condition (1.1) if there is no 2-torsion and once λ is divisible by two. 3. We rewrite condition (6.4) in turn in a suggestive way as λ − 2 ([G4 ] − a) = 0, so that we identify α := 2([G4 ] − a) as the twist of the String structure [Wa] [SSS3]. 4. Alternatively, we can work not with twisted String structure but rather with what was called twisted F 4 -structure in [SSS3] to account for the factor of 2 dividing λ. There is no canonical description of F 4 yet except through BO 8. We can consider 3 different cases: • the case when a = 0 so that the E8 bundle is trivial and the twist is provided by [G4 ], • the case [G4 ] = 0 so that the flux is G4 = dC3 and the twist is provided by the E8 class a, • the general case, where the twist is provided by α. To make a comparison, let us briefly recall the model of [DFM]. The field strength in M-theory is geometrically described as a shifted differential character [DFM] in the sense of [HS]. A shifted differential character is the equivalence class of a differential cocycle which trivializes a specific differential 5-cocycle related to the integral Stiefel-Whitney class W5 (Y ). The Stiefel-Whitney class w4 (Y ) ∈ H 4 (Y ; Z2 ) defines a differential cohomology class w ˇ4 via the inclusion H 4 (Y ; Z2 ) → 4 4 H (Y ; R/Z) → H (Y ; Z). On a Spin manifold, W5 (Y ) = 0 is satisfied since λ is an integral lift of w4 (Y ). In this case, the differential cohomology class w ˇ4

ˇ 5 (Y ) = 0, 1 λ, 0 ∈ Zˇ 5 (Y ) ⊂ can be lifted to a differential cocycle by defining W 2 C 5 (Y ; Z) × C 4 (Y ; Z) × Ω5 (Y ), and the C-field can be defined as the differential ˇ 5 , δ Cˇ = W ˇ 5, cochain Cˇ = (a, h, ω) ∈ C 4 (Y ; Z) × C 3 (Y ; Z) × Ω4 (Y ) trivializing W i.e. in components [DFM] 1 δh = ω − aR + λ, 2 It was proposed in in [DFM] that G4 lives in (6.5)

δa = 0,

(6.6)

ˇ 41 (Y 11 ), H λ

the space of shifted characters on Y worldvolume.

11

dω = 0 .

2

with shift 12 λ, and similarly on the fivebrane

Remark 6.12. In defining G4 to live in (6.6), the authors of [DFM] are taking the point of view that the class 12 λ acts as a twist for the differential character. The point of view we would like to take here is that 12 λ is what is being twisted, and hence plays a more central role. After all, natural structures, i.e. ones related to the tangent bundle, should be in a sense more fundamental for describing structures on a manifold that are extra or auxiliary structures such as bundles not related to the

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tangent bundle. The E8 gauge theory can then be seen as responsible for the twist of 12 λ. This is the point of view also adopted in [SSS3], building on the definition in [Wa]. Therefore " # 1 (6.7) λ − shifted E8 structure =⇒ {E8 − twisted String structure} . 2 6.4. The M5-brane charge. There are two definitions for the the M5-brane partition function [W8] [W9] [HS], and hence for the M5-brane charge [DFM]. One is intrinsic and uses the theory on the worldvolume. The other is extrinsic and uses anomaly inflow and hence the embedding in eleven dimensions. We will use the extrinsic definition for the M5-brane charge as this is the one that uses the Thom isomorphism and the push-forward (in the appropriate theory). Some aspects of the intrinsic approach were used in [SSS3] to relate twisted Fivebrane structures to the worldvolume theory of the fivebrane. Consider the embedding ι : W → Y of the fivebrane with six-dimensional worldvolume W into eleven-dimensional spacetime Y . Consider the ten-dimensional unit sphere bundle π : V → W of W with fiber S 4 associated to the normal bundle N → W of the embedding ι. There is a corresponding 11-manifold Yr with boundary V obtained by removing the disk bundle of radius r, Yr = Y − Dr (N ) [DFM]. Corresponding to the sphere bundle V there is the Gysin sequence. Using that the normal bundle has vanishing Euler class e(N ) = 0, one can deduce that [DFM]

(6.8)

H 3 (V ; Z) ∼ = 4 H (V ; Z) ∼ =

H 3 (W; Z) H 4 (W; Z) ⊕ Z (noncanonically).

so that H 3 (V, U (1)) ∼ = H 3 (W, U (1)). This relates G4 on the worldvolume to that on the normal bundle, i.e. the tangential components to the transverse components. The M5-brane is magnetically charged under the C-field, i.e. the former acts as a source for the latter. The charge is then measured by the value of the integral of G4 over the linking sphere S 4 of W in Y  G4 = k ∈ Z . (6.9) QM 5 = S4

Since the total Pontrjagin class of S 4 is 1, then QM5 is equal to the instanton number of an E8 instanton of the E8 gauge theory with four-class a: S 4 a = k ∈ Z. The description in terms of instantons is further given in [ES1]. Note that the charge defined this way highlights the role of E8 . If the E8 bundle is trivial then the charge is zero. The anomaly inflow cancellation allows for a partition function of the M5-brane to be well-defined. In order for this to be made precise, the C-fields on W must be related to the C-field on V . The gauge equivalence classes of C-fields on W is ˇ 41 (W ), and similarly for W . This then the shifted differential character [C] ∈ H 2λ requires the existence of the map (6.10)

ˇ 41 (W ) → H ˇ 41 (X). i:H λW λX 2

2

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In [DFM], a C-field Cˇ0 on X was chosen such that via the map i the C-fields are related as (6.11)

i[C] = [Cˇ0 ] + π ∗ [C].

We now give our main proposal. Proposal 6.13. (1) As we mentioned earlier, the shifted differential character gives the minor role to 12 λ. This obscures an interpretation in terms of generalized cohomology. We would like to replace (6.10) with   1 1 (6.12) ρ: λ + α (W) −→ λ + α (V ) . 2 2 (2) In order to properly define the M5-brane charge, what we need is then a Thom isomorphism and a push-forward map in the appropriate theory. Our proposal is that the desired theory is TMF and hence we apply the Thom isomorphism and the push-forward in TMF to obtain the M5-brane charge. This way, the analog of (6.11) will be canonical. If X is a space, then the twisted forms of K ∗ (X) correspond to homotopy classes of maps [X, BGL1 K]. The third homotopy group of the parametrizing space is π0 BGL1 K = Z since K 2 (pt) = Z. This is the determinantal twist in K-theory an example of which being the NS H-field in string theory. Twists of TMF are classified by BGL1 T M F and there is a corresponding map K(Z, 4) → BGL1 T M F . The proof of the following theorem is explained by Matthew Ando. Theorem 6.14. A class α ∈ H 4 (X; Z) gives rise to a twist tmfα∗ (X) of tmf (X). Moreover, if V is a (virtual) spin vector bundle over X with half-Pontrjagin class λ, then tmfλ∗ (X) ∼ = tmf ∗ (X V ) as modules over tmf ∗ (X). Then, armed with a Thom isomorphism and a pushforward map (see Ando’s contribution to these proceedings [ABG]), the main application is Definition/Theorem 6.15. Given an embedding ι : W → Y , the charge of the M5-brane is given by (6.13)

