Geometry and Quantization of Moduli Spaces 978-3-319-33578-0, 3319335782, 978-3-319-33577-3

Table of contents :
Front Matter ....Pages i-x
Loop Groups, Clusters, Dimers and Integrable Systems (Vladimir V. Fock, Andrey Marshakov)....Pages 1-65
Lectures on Klein Surfaces and Their Fundamental Group (Florent Schaffhauser)....Pages 67-108
Five Lectures on Topological Field Theory (Constantin Teleman)....Pages 109-164
Higgs Bundles and Local Systems on Riemann Surfaces (Richard Wentworth)....Pages 165-219
Back Matter ....Pages 220-220

Citation preview

Advanced Courses in Mathematics CRM Barcelona

Vladimir Fock Andrey Marshakov Florent Schaffhauser Constantin Teleman Richard Wentworth

Geometry and Quantization of Moduli Spaces

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Enric Ventura

More information about this series at http://www.springer.com/series/5038

Vladimir Fock • Andrey Marshakov Florent Schaffhauser • Constantin Teleman Richard Wentworth

Geometry and Quantization of Moduli Spaces Editors for this volume: Luis Álvarez-Cónsul, Universidad Autónoma de Madrid Jørgen Ellegaard Andersen, Aarhus University Ignasi Mundet i Riera, Universitat de Barcelona

Vladimir Fock Institut de Recherche Mathématique Avancée Université de Strasbourg Strasbourg, France

Andrey Marshakov Theory Department Lebedev Physics Institute Moscow, Russia

Florent Schaffhauser Departamento de Matemáticas Universidad de Los Andes Bogotá, Colombia

Constantin Teleman Department of Mathematics University of California Berkeley, California, USA

Richard Wentworth Department of Mathematics University of Maryland College Park, Maryland, USA

ISSN 2297-0304 ISSN 2297-0312 (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-319-33577-3 ISBN 978-3-319-33578-0 (eBook) DOI 10.1007/978-3-319-33578-0 Library of Congress Control Number: 2016962053

Mathematics Subject Classification (2010): 14D20, 32G13, 57R56, 14D21, 14D22, 37K30, 53D45 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword This book is based on four advanced courses given during a semester on The Geometry and Quantization of Moduli Spaces, held at the Centre de Recerca Matem`atica (CRM) in Bellaterra, Barcelona, from March to June 2012. Besides their important role in many areas of mathematics, in the last decades moduli spaces turned out to be crucial for the understanding of phenomena in high energy physics and, as such, have led to an interplay between mathematics and physics that has been amazingly fruitful for both sciences. We hope the reader will appreciate and enjoy the beauty of this interaction in this volume. The courses were devoted to several topics that are experiencing extraordinary growth within the broad research area of moduli spaces, and reflected a recurrent thread in the research semester, namely the moduli space of local systems on a Riemann surface, both from classical and quantum perspectives. Chapter 1, written by V.V. Fock (IRMA, Strasbourg, France) and A. Marshakov (ITEP, Moscow, Russia) and entitled “Loop Groups, Clusters, Dimers and Integrable Systems”, is framed in the context of cluster integrable systems, more precisely, the class of integrable systems constructed by A.B. Goncharov and R. Kenyon using the theory of dimer models in the statistical physics of bipartite graphs on a 2-dimensional torus. This research article describes an isomorphism between these integrable systems and standard ones constructed on appropriate affine Poisson–Lie groups, hence providing a new construction of the former and new viewpoints on the latter. The chapter includes very interesting examples, and points out at exciting generalizations. Chapter 2, written by F. Schaffhauser (Universidad de los Andes, Bogot´a, Colombia), and entitled “Lectures on Klein Surfaces and Their Fundamental Group”, is an expository survey of the fundamental group of a Klein surface. It includes an introduction to Klein surfaces and real algebraic curves, a careful account of the fundamental group of a Klein surface, and a discussion of linear and unitary representations of the fundamental group of a Klein surface, and the unitary representation varieties. Many definitions and results are motivated or illustrated with simple but descriptive examples, and each section has interesting exercises that the reader can use to reinforce the material. Next is Chapter 3, written by C. Teleman (University of Oxford, Oxford, UK) and entitled “Five Lectures on Topological Field Theory”. Beginning from v

vi

Foreword

the classic definition of topological quantum field theories (TQFTs) systematized by M.F. Atiyah, it goes on through different examples to motivate the additional structure which these theories are now known to enjoy in many cases, as predicted by the so-called cobordism hypothesis proved recently by J. Lurie. Rather than giving a fully detailed definition of extended TQFTs, that the reader will find in Lurie’s paper, the author explains, using well chosen examples, the substantial aspects of the notion, as well as the difficulties to prove its consistency and the cobordism hypothesis. This makes this text, in our opinion, an excellent companion to Lurie’s paper. The fourth and last is Chapter 4, written by R.A. Wentworth (University of Maryland, College Park, USA) and entitled “Higgs Bundles and Local Systems on Riemann Surfaces”. Higgs bundles, introduced by N. Hitchin thirty years ago, have proved to be of central importance in different areas, such as low dimensional topology, algebraic geometry, mathematical physics, or even number theory (e.g., in the proof of the fundamental lemma in the Langlands program by B.C. Ngˆo a few years ago). This chapter gives a pedagogical introduction to the notion restricted to Riemann surfaces (which was the original context in Hitchin’s papers). It gives an almost self-contained proof of the correspondence between the moduli spaces of Higgs bundles and local systems, and then applies it to study the oper moduli space, an object playing a crucial role in the recent developments on the geometric Langlands program. We would like to thank the authors of the present book for their great work, both in preparing a very interesting set of lectures and for writing the present notes. We are also grateful to the referees who helped us uncompromisingly. The semester on The Geometry and Quantization of Moduli Spaces was made possible by the generous financial support of different institutions. We would like to thank for this reason the Centre de Recerca Matem`atica, the European Research Foundation through the research network ITGP (Interactions between Topology, Geometry and Physics), the IMUB (Institut de Matem`atiques de la Universitat de Barcelona) and the SCM (Societat Catalana de Matem`atiques). Finally, we would like to thank the staff of the CRM for their efficiency and help during the preparation and running of the program.

Madrid, Aarhus, Barcelona,

´ Luis Alvarez-C´ onsul (ICMAT) Jørgen Ellegaard Andersen (QGM) Ignasi Mundet i Riera (UB)

Contents Foreword 1

Loop Groups, Clusters, Dimers and Integrable Systems Vladimir V. Fock and Andrey Marshakov 1.1