ι! (E) ∈ T M Fα∗ (Y )

Remarks 6.16. We consider the Witten quantization condition for the worldvolume, normal bundle, and target for both the M2-brane and the M5-brane. (1) M2-brane: The condition 12 λ(M 3) + [G4 ]|M 3 − aE8 |M 3 = 0 is satisfied on the M2-brane worldvolume, by dimension reasons. Therefore, the condition 12 λ(N ) + [G4 ]|N − aE8 |N = 0 on the normal bundle N is equivalent to the same condition on the target. This is indeed what enters in Witten’s derivation of the quantization condition (1.1). (2) M5-brane: Assuming as above that the condition holds for the normal bundle to the fivebrane, then this implies that the condition on the worldvolume is equivalent to that on the target. So we have 1 λ(W) + [G4 ]|W − aE8 |W = 0 2

=⇒

1 λ(Y ) + [G4 ]|Y − aE8 |Y = 0 . 2

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Discussion and further evidence for the interpretation of M5-brane charge. 1. Two-gerbes. The system of M2-branes ending on M5-branes is an Mtheoretic realization of the system of strings ending on D-branes. In the latter, there is a gauge field, or connection one form A, on the string boundary ∂Σ ⊂ Q. In addition, there is the B-field on Σ, which is acts as a twist for the Chan-Paton bundle, whose connection is A and curvature is F . Now in the case of M-theory, the membrane boundary ∂M2 ⊂ M5 (written schematically) has a degree two potential, essentially a B-field, which represents a gerbe made nonanbelian by the presence of the pullback of the C-field. This is equivalent to a 2-gerbe system. 2. Loop variables for the membrane. The system of multiple M5-branes, generalizing the system of n D-branes leading to U (n) nonabelian gauge symmetry,  gerbes, i.e. twisted gerbes for the universal central can be described by twisted ΩGextension of the based loop group ΩG, where G is any of the Lie groups Spin(n), n ≥ 7, E6 , E7 , E8 , F4 , and G2 [AJ]. Indeed, it has been shown explicitly in [G3] that classical membrane fields are loops. 3. Fivebrane in loop space. The generalization of the abelian system of fields, called tensor multiplet, to the nonabelian case leading to the nonabelian tensor multiplet, which appears in the worldvolume theory of the M5-brane, requires loop space variables and a formulation in loop space [G1] [G2]. Given the general principle that degree n phenomena in a space are captured by degree n−1 phenomena on its loop space, the situation for n = 3 suggests a relation between (twisted) K-theroy in loop space to be related to (twisted) T M F -cohomology of the space. 4. Relating TMF in M-theory to twisted K-theory in string theory. As we have reviewed it is known that twisted K-theory classifies D-branes and their charges in the presence of the NS B-field on a ten-dimensional space X 10 . We have also seen how M-branes and their charges should take values in TMF on an eleven-dimensional space Y 11 . Given the relation between M-theory and type IIA string theory, the situation when Y 11 is a (possibly trivial) circle bundle over X 10 , there should be a relation between the TMF description and the twisted K-theory description, in the sense that (possibly S 1 -equivariant twisted )TMF of X 10 × S 1 should give rise to twisted K-theory of X 10 . Current discussions with Matthew Ando and with Christopher Douglas suggest schematically the following

Conjecture 6.17. There is a map Ωtmf × S 1 /S 1 → k/∗. 5. Capturing the fields of degree 4k in M-theory. Twisted K-theory, under the twisted Chern character map that lands in rational cohomology, leads to differential forms of all even degree up to the dimension of the manifold. These forms are the components of the RR field in the classical supergravity approximation. In eleven dimensions, then, one should ask about some form of classical limit of the TMF description. What replaces the Chern chracter map should be a version for TMF (cf. the Miller character) of the Pontrjagin character map in KO-theory (6.14)

⊗C ch P h : KO ∗ (X) −→ K ∗ (X) −→ H ∗∗ (X; Q) .

The range is degree 4k cohomology, which indeed captures the field G4 and its dual G8 (or Θ). We think of this as the combination v G4 + w G8 , for suitably identified generators v and w, which make H ∗∗ (X; Q). Some aspects of this in connection to Spin K-theory have been discussed in [S7] (see also section 3.4). 6. Infinite number of fields. We know that, physically, an element in differential K-theory corresponds to a collection of physical fields: all the RR fields.

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Analogously, an element in (differential twisted) elliptic cohomology or (differential twisted) T M F should be given by a large collection of physical fields. At the level of differential forms, the large number of fields should be seen in the same way that the K-theoretic RR fields are seen at the level of differential forms via supergravity fields and their Hodge duals. But eleven-dimensional supergravity as traditionally known features only a single candidate field: the 3-form C-field (and its Hodge dual). This certainly cannot model generic elements in T M F by itself. Therefore, our previous discussion suggests a considerably richer structure hidden within and beyond eleven-dimensional supergravity. Recall that such a rich structure is also suggested by hidden symmetries: First, at the level of differential forms: • In the dimensional reduction of eleven-dimensional supergravity (and hence type IIA supergravity) on tori T n with fluxes one gets the Cremmer-Julia [CJ] exceptional groups En(n) , which are infinite-dimensional Kac-Moody groups for n ≥ 9 (cf. [J]). In the latter case, hence, there are an infinite number of fields at the classical level. Passing to the quantum theory, one has the U -duality groups En(n) (Z), the Z-forms of the above non-compact groups, and so we still have an infinite number of fields. • Already in eleven dimensions, the classic works of de Wit, Nicolai ( reviewed in [dWN]), and conjecture of [Du], imply the existence of Cremmer-Julia groups without compactification. One recent striking proposal is that of [We] in which the Lorentzian Kac-Moody algebra E11 is proposed as a symmetry in M-theory. This proposal has withstood many checks. The algebra e11 admits an infinite Z-grading as e11 = · · · ⊕ e8 ⊕ · · · . Second, integrally: The above fields should have refinements at the quantum level to whichever (generalized) cohomology theory ends up arising. 7. The topological term via higher classes. The topological part of the Mtheory action is written in a suggestive compact form when lifted to the bounding twelve-manifold Z 12 in [S1] [S2] [S3]. To do so, the total String class (in the notation of those papers) λ = 1 + λ1 + λ2 + · · · is introduced, where λ1 = p1 /2 is the usual ‘String class’, and λ2 = p2 /2 which is well-defined for Spin manifolds. The interpretation of the class and the characters is as degree 4k analogs of the Chern class and the Chern character, mapping from the cohomology theory describing M-theory to degree 4k cohomology. These are essentially the Spin characteristic classes defined in [Th] and were precursors to the discussion of Fivebrane structures in [SSS1] [SSS2] [SSS3]. Theorem 6.18. [S1] [S2] [S3][S5] (1) The M-theory fields are elements of a unified field strength. The quantization formula on the total M-theory field reproduces the quantization on G4 and its dual G8 . (2) G4 can be viewed as an index 2-gerbe. 8. Twisted cohomology in M-theory. The characters can be extended to include the dual fields and to account for their dynamics. For the dual field, one can pick either the straightforward degree seven Hodge dual or its differential, the degree eight field Θ. The dual formulation favors the degree four/eight combination whereas the duality-symmetric dynamics favors the degree four/seven combination. The second combination hints at a role for the prime p = 3 in M-theory analogous

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to the role played by the prime p = 2 in K-theory, and consequently in string theory. The combination leads to twisted (generalized) cohomology. However, unlike the case in K-theory, the twist is given by a degree four class, suggesting relation to TMF, since a twist of the latter can be seen, at least heuristically, as a K-theory (degree three) twist on the loop space. Theorem 6.19. [S2] [S3] There is a twisted (graded) cohomology on the Mtheory fields with a twist given by a degree four class. 9. The q-expansions. The q-expansions from the Miller character T M F → K[[q]] → H ∗ Q[[q]] will come from the comparison to type IIA string theory. We have the following diagram =

(6.15)

S1

u

/ W3   π

 

Σ2

/ Y 11 o 

S1 .

π

/ X 10

There are two principal circle bundles: π and π  . In the dimensional reduction from M-theory to type IIA string theory the two fibers are identified and from a membrane in eleven dimensions we get a string in ten dimensions. At the level of partition functions of the targets (the fiber bundle π in the diagram) it was observed in [KS1] that the resulting partition function, formulated in elliptic cohomology, is a q-expansion of the K-theoretic partition function. There q was built out of the generators v1 and v2 at the prime 2, q = v13 v2−1 . This essentially came from the fact that EO2 (pt) = Z2 [[q]]. The E-theoretic quantization. In the untwisted case, the total % field strength ˆ F (x) of type II string theory as described by K-theory is 2π times A(X)ch(x). The refinement of the description of the fields to elliptic cohomology is For any elliptic cohomology theory E, there is a canonical map E → K((q)) (where q is as above), so% compose with the Chern character to get a map chE : E → H ∗ ((q)). ˆ The term A(X) should be replaced by an analogous term related to the Witten genus σ(X)1/2 where Theorem 6.20 ([KS3]). The E-theoretic quantization condition for the RR fields is given by the formula for the elliptic field strength associated with x: F (x) = σ(X)1/2 chE (X), where σ(X) is the characteristic class of X associated with the power series & (1 − q n ez )(1 − q n e−z ) σ(z) = (ez/2 − e−z/2 ) . n≥1 (1 − q n )2 The σ-function, in the q → 0 limit, reduces to the characteristic function of ˆ the A-genus, thus reducing this field strength to the type II field strength in the 10-dimensional limit. The connection of this to charges is as follows. Considering the bundle π  , we get q-expansions on Σ2 upon taking Fourier modes of the circle. The boundary ∂Σ2 has

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Chan-Paton charges on it. Since the boundary of the string ends on a D-brane then certainly the q-expansions will be seen by the D-brane as well. The result is that the Chan-Paton bundle on the D-brane gets replaced by its appropriate q-expansions E → S • E, Λ• E. The charges of D-branes have been previously calculated at q = 1, i.e. at Λ• = S • = 1, as they are essentially the Chern character (6.16)

K ∗ (q = 1) −→ H ∗ Q(q = 1) .

The q-refinement of the D-brane charge formula should then be (6.17)

Q = ΦW (X 10 )chell (E) ,

the Witten genus of X 10 twisted by the appropriate exterior and symmetric powers of the Chan-Paton bundle. The higher modes for the supermultipet. In an orthogonal discussion to [KS1], we could also view q as coming from Fourier modes on the circle, i.e. KaluzaKlein modes and interpret q accordingly. Consider the supergravity fields (g, C3 , ψ). The coupling to vector bundles V gives V ⊗ Lk ). For example V is an E8 bundle with characteristic class a and set c1 (L). Consider the connection A on the vector bundle V . Coupling it to Lk leads to the connection A ⊗ e−ikθ . Then the C-field, which is essentially the Chern-Simons form of the E8 bundle, will also be also have Fourier components as C3 ⊗ e−ikθ . This is of course compatible with, and is in fact the same as, just the dimensional reduction of the C-field directly from eleven dimensions without use of the E8 bundle. 10. Concluding remark. We have reviewed and indicated further development that a closer examination of the deep structures involved in string- and M-theory indicates and shows that very rich cohomological phenomena are at work in the background. While it is now well-known that (differential) K-theory encodes much of the interesting structure of Type II string theory, we have shown and argued that by further reasoning along such lines one finds various generalized cohomological structures that go beyond K-theory. This includes Morava K-theory and E-theory, elliptic cohomology, and T M F . In addition, we have considered higher structures generalizing Spin structures, such as String and Fivebrane structures, and have emphasized a delicate interplay between all these.

Acknowledgements I would like to thank Matthew Ando, David Lipsky, Corbett Redden, Urs Schreiber, and Jim Stasheff for very useful discussions, and Dan Freed for useful remarks. I also thank the organizers of the CBMS conference on “Topology, C ∗ -algebras, and String Duality”, Robert Doran and Greg Friedman, for the kind invitation to the conference. Special thanks are due to the anonymous referee, who kindly gave many helpful remarks and suggestions for improving the presentation and strengthening the points made in the article. I also thank Arthur Greenspoon for editorial suggestions. References [AKMW] O. Alvarez, T. P. Killingback, M. L. Mangano, and P. Windey, String theory and loop space index theorems, Commun. Math. Phys. 111 (1987) 1-10. [ASi] O. Alvarez and I. M. Singer, Beyond the elliptic genus, Nuclear Phys. B 633 (2002) 309-344, [arXiv:hep-th/0104199].