1.2 1.3

1.4

1.5

1.6 1.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Integrable Systems on Poisson–Lie Groups . . . . . . . . . . 1.1.2 Goncharov–Kenyon Integrable Systems . . . . . . . . . . . . 1.1.3 Relations Between the Two Approaches . . . . . . . . . . . Integrable Systems and r-Matrices . . . . . . . . . . . . . . . . . . Cluster Parametrisation of Double Bruhat Cells: Simple Groups . . 1.3.1 Cartan–Weyl Generators of a Simple Group . . . . . . . . . 1.3.2 Construction of the Cluster Seeds . . . . . . . . . . . . . . . 1.3.3 Generators and Thurston Diagrams for the Group PGL(N ) 1.3.4 Example: Poisson submanifolds of PGL(3) . . . . . . . . . . Cluster Parameterisation of Double Bruhat Cells: Loop Groups . . 1.4.1 The Coextended Affine Weyl Group, Wiring, and Thurston Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Realisations of the Coextended Loop Group . . . . . . . . . 1.4.3 Integrable Systems on Double Bruhat Cells . . . . . . . . . Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Recollection About Dimers . . . . . . . . . . . . . . . . . . 1.5.2 Matrices and Dimers on a Disk . . . . . . . . . . . . . . . . 1.5.3 Dimer Partition Functions with Signs . . . . . . . . . . . .  1.5.4 PGL(N ) and Dimers on a Cylinder . . . . . . . . . . . . . . 1.5.5 Dimers and the Discrete Dirac Operator . . . . . . . . . . . 1.5.6 Spectral Submanifold and Face Partition Function . . . . . 1.5.7 Dual Surface, Double Partition Function, and Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimers and Integrable Systems for the Loop Groups . . . . . . . . Mutations and Discrete Flows . . . . . . . . . . . . . . . . . . . . . 1.7.1 Equivalence of Bipartite Graphs . . . . . . . . . . . . . . . 1.7.2 Discrete Flow τ . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents 1.7.3 General Discrete Flows . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 The Simplest Relativistic Toda Chain . . . . . . . . . . . . 1.8.3 Degeneration to non-Affine Toda System . . . . . . . . . . . 1.8.4 Relativistic Toda Chain of Rank Two . . . . . . . . . . . . 1.8.5 Parallelograms of Arbitrary Size and the Pentagram Map . 1.9 Appendix A: Cluster Varieties of Type X . . . . . . . . . . . . . . 1.10 Appendix B: Relations among the Generators of a Simply Laced Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Appendix C: Exchange Graphs and Decompositions of u ∈ W ×W 1.12 Appendix D: Thurston Diagrams . . . . . . . . . . . . . . . . . . . 1.13 Appendix E: Proofs of the Properties of the Minors Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Appendix F: Schwartz Coordinates and the Pentagram Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8

2

Lectures on Klein Surfaces and Their Fundamental Group Florent Schaffhauser Klein Surfaces and Real Algebraic Curves . . . . . . . . . . . . . . 2.1.1 Algebraic Curves and Two-Dimensional Manifolds . . . . . 2.1.2 Topological Types of Real Curves . . . . . . . . . . . . . . . 2.1.3 Dianalytic Structures on Surfaces . . . . . . . . . . . . . . . 2.2 The Fundamental Group of a Real Algebraic Curve . . . . . . . . . 2.2.1 A Short Reminder on the Fundamental Group of a Riemann Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Fundamental Group of a Klein Surface . . . . . . . . . 2.2.3 Galois Theory of Real Covering Spaces . . . . . . . . . . . . 2.3 Representations of the Fundamental Group of a Klein Surface . . . 2.3.1 Linear Representations of Fundamental Groups of Real Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Real Structure of the Usual Representation Variety . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Five Lectures on Topological Field Theory Constantin Teleman 3.1

Introduction and Examples . . . . . . . . . . 3.1.1 Definition . . . . . . . . . . . . . . . . 3.1.2 Example: Finite Group Gauge Theory 3.1.3 Baby Classification, D = 1 . . . . . . 3.1.4 D = 2 and Frobenius Algebras . . . . 3.1.5 Finite Group Gauge Theory in 2D . .