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Department of Mathematics, Yale University, New Haven, Connecticut 06511 Current address: Department of Mathematics, University of Maryland, College Park, Maryland 20742 E-mail address: [email protected]

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http://dx.doi.org/10.1090/pspum/081/2681766

Proceedings of Symposia in Pure Mathematics Volume 81, 2010

Landau-Ginzburg Models, Gerbes, and Kuznetsov’s Homological Projective Duality Eric Sharpe Abstract. In this talk we briefly outline several recent developments in LandauGinzburg models and associated areas. We begin by discussing recent work on A- and B-twisted Landau-Ginzburg models – certain two-dimensional topological field theories derived from Landau-Ginzburg models. After briefly outlining some pertinent work on string propagation on stacks and gerbes, including the decomposition conjecture describing how strings on gerbes are equivalent to strings on disjoint unions of spaces, we discuss how certain Landau-Ginzburg models on total spaces of bundles over gerbes are equivalent to strings on branched double covers and noncommutative resolutions thereof, and realize Kuznetsov’s “homological projective duality.”

1. Introduction Landau-Ginzburg models, certain non-scale-invariant two-dimensional quantum field theories, have historically been a fruitful arena for many issues in string compactifications. Roughly, a Landau-Ginzburg model consists of a string propagating on a space, together with a potential function over that space (which in a supersymmetric theory is defined by a holomorphic function, known as a “superpotential”). Historically, most Landau-Ginzburg models considered described strings propagating on vector spaces (plus a superpotential), or finite group quotients thereof, rather than a more general space or stack. One reason for this is that many such examples are closely related to strings on nontrivial spaces, but perhaps a more honest reason is that a string on a vector space (plus superpotential) is technically much easier to analyze than a string on a nontrivial space. Technology is now finally being developed to allow the analysis of LandauGinzburg models on nontrivial spaces. This is technically more difficult, but, one can also get more interesting results. In this talk we will summarize several recent developments in Landau-Ginzburg models and related areas. We will begin by outlining A- and B-type topological 2010 Mathematics Subject Classification. Primary 81T45, Secondary 14D23, 14F05, 53C08, 53D37, 14N35. Key words and phrases. Landau-Ginzburg, gerbes, homological mirror symmetry, stacks. The author was supported in part by NSF grants DMS-0705381, PHY-0755614. c Mathematical 0000 (copyright Society holder) c 2010 American

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field theories based on Landau-Ginzburg models. We then study certain examples of Landau-Ginzburg models defined over total spaces of bundles on gerbes. Applying results from the study of strings on stacks and gerbes, we find that those Landau-Ginzburg models are equivalent to strings on branched double covers and, sometimes, noncommutative resolutions thereof, giving a physical realization of such noncommutative spaces as well as new conformal field theories. We conclude with a discussion of how these Landau-Ginzburg models fit into families of physical theories, and give a physical realization of Kuznetsov’s “homological projective duality.” 2. A-, B-topological Twists of Landau-Ginzburg Models on Nontrivial Spaces We begin by briefly outlining the construction of the A- and B-model topological field theories, and their analogues for Landau-Ginzburg models. In a quantum field theory, one computes ‘correlation functions,’ closely analogous to correlation functions in statistics, in which instead of summing over events and weighting by probabilities, one performs some infinite-dimensional integral, and weights by an exponential. Schematically:  [Dφ · · · ] exp(−S(φ, · · · ))O1 · · · On  O1 · · · On  = . [Dφ · · · ] exp(−S(φ, · · · )) In a topological field theory, there is a nilpotent scalar symmetry, known as the BRST symmetry, and the Oi live in the cohomology of the generator of that symmetry, called the BRST operator and typically denoted Q. The quantity S is known as the ‘action.’ For strings propagating on a space or stack X, the action is an integral over the worldsheet Σ of a quantity constructed from maps √ φ : Σ → X, and i,ı various Grassmann-valued quantities ψ± that are sections of KΣ tensored with a pullback of some part of the complexified tangent bundle of X. In the analogy with statistics, the exponential exp(−S) acts as an unnormalized probability for i,ı any given set of φ, ψ± . The action for a Landau-Ginzburg model has the form   j j i i i j k  S = d2 x gij ∂φi ∂φj + igij ψ+ Dz ψ+ + igij ψ− Dz ψ− + Rijk ψ+ ψ+ ψ− ψ− Σ  i j ı j + g ij ∂i W ∂j W + ψ+ ψ− Di ∂j W + ψ+ ψ− Dı ∂j W , where W : X → C is a holomorphic function known as the superpotential. The superpotential W : X → C is holomorphic (so Landau-Ginzburg models are only interesting when X is noncompact). A nonlinear sigma model is a special case of a Landau-Ginzburg model, specifically the case that the superpotential W is identically zero. Nonlinear sigma models describe strings propagating on X; a Landau-Ginzburg model describes something slightly more complicated. For ordinary nonlinear sigma models, there are two topological twists, known as the A- and B-models (see e.g. [W91]). The A-model is a topological field theory whose correlation functions turn out to be independent of the complex structure on X. To build the A-model from a i ı nonlinear sigma model, we modify the worldsheet spinors ψ+ and ψ− to be scalars: i ı ψ+ ∈ Γ(φ∗ (T 1,0 X)) → χi , ψ− ∈ Γ(φ∗ (T 0,1 X)) → χı .

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The BRST operator Q acts as Q · φi = χi , Q · φı = χı , Q · χ = 0, Q2 = 0, and so we identify χμ ∼ dxμ , Q ∼ d. The states of this theory are bμ···ν χμ · · · χν ↔ H ·,· (X). The B-model is a topological field theory whose correlation functions turn out to be independent of the K¨ahler or symplectic structure on X. To build the Bı model from a nonlinear sigma model, we modify the worldsheet spinors ψ± to be worldsheet scalars. It is convenient to define   j j ı ı , η ı = ψ+ + ψ− , θi = gij ψ+ − ψ− in terms of which the BRST operator Q acts as Q · φi = 0, Q · φı = η ı , Q · η ı = 0, Q · θj = 0, Q2 = 0. We identify η ı ↔ dz ı , θj ↔

∂ , Q ↔ ∂, ∂z j

and so the states are ···jm ı1 bjı11···ı η · · · η ın θj1 · · · θjm ↔ H n (X, Λm T X) . n

Next, we shall outline the A- and B-topological twists of Landau-Ginzburg models (i.e. nonlinear sigma models with nonvanishing superpotential W ), begini,ı ning with the Landau-Ginzburg B-model. We redefine the ψ± in exactly the same fashion as for the B-model previously, and define η and θ as before. The only essential modification is that the action of the BRST operator changes to Q · φi = 0, Q · φı = η ı , Q · η ı = 0, Q · θj = ∂j W, Q2 = 0. Specifically, the BRST variation of θj is no longer zero, but rather is proportional to a derivative of the superpotential. We can identify η ı ↔ dz ı , θj ↔

∂ , Q ↔ ∂, ∂z j

and the states of the theory ···jm ı1 η · · · η ı n θj1 · · · θjm b(φ)jı11···ı n

are now interpreted as elements of the hypercohomology group   dW dW H· X, · · · −→ Λ2 T X −→ T X −→ OX . Since B-twisted Landau-Ginzburg models over nontrivial spaces are rarely discussed in the literature, let us take a moment to check that the results above correctly generalize results for both the ordinary B-twisted nonlinear sigma models, as well as Landau-Ginzburg models on vector spaces. (1) First, consider the standard B-model obtained by setting the superpotential to zero. In this case, the hypercohomology group above trivially reduces to H · (X, Λ· T X), reproducing the result discussed earlier for this case.