53 53 55 60 61 65

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3.2

3.3

3.4

3.5

3.1.6 Finite Higher-Groupoid Theories . . . . . . . . . 3.1.7 Yang–Mills Theory in 2D . . . . . . . . . . . . . 3.1.8 Variant of TQFT: Cohomological Field Theories Two-Dimensional Gauge Theory . . . . . . . . . . . . . 3.2.1 Interpretation in K-Theory . . . . . . . . . . . . 3.2.2 Integration from the Index . . . . . . . . . . . . 3.2.3 Index Formulas on Φ(Σ; G) . . . . . . . . . . . . 3.2.4 The Verlinde Ring . . . . . . . . . . . . . . . . . 3.2.5 Twisted K-Theory, a Crash Course . . . . . . . . 3.2.6 Twisted KG (G) . . . . . . . . . . . . . . . . . . . 3.2.7 Generalizations: Higgs Bundles as an Example . Extended TQFT and Higher Categories . . . . . . . . . 3.3.1 Higher Categories . . . . . . . . . . . . . . . . . 3.3.2 Strict Versus Weak Categories . . . . . . . . . . 3.3.3 Finite Group Gauge Theory in 2D . . . . . . . . 3.3.4 The Correspondence 2-Category . . . . . . . . . 3.3.5 Inadequacy of Strict Categories . . . . . . . . . . 3.3.6 The Quadratic Map π2 → π3 . . . . . . . . . . . 3.3.7 A Braided Tensor Category from X . . . . . . . 3.3.8 Ribbon Structure . . . . . . . . . . . . . . . . . . 3.3.9 Finite Homotopy Types . . . . . . . . . . . . . . The Cobordism Hypothesis in Dimensions 1 and 2 . . . 3.4.1 Duals and 1D framed TQFT’s . . . . . . . . . . 3.4.2 Cobordism Hypothesis in 1D . . . . . . . . . . . 3.4.3 O(1) Action on Dualizable Objects . . . . . . . . 3.4.4 Cobordism Hypothesis in 2D . . . . . . . . . . . 3.4.5 The Serre Twist . . . . . . . . . . . . . . . . . . 3.4.6 Oriented and r-Spin Theories . . . . . . . . . . . 3.4.7 Adjunction in Pictures: Oriented Handles . . . . 3.4.8 Framed Handles . . . . . . . . . . . . . . . . . . 3.4.9 Adjunction: Algebraic Conditions . . . . . . . . . 3.4.10 Oriented TQFT’s from Frobenius Algebras . . . 3.4.11 Finite Gauge Theory Revisited . . . . . . . . . . 3.4.12 The Serre Functor on a Scheme . . . . . . . . . . 3.4.13 Vector Spaces Associated to the Circle . . . . . . Cobordism Hypothesis in Higher Dimension . . . . . . . 3.5.1 Reduction to co-Dimension 2 . . . . . . . . . . . 3.5.2 3D Example . . . . . . . . . . . . . . . . . . . . . 3.5.3 Tensor and Module Categories . . . . . . . . . . 3.5.4 Tensor Product of Categories . . . . . . . . . . . 3.5.5 2-Dualizability . . . . . . . . . . . . . . . . . . . 3.5.6 Drinfeld Center and Hochschild Homology . . . . 3.5.7 Fusion Categories . . . . . . . . . . . . . . . . . . 3.5.8 The Serre Automorphism T ∨ . . . . . . . . . . .

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113 114 115 118 119 120 121 122 122 124 126 127 128 129 129 130 132 133 134 135 136 137 139 141 141 142 143 144 144 145 147 149 149 150 150 151 152 153 157 157 158 158 160 161

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Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4

Higgs Bundles and Local Systems on Riemann Surfaces Richard A. Wentworth 4.1 4.2

Preface . . . . . . . . . . . . . . . . . . . . . . . . . The Dolbeault Moduli Space . . . . . . . . . . . . . 4.2.1 Higgs Bundles . . . . . . . . . . . . . . . . . 4.2.2 The Moduli Space . . . . . . . . . . . . . . . 4.2.3 The Hitchin–Kobayashi Correspondence . . . 4.3 The Betti Moduli Space . . . . . . . . . . . . . . . . 4.3.1 Representation Varieties . . . . . . . . . . . . 4.3.2 Local Systems and Holomorphic Connections 4.3.3 The Corlette–Donaldson Theorem . . . . . . 4.3.4 Hyperk¨ ahler Reduction . . . . . . . . . . . . 4.4 Differential Equations . . . . . . . . . . . . . . . . . 4.4.1 Uniformization . . . . . . . . . . . . . . . . . 4.4.2 Higher Order Equations . . . . . . . . . . . . 4.4.3 Opers . . . . . . . . . . . . . . . . . . . . . . 4.4.4 The Eichler–Shimura Isomorphism . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

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165 168 168 173 178 188 188 188 192 197 200 200 202 205 213 215

Chapter 1

Loop Groups, Clusters, Dimers and Integrable Systems Vladimir V. Fock and Andrey Marshakov1 1.1 Introduction The main idea of this work is to demonstrate the equivalence of two a priori different methods of construction and description of a wide class of integrable models, and thus, to propose a unified approach for their investigation. In the first, well-known method [24], the phase space is taken as a quotient of double Bruhat cells of a Kac–Moody Lie group, with the Poisson structure defined by a classical r-matrix, and the integrals of motion are just the Ad-invariant functions. The second method was suggested recently by A. Goncharov and R. Kenyon in [11], and it emerged from the study of dimer models of statistical physics on bipartite graphs on a two-dimensional torus. We are going to show that in fact the latter class of integrable systems is a particular case of the former, corresponding to the affine group of type AˆN −1 . The best known example of integrable system of this class [6] is the relativistic Toda chain, discovered by S. Ruijsenaars [25] and studied by Y. Suris [28] and many others, which gives the common Toda chain in a certain limit, corresponding in our terms to passing from Lie groups to Lie algebras. Another known 1 We

would like to thank the QGM, University of Aarhus, the Max Planck Institute, and the Haussdorf Institute for Mathematics in Bonn where essential parts of this work have been done. We are indebted to the referee for carefully reading the manuscript and correcting many errors. The work of V.F. has been partially supported by ANR GTAA and ANR ETTT grants. The work of A.M. has been partially supported by the research grant 13-05-0006 of NRU HSE, by the joint RFBR project 12-02-92108, by the Program of Support of Scientific Schools (NSh3349.2012.2), and by the Russian Ministry of Education under the contract 8207.

© Springer International Publishing Switzerland 2016 V. Fock et al., Geometry and Quantization of Moduli Spaces, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-319-33578-0_1

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Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

example is the pentagram map – a discrete integrable system on the space of broken lines in a projective plane, discovered by R. Schwartz [26] and studied by him with V. Ovsienko and S. Tabachnikov [22, 23], and recently by M. Glick [10], B. Khesin and F. Soloviev [17, 18, 27], M. Shapiro, M. Gekhtman, A. Vainshtein, and S. Tabachnikov [9]. On the other hand, V. Ovsienko and S. Tabachnikov have shown in [22] that the discrete flow of the pentagram map gives the Boussinesq flow in the continuum limit. This observation, generalised in [17] for other dimensions, suggests that the technique proposed in [22] can be also applied to the study of continuous integrable systems, such as n-KdV hierarchies.

1.1.1 Integrable Systems on Poisson–Lie Groups Our starting observation is that on a Poisson–Lie group, with the Poisson bracket defined by the classical r-matrix, the Ad-invariant functions Poisson commute with each other. For a finite-dimensional simple group, the number of linearly independent Ad-invariant functions is equal to the rank of the group and, thus, the corresponding integrable system can arise on a symplectic leaf of dimension not larger than twice the rank, see [6]. However, for affine groups the number of linearly independent Ad-invariant functions is infinite, though all Poisson submanifolds are still finite-dimensional, and thus the set of integrable systems one gets in this way is much larger. Such integrable systems can be constructed on any affine ˆ but here we will need only the systems on the groups of Poisson–Lie group G, type Aˆ with trivial center. This group can be realised as a group of matrix-valued Laurent polynomials A(λ) of a single variable with nonzero constant determinant and considered up to multiplication by a nonzero constant. For a given A(λ), the set  Hij λi μj = 0} {(λ, μ) | det(A(λ) − μ) = ij × 2

is an algebraic curve in (C ) , called spectral curve, embedded into the torus (C× )2 . This curve, considered up to the torus automorphisms, is a conjugacy class invariant. The spectral curve comes together with the line bundle given by the ˆ to the space of curves is called kernel of A(λ) − μ · Id. The map from the group G the action map, and it is a Poisson map if we take a trivial Poisson bracket on the space of curves. The functions Hij themselves are not well defined, since changing them by Hij → Hij αi β j γ would correspond to the same curve; however, one can use this freedom to make any three nonvanishing coefficients to be equal to one. With this condition, Hij become well defined and do Poisson commute. The map to the pair (curve, line bundle on it) is called the action-angle map. The flows generated by the Poisson commuting integrals of motion or Hamiltonians amount to the constant flow of the line bundle along the Jacobian of the spectral curve. We describe such integrable systems in more detail in Section 1.2. ˆ does not have a cluster variety structure. However, it is A loop group G ˆ  admitting a embedded as a Poisson submanifold into a central coextension G