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(2) Next, consider the B-twist of a Landau-Ginzburg model over X = Cn with superpotential W a quasi-homogeneous polynomial. In this case, the coefficient sequence of the hypercohomology group is a Koszul resolution of the fat point {dW = 0}, so the hypercohomology reduces to C[x1 , · · · , xn ]/(dW ) which is the standard result for this case (see e.g. [V91]). See [GS08a] for more information on B-twisted Landau-Ginzburg models on nontrivial spaces. Next, let us consider A-twists of Landau-Ginzburg models. Here, the nonlinear sigma model twist does not by itself give a well-defined result. In particular, the Aj i twist of a nonlinear sigma model makes ψ+ a worldsheet scalar and ψ− a worldsheet one-form, but a Landau-Ginzburg model action contains a term  i j ψ+ ψ− Di ∂j W. Σ

If we were to perform the same A-twist as in a nonlinear sigma model, then the term above would involve integrating a one-form over the worldsheet Σ, which does not make sense. Therefore, the A-twist must be something slightly different than in a nonlinear sigma model. There are at least two different ways to fix this particular problem. • One way is to multiply offending terms in the action by another 1-form, e.g. multiply the superpotential W by a holomorphic section of KΣ . This is the approach used to solve a closely related problem in [W94]. • Another way is to combine the twist with a U (1) action, resulting in fermions coupling to different bundles. The second method is advantageous for physics, so it is the method that we shall follow, but, it has the disadvantage that not all Landau-Ginzburg models will admit an A-twist in this prescription – only those such that the space X admits a U (1) action, with respect to which the superpotential is quasi-homogeneous. Let us assume that such a U (1) action exists. Let Q(ψi ) be a set of numbers encoding the quasi-homogeneity of the superpotential under the U (1) action:   W λQ(ψi ) φi = λW (φi ). Then, to define this A-twist, we change the bundles to which fermions couple schematically as follows:   −(1/2)QL −(1/2)QR ⊗ KΣ ψ → Γ original ⊗ KΣ where

⎧ i ,R ⎨ 1 ψ = ψ+ i 1 ψ = ψ− ,L . QR,L (ψ) = Q(ψ) + ⎩ 0 else

For example, consider the case of a Landau-Ginzburg model over X = Cn , with W a quasi-homogeneous polynomial of degree d. In this case, to perform the 1/(2d) operation above, we would need to make sense of e.g. KΣ . Since on a genus g worldsheet, c1 (KΣ ) is not divisible by 2d for d > 1 in general, we have a problem. There are two ways to proceed:

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• One way is to couple to topological gravity. After doing so, KΣ must be modified to take into account punctures and so forth. In this case, on a worldsheet of essentially any genus, one can find a suitable set of 1/(2d) punctures so that KΣ makes sense for d > 1. This is the approach implicitly followed by [FJR07a, FJR07b]. • If we do not couple to topological gravity, then we cannot make sense of 1/(2d) KΣ in general, and so we cannot make sense of the A-twist in this particular example. This is the route followed in [GS08a, GS08b]. We shall work with the latter case. As a result, there will implicitly be a selection rule that further restricts twistable examples, beyond merely requiring the existence of a suitable U (1) action. Now, let us consider an example which is twistable if one does not couple to topological gravity. Specifically, consider a Landau-Ginzburg model on   π X = Tot E ∨ −→ B (E a holomorphic vector bundle over B) with superpotential W = pπ ∗ s, p a fiber coordinate1 on E ∨ , and s a holomorphic section of E. The U (1) action acts as phases on the fibers of the bundle. It can be shown that correlation functions in this theory match those in a nonlinear sigma model on {s = 0} ⊂ B. We see this as follows. In prototypical cases, they can be written schematically as  O1 · · · On  = ω1 ∧ · · · ∧ ωn M 

· dχp dχp exp −|s|2 − χp dz i Di s − c.c. − Fij dz i dz j χp χp .   Mathai−Quillen form

where M is some (compactified) moduli space of holomorphic maps into B. The Mathai-Quillen form is a representative of a Thom class, so  O1 · · · On  = M ω1 ∧ · · · ∧ ωn ∧ Eul(N{s=0}/M ) = {s=0} ω1 ∧ · · · ∧ ωn , which matches correlation functions in a nonlinear sigma model on {s = 0}. This is not a coincidence, as we shall see shortly. To understand the matching above, we must take a moment to explain the renormalization group. The renormalization group (strictly speaking, a semigroup, not a group) constructs a series of quantum field theories in which each element is an approximation to the previous one, valid at longer distance scales. The effect of the renormalization group is much like starting with a picture and standing further and further away from it, to get successive approximations. The final result might look very different from the starting point. By repeating this approximation many times, one constructs a sequence of theories, said to lie along the flow of the renormalization group. The renormalization group is a very useful, and widely used, idea in quantum field theory. Unfortunately, as a practical matter, it is usually impossible to follow it completely explicitly. Typically, the best one can do is construct an asymptotic 1i.e. a tautological section of π ∗ E ∨ .

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series approximation to the tangent vector to the renormalization group at any given point in its motion. The renormalization group is pertinent here because it has the property that it preserves topological field theory structures. In other words, if two physical theories are related by renormalization group flow, then correlation functions in a topological twist of one should match correlation functions in a corresponding topological twist of the other. For example, in principle the Landau-Ginzburg model discussed above, on   π X = Tot E ∨ −→ B with W = pπ ∗ s, flows under the action of the renormalization group to a nonlinear sigma model on {s = 0} ⊂ B. This is the basic reason why the A-twisted correlation functions in the Landau-Ginzburg model should match those in the nonlinear sigma model on {s = 0}. This observation can have computational benefits. For example, consider curvecounting in a degree 5 (quintic) hypersurface in P4 . To compute A-model correlation functions directly, one needs to know the moduli space of curves in the quintic, which can be somewhat complicated. If we apply the idea of the renormalization group, then we can replace the nonlinear sigma model on the quintic with a Landau-Ginzburg model on