1.1. Introduction

3

ˆ  )u , standard decomposition into the disjoint union of Poisson submanifolds (G called double Bruhat cells in [19], which already are cluster varieties. These cells ×W  ) of the square of the are enumerated by the elements u of a coextension (W  of G ˆ by the automorphism group of the Dynkin diagram, which is Weyl group W a cyclic group for the series AˆN −1 . ˆ  with G, ˆ quotiented by conjugation by Intersections of the Bruhat cells of G the finite-dimensional Cartan subgroup H (we also call them double Bruhat cells, ˆ u /AdH of our integrable systems. The ˆ u ) are the phase spaces G and denote by G dimension of such a phase space is (in the case of affine groups) one less than the length of u. Given a presentation of u as a reduced word in the standard generators, one can define the cluster coordinates x = {xf }, enumerated by theletters of the word (except for the generator of the coextension) and subject to xf = 1. For a given Bruhat cell, the Laurent polynomial det(A(x, λ, μ)) = Hij (x)λi μj has a fixed Newton polygon Δ, and the number of Poisson commuting Hamiltonians nonvanishing on the cell is three less than the number of integral points of Δ. In order for this system to be integrable, these integrals of motion must be indepenˆ u )/2 ˆ u + corank G dent, and their number should be the maximal possible (dim G for the given dimension of the cell and given rank of the Poisson bracket. This condition is satisfied on double Bruhat cells, corresponding to u having minimal length in its conjugacy class (such elements are also called cyclically irreducible). Following [4], in Section 1.4 we introduce cluster coordinates (cluster seeds) ˆ  and on their quotients by conjugation by on the double Bruhat cells of the group G the Cartan subgroup (describing them first for the simple groups in Section 1.3 and generalising then to the affine case). Given a set of coordinates x, corresponding to a reduced word, we construct the corresponding matrix polynomial A(x, λ) as a product of elementary matrices, each of which is either a constant or depends on just a single coordinate xf . Then, we formulate our integrable systems in terms of these cluster coordinates. Coordinates corresponding to different decompositions of the word u are related by cluster transformations.

1.1.2 Goncharov–Kenyon Integrable Systems Recall that a cluster variety is an algebraic variety, covered by charts isomorphic to algebraic tori (C× )N with transition functions being compositions of special birational transformations called mutations (see Appendix 1.9). The Goncharov– Kenyon (GK) approach proposes an integrable system, associated with a dimer model on a graph on a two-dimensional torus. The integrable system structure turns out to be compatible with the structure of the cluster variety, implying duality, discrete group action, quantisation, tropical limit, and many other attractive features. Moreover, the mutually Poisson commuting integrals of motion can be chosen to be the cluster functions. Many such systems admit an abelian group of discrete cluster transformations, commuting with the integrable flows.