Tot O(−5) −→ P4 . In this Landau-Ginzburg model, curve-counting involves moduli spaces of curves in P4 , which are much simpler than moduli spaces of curves in the quintic. In the mathematics literature, this is a standard mathematical trick for curve-counting (see e.g. [Kont94]), though its physical realization in Landau-Ginzburg models seems to be novel. So far we have outlined the A- and B-topological twists of Landau-Ginzburg models, and talked about how the renormalization group is realized mathematically in these examples via a Thom class computation. Something closely analogous happens in elliptic genus computations. For example, an elliptic genus of a LandauGinzburg model on   X = Tot E ∨ −→ B π

with superpotential as before, is given by [AndoS09]   Td(T B) ∧ ch Λ−1 (T B) ⊗ Λ−1 (E ∨ ) B

 n=1,2,3,···

 n=1,2,3,···



Sqn ((T B)C )

Sqn ((E ∨ )C )

n=0,1,2,··· C

Λ−qn ((T B) )



 ∨ C

Λqn ((E ) ) .

n=1,2,3,···

It can be shown [AndoS09] that the elliptic genus above matches the Witten genus of {s = 0} ⊂ B, by virtue of a Thom class computation. In the case of A-twisted correlation functions, we saw the renormalization group was realized mathematically via a Mathai-Quillen representative of a Thom form. It is true in general, as in the example above, that something analogous happens

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in elliptic genera: elliptic genera of Landau-Ginzburg and nonlinear sigma models related physically by renormalization group flow, are related mathematically by Thom forms. This suggests a mathematical interpretation of the renormalization group in twisted theories as a Thom class, possibly arising ultimately from an underlying Atiyah-Jeffrey/Baulieu-Singer-type of description [AJ90, Bau-Singer]. 3. Landau-Ginzburg Models on Gerbes 3.1. Strings on stacks and gerbes. There are two basic motivations for considering stacks in physics: first, the possibility of building new string compactifications, new conformal field theories, and second, to better understand certain existing string compactifications (such as orbifolds). The first question a physicist must ask is, how does one construct quantum field theories for strings propagating on stacks? Put another way, how to make sense of strings on stacks concretely? To do this, we can use the fact that smooth, separated, generically tame Deligne-Mumford stacks of finite type over a field can be presented as a global quotient [X/G] for X a space and G a group. (G need not be finite, and need not act effectively.) (See for example [Kresch05][theorem 4.4] or [Tot02].) To such a presentation, one associates a “G-gauged sigma model on X” [PSa, PSb, PSc]. The first problem is that such presentations of a given stack are not unique, and those presentations can have very different physics. This problem can be fixed with the renormalization group: we conjecture that stacks are associated to certain equivalence classes (known as “universality classes”) of renormalization group flow in gauged sigma models. Analogous issues arise in other relatively recent developments in physics, such as in the physical realization of derived categories. There, localization on quasiisomorphisms is realized physically by (boundary) renormalization group flow, and fixes a closely analogous potential presentation-dependence issue. There are various other technicalities, aside from the potential presentationdependence issue. For example, let us specialize to gerbes, i.e. quotients by noneffectively-acting groups. One problem with gerbes is that the massless spectrum computation which gives sensible results for other stacks, for gerbes results in multiple dimension zero states. The existence of multiple dimension zero states signals a violation of one of the foundational axioms of quantum field theory, known as cluster decomposition. There is, to our knowledge, a single known loophole: nonlinear sigma models with disconnected target spaces also have multiple dimension zero operators, counting the number of components. In such a case, the quantum field theory is nevertheless well-behaved, despite violating cluster decomposition, because cluster decomposition has been violated in the mildest fashion possible. We believe that is what is happening in strings on gerbes. A little more specifically, we conjecture that strings on gerbes are equivalent to strings on disjoint unions of spaces. Consider [X/H], where 1 −→ G −→ H −→ K −→ 1 and G acts trivially on X (and is assumed finite). We and the other authors of [HHPSA] conjecture that   ˆ CFT([X/H]) = CFT (X × G)/K

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ˆ is the set of irreducible representations of G, and the K action can be where G obtained by diagram chasing (or see [HHPSA]). The elements of the disjoint union on the right-hand side above have various B fields2 we have here suppressed for brevity. We and the other authors of [HHPSA] call this our decomposition conjecture. ˆ In this For special gerbes known as “banded” gerbes, K acts trivially upon G. case, the decomposition conjecture reduces to ⎛ ⎞  CFT(G − gerbe on X) = CFT ⎝ X ⎠ ˆ G

where the various copies of X on the right-hand side above have different B fields. ˆ correspondThe B field on a copy of X corresponding to a particular element of G, ing to a map Z(G) → U (1), is determined by the image of the characteristic class of the gerbe: Z(G)→U(1)

H 2 (X, Z(G)) −→ H 2 (X, U (1)). There are a number of checks and implications of the decomposition conjecture above. Very briefly, • For global quotients by finite groups, one can compute physical quantities known as partition functions (related to elliptic genera) exactly at arbitrary worldsheet genus, and check that the decomposition conjecture makes correct statements about those partition functions. • Implies that ˆ KH (X) = twisted KK (X × G) which can be checked independently. • Implies known facts about sheaf theory on gerbes. • Has nontrivial implications for Gromov-Witten theory of stacks, implications which are currently being checked in [AJT08, AJT09a, AJT09b]. 3.2. Landau-Ginzburg models. Now, let us apply the decomposition conjecture above to some examples of Landau-Ginzburg models. Consider, for example, a Landau-Ginzburg model over   X = Tot O(−1)⊕2n+2 −→ Pn[2,2,··· ,2] with superpotential W =

 a

pa Ga (φ) =



φi Aij (p)φj

ij

where the Ga (φ) are quadric polynomials in the fiber coordinates φ, and the p’s are homogeneous coordinates on Pn[2,2,··· ,2] , a Z2 gerbe over Pn . The matrix Aij (p) has entries that are degree one in the p’s, and is a rewriting of the four quadrics Ga . Writing the superpotential in the form above emphasizes that it is giving a mass to the φ fields, at least away from the locus {det A = 0}. Thus, at least away from that locus, this theory appears to be that of a nonlinear sigma model on Pn[2,2,··· ,2] , a Z2 gerbe, which by the decomposition conjecture, physics will see as a double cover. A more detailed analysis of the physics [CDHPS, DS, HHPSA] 2On a space Y , the “B field” is an element of H 2 (Y, U (1)).

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reveals a ‘Berry phase’ about the locus {det A = 0}, which seems to imply that the physics sees a branched double cover of Pn , branched over the degree 2n + 2 locus {det A = 0}, which would make it a Calabi-Yau. In the cases n = 1, 2, our story ends with the branched double cover, describing (for n = 1) an elliptic curve, and (for n = 2) a K3 surface. However, beginning at n = 3 (for which the branched double cover would be Clemens’ octic double solid), we find something more interesting. In particular, the branched double cover in this example has singularities, but an analysis of the physics reveals that it does not see those singularities, it behaves as if instead the space were smooth. We believe that in this case, this Landau-Ginzburg model is actually describing a “noncommutative resolution” of the branched double cover worked out by Kuznetsov [Kuz1, Kuz2, Kuz3] and others (see for example [Kont98, Soi03, Cos04, VdB1, VdB2, VdB3]). There are several notions of noncommutative geometry appearing in the physics literature, see e.g. [SW, RW]. However, this notion is (to our knowledge) distinct from other notions of noncommutative geometry that have so far appeared in physics. A noncommutative space, in this context, is defined by its category of sheaves. In the present case, the noncommutative resolution in question is defined by the pair (P3 , B) where B is the sheaf of even parts of Clifford algebras associated with the universal quadric over P3 :  pa Ga (φ) a

– in other words, the GLSM superpotential. On the noncommutative space so defined, B acts as the structure sheaf, and other sheaves are sheaves of B-modules. Equivalently, instead of working with (P3 , B), we could work with the branched double cover f : Z → P3 together with a sheaf of algebras A for which f∗ A = B. One can work with either the branched double cover or (P3 , B); that said, it can be convenient to use the description (P3 , B), and so we shall do so here. Physically, the sheaf B above, and sheaves of B-modules, are interpreted as “Dbranes” in the Landau-Ginzburg model. In general, a D-brane is some propagating multidimensional object, defined by boundary conditions on an open string plus additional data confined to the boundary. In the case of a Landau-Ginzburg model, that extra data takes the form of a ‘matrix factorization’: a submanifold S with vector bundles E, F → S and maps F : E → F, G : F → E such that F ◦ G and G ◦ F are both W |S times the identity maps on F and E, respectively. Intuitively, we can see that matrix factorizations in this Landau-Ginzburg model match the sheaves of B-modules of Kuznetsov’s noncommutative resolution as follows. Work locally over the P3 (physically, in a Born-Oppenheimer approximation; mathematically, in families). At a point on P3 , the superpotential is a quadratic polynomial. Physically, such a superpotential has no closed string modes, but it is known that one still has D0-branes with a Clifford algebra structure [KapLi]. Here, we have a Landau-Ginzburg model fibered over P3 , which gives sheaves of Clifford algebras (determined by the universal quadric / superpotential) and modules thereof. Thus, we see that matrix factorizations/D-branes duplicate Kuznetsov’s definition of his noncommutative resolution, and so we interpret this model as living on the noncommutative resolution. The picture above is of interest to physicists for a variety of reasons, including:

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• Geometry is being realized physically in a novel fashion. Ordinarily when nonlinear sigma models are constructed as the endpoint of RG flow, the Landau-Ginzburg model in question is realizing a complete intersection, arising as the critical locus of the superpotential. Here, the geometry is being realized physically in a very different fashion, via nonperturbative effects in a gauge theory. • This is the first physical realization of a noncommutative resolution (for this particular notion of noncommutative). In particular, this is a new kind of conformal field theory, not previously analyzed by physicists. • These Landau-Ginzburg models appear as limit points in moduli spaces of “gauged linear sigma models,” and there give counterexamples to old unproven lore that all such limit points should be related by birational transformations. More recent work on this subject includes an analysis of “D-brane probes” in these backgrounds. On general principles, the “D-brane probe moduli space” should be a (possibly non-K¨ ahler) small resolution of the singular space, and that is what is found in [Add09a, Add09b].

4. Homological Projective Duality Kuznetsov describes [Kuz1, Kuz2, Kuz3] a notion of “homological projective duality” (induced on linear sections) that relates different spaces and/or noncommutative resolutions, including the noncommutative resolutions above. His examples have physical realizations in gauged linear sigma models, which reduce to LandauGinzburg models such as the ones above at various endpoints. In more detail, given a gauged linear sigma model, one can apply renormalization group flow to obtain a Landau-Ginzburg model (and sometimes a nonlinear sigma model after further renormalization group flow). By altering parameters in the gauged linear sigma model, one can obtain Landau-Ginzburg models over different spaces, realizing the duality. We list below several examples of homological projective duals, and corresponding Landau-Ginzburg models: P3 [2, 2] LG model on Tot(O(−2)⊕2 → P3 ) P5 [2, 2, 2] LG model on Tot(O(−2)⊕3 → P5 ) P7 [2, 2, 2, 2] LG model on Tot(O(−2)⊕4 → P7 P2g+1 [2, 2] LG model on Tot(O(−2)⊕2

branched double cover of P1 , branched over a degree 4 locus LG model on Tot(O(−1)⊕4 → P1[2,2] ) branched double cover of P2 , branched over a degree 6 locus LG model on Tot(O(−1)⊕6 → P2[2,2,2] ) nc res’n of branched double cover of P3 , branched over a degree 8 locus LG model on Tot(O(−1)⊕8 → P3[2,2,2,2]

branched double cover of P1 , branched over a degree 2g + 2 locus → P2g+1 ) LG model on Tot(O(−1)⊕2g+2 → P1[2,2] )

Additional examples, including additional noncommutative resolutions, are listed in [CDHPS, DS, HT]. In each case above, the Landau-Ginzburg superpotential

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can be written in the form W =



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xi xj Aij (p)

ij

and so we have omitted it from the tables above. Further examples of the same form are listed in [CDHPS]. There are additional examples of homological projective duals physically realized by gauged linear sigma models in [HT, DS], but some of their LandauGinzburg models are not as well understood, so we do not list them here. Note that in every example in the list above (in fact, in every example in [CDHPS]), the dual Landau-Ginzburg models are defined over birational spaces. With this in mind, we and the other authors of [CDHPS] conjecture that [CDHPS] All examples of homological projective duality, induced on linear sections, are equivalent to Orlov-type [Orlov03, Wal04] equivalences between matrix factorizations in Landau-Ginzburg models over birational spaces, with compatible superpotentials. We and the other authors of [CDHPS] also conjecture that in all examples of gauged linear sigma models, the endpoints of the K¨ahler moduli spaces are related by Kuznetsov’s homological projective duality [CDHPS]. This certainly holds true in the examples here, but is very difficult to check more generally, so we leave it as a conjecture. 5. Summary In this talk we have summarized several recent developments in Landau-Ginzburg models and related areas. We began by outlining A- and B-type topological field theories based on Landau-Ginzburg models, including results from B-twisted LandauGinzburg models on nontrivial spaces, and outlined multiple possible A-twists, including details of one particular notion of A-twist that is of physical interest. We then briefly outlined how to make sense of strings propagating on stacks and gerbes and various technical issues arising in such notions. For strings on gerbes, we outlined the decomposition conjecture, relating a string on a gerbe to a string on a disjoint union. We then applied the decomposition conjecture to understand the physics of certain Landau-Ginzburg models on total spaces of bundles over gerbes, which we then interpreted in terms of branched double covers and, sometimes, noncommutative resolutions thereof, which gave us novel physical realizations of geometry, as well as new conformal field theories. References [Add09a] [Add09b] [AJ90] [AJT08] [AJT09a] [AJT09b]

N. Addington, “The derived category of the intersection of four quadrics,” arXiv: 0904.1764. N. Addington, “Spinor sheaves on singular quadrics,” arXiv: 0904.1766. M. Atiyah, L. Jeffrey, “Topological Lagrangians and cohomology,” J. Geom. Phys. 7 (1990) 119-136. E. Andreini, Y. Jiang, H.-H. Tseng, “On Gromov-Witten theory of root gerbes,” arXiv: 0812.4477. E. Andreini, Y. Jiang, H.-H. Tseng, “Gromov-Witten theory of product stacks,” arXiv: 0905.2258. E. Andreini, Y. Jiang, H.-H. Tseng, “Gromov-Witten theory of etale gerbes, I: root gerbes,” arXiv: 0907.2087.

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248 12

ERIC SHARPE

[AndoS09]

M. Ando, E. Sharpe, “Elliptic genera of Landau-Ginzburg models over nontrivial spaces,” arXiv: 0905.1285. [Bau-Singer] L. Baulieu, I. Singer, “The topological sigma model,” Comm. Math. Phys. 125 (1989) 227-237. [CDHPS] A. Caldararu, J. Distler, S. Hellerman, T. Pantev, E. Sharpe, “Non-birational twisted derived equivalences in abelian GLSM’s,” arXiv: 0709.3855, to appear in Comm. Math. Phys. [Cos04] K. Costello, “Topological conformal field theories and Calabi-Yau categories,” Adv. Math. 210 (2007) 165-214, math.QA/0412149. [DS] R. Donagi, E. Sharpe, “GLSMs for partial flag manifolds,” J. Geom. Phys. 58 (2008) 1662-1692, arXiv: 0704.1761. [FJR07a] H. Fan, T. Jarvis, Y. Ruan, “The Witten equation, mirror symmetry, and quantum singularity theory,” arXiv: 0712.4021. [FJR07b] H. Fan, T. Jarvis, Y. Ruan, “The Witten equation and its virtual fundamental cycle,” arXiv: 0712.4025. [GS08a] J. Guffin, E. Sharpe, “A-twisted Landau-Ginzburg models,” arXiv: 0801.3836, to appear in J. Geom. Phys. [GS08b] J. Guffin, E. Sharpe, “A-twisted heterotic Landau-Ginzburg models,” arXiv: 0801.3955, to appear in J. Geom. Phys. [HHPSA] S. Hellerman, A. Henriques, T. Pantev, E. Sharpe, M. Ando, “Cluster decomposition, T-duality, and gerby CFT’s,” Adv. Theor. Math. Phys. 11 (2007) 751-818, hep-th/0606034. [HT] K. Hori, D. Tong, “Aspects of non-Abelian gauge dynamics in two-dimensional N = (2, 2) theories,” JHEP 0705 (2007) 079, hep-th/0609032. [KapLi] A. Kapustin, Y. Li, “D-branes in Landau-Ginzburg models and algebraic geometry,” JHEP 0312 (2003) 005, hep-th/0210296. [Kont94] M. Kontsevich, “Enumeration of rational curves via torus actions,” pp. 335-368 in The moduli spaces of curves (Texel Island, 1994) (R. Dijkgraaf, C. Faber, G. van der Geer, eds.), Progress in Math. 129, Birkh¨ auser, Boston-Basel-Berlin, 1995, hep-th/9405035. [Kont98] M. Kontsevich, “Course on non-commutative geometry,” ENS, 1998, lecture notes at http://www.math.uchicago.edu/~arinkin/langlands/kontsevich.ps. [Kresch05] A. Kresch, “On the geometry of Deligne-Mumford stacks,” pp. 259-271 in Algebraic geometry (Seattle, 2005), Proc. Sympos. Pure. Math. 80, part 1, Amer. Math. Soc., Providence, RI, 2009. [Kuz1] A. Kuznetsov, “Homological projective duality,” Publ. Math. Inst. Hautes Etudes Sci. 105 (2007) 157-220, math.AG/0507292. [Kuz2] A. Kuznetsov, “Derived categories of quadric fibrations and intersections of quadrics,” Adv. Math. 218 (2008) 1340-1369, math.AG/0510670. [Kuz3] A. Kuznetsov, “Homological projective duality for Grassmannians of lines,” math.AG/0610957. [Orlov03] D. Orlov, “Triangulated categories of singularities and D-branes in Landau-Ginzburg models,” Proc. Steklov Inst. Math 246 (2004) 227-248, math.AG/0302304. [PSa] T. Pantev, E. Sharpe, “Notes on gauging noneffective group actions,” hep-th/0502027. [PSb] T. Pantev, E. Sharpe, “String compactifications on Calabi-Yau stacks,” Nucl. Phys. B733 (2006) 233-296, hep-th/0502044. [PSc] T. Pantev, E. Sharpe, “GLSMs for gerbes (and other toric stacks),” Adv. Theor. Math. Phys. 10 (2006) 77-121, hep-th/0502053. [RW] D. Roggenkamp, K. Wendland, “Limits and degenerations of unitary conformal field theories,” Comm. Math. Phys. 251 (2004) 589-643, hep-th/0308143. [Soi03] Y. Soibelman, “Lectures on deformation theory and mirror symmetry,” IPAM, 2003, http://www.math.ksu.edu/~soibel/ipam-final.ps. [SW] N. Seiberg, E. Witten, “String theory and noncommutative geometry,” JHEP 9909 (1999) 032, hep-th/9908142. [Tot02] B. Totaro, “The resolution property for schemes and stacks,” math/0207210. [VdB1] M. van den Bergh, “Three-dimensional flops and noncommutative rings,” Duke Math. J. 122 (2004) 423-455.

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LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D.

[VdB2] [VdB3] [V91] [Wal04] [W91]

[W94]

249 13

M. van den Bergh, “Non-commutative crepant resolutions,” pp. 749-770 in The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002, Springer, 2004. M. van den Bergh, “Non-commutative quadrics,” arXiv: 0807.3753. C. Vafa, “Topological Landau-Ginzburg models,” Mod. Phys. Lett. A6 (1991) 337346. J. Walcher, “Stability of Landau-Ginzburg branes,” J. Math. Phys. 46 (2005) 082305, hep-th/0412274. E. Witten, “Mirror manifolds and topological field theory,” pp. 121-160 in Mirror Symmetry I, ed. S.-T. Yau, Amer. Math. Society, International Press, 1998, hep-th/9112056. E. Witten, “Supersymmetric Yang-Mills theory on a four-manifold,” J. Math. Phys. 35 (1994) 5101-5135, hep-th/9403195.

Physics Department, Robeson Hall (0435), Virginia Tech, Blacksburg, VA 24061 E-mail address: [email protected]

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