4

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

The scheme of the construction of the GK integrable system, described in detail in Section 1.5, is roughly as follows. The starting point is a bipartite graph on a closed surface Σ, satisfying certain minimality and non-triviality conditions (following [11], we assume the surface Σ to be of genus one). Many aspects of the construction can be generalised for surfaces of higher genus, but we postpone this generalisation for a forthcoming publication. Consider the space of discrete connections on the graph Γ with values in a multiplicative group (to be specific we assume it to be the multiplicative group of non-zero complex numbers C× ). Since every edge of a bipartite graph can be canonically oriented, say from a white to a black vertex, we can interpret this space as the multiplicative group of 1-cochains C 1 (Γ), which is just the product of C× factors, corresponding to each edge of Γ. The quotient of this space by discrete gauge transformations can be interpreted as the cohomology group H 1 (Γ). This group is an extension of the group of coboundaries B 2 (Σ) by the cohomology group of the torus, H 1 (Σ) = C× × C× . The elements of B 2 (Σ) are collections of numbers on faces of Σ, with their product equal to the unity. The differential ∂ : H 1 (Γ) → B 2 (Σ) (given by monodromies around the faces) defines on H 1 (Γ) the structure of a principal H 1 (Γ)-bundle over B 2 (Σ). For every discrete connection A = {Ae } ∈ C 1 (Γ) and for every choice of a spin structure on the torus Σ, we define in Subsection 1.5.5 a discrete Dirac operator D(A) : CB → CW from the functions on black vertices to the functions on white vertices. This operator degenerates on a subvariety of C 1 (Γ) which is the vanishing locus of the determinant det D(A). This variety is gauge invariant and thus it defines a subvariety in H 1 (Γ). The intersection of this locus with a fibre over a point x = {xf } ∈ B 2 (Σ) gives an algebraic curve with line bundles on it given by the kernel of D(A). Observe that the determinant of D(A) is a sum of monomials over the perfect matchings of the white and black vertices, and this is how the dimer configurations on the graph Γ come into play. The spin structure permits to control signs of the monomials. This construction therefore defines the action map of the phase space of our integrable system B 2 (Σ) on the space of plane algebraic curves and the action-angle map to the pairs (plane curve, line bundle on it) which is a birational isomorphism. In coordinates, the action map reads as follows. Choose a trivialisation of the bundle ∂ which amounts to an isomorphism H 1 (Γ) = B 2 (Σ) × H 1 (Σ), and then choose a lift of H 1 (Γ) to C 1 (Γ). Under these identifications we can associate (λ, μ) ∈ H 1 (Σ). The spectral a connection A(x, λ) to any x ∈ B 1 (Σ) and λ =  curve is defined by the equation det D(A(x, λ)) = Hij (x)λi μj = 0 and it does not depend on the choices made, if we consider the curve up to automorphisms of H 1 (Σ) = C× × C× . The coefficients Hij are defined up to a transformation Hij → Hij αi β j γ and we can use this degree of freedom to make three of them to be equal to one. The remaining coefficients give a collection of Poisson commuting functions with respect to a natural Poisson structure on B 2 (Σ), which is maximal if the graph satisfies a certain minimality condition.

1.1. Introduction

5

The space B 2 (Σ) possesses a canonical log-constant Poisson bracket introduced in [11] as follows. Embedding the graph Γ into the surface Σ induces a cyclic order of ends of the edges at every vertex (a fat graph structure). Consider ˜ corresponding to the same bipartite graph Γ, but with the cyclic order a surface Σ changed to the opposite in white vertices and kept unchanged in the black ones. ˜ we have the map H 1 (Σ) ˜ → B 2 (Σ), which Since the graph Γ is embedded into Σ, is the composition of the standard embedding with the coboundary operator. The ˜ has a canonical Poisson structure, coming from the intersection index space H 1 (Σ) ˜ on Σ, and the map to B 2 (Σ) induces the Poisson bracket on the latter. This bracket can be extended by multiplication invariance to the space B 2 (Σ) of collections of nonzero numbers attached to the faces of the graph, and defines a cluster seed with the skew-symmetric exchange matrix defined by the Poisson bracket. In [11] it is observed that graphs admit elementary transformations, called spider moves, such that the integrable systems corresponding to them are isomorphic provided the phase spaces are related by a cluster mutation. Equivalence classes of integrable systems under such transformations are enumerated by Newton polygons of det D(x, λ), and the number of independent integrals of motion is just the number of integral points strictly inside these Newton polygons.

1.1.3 Relations Between the Two Approaches We claim in Section 1.6 that GK integrable systems coincide with the integrable  ). The isomorphism identifies not systems on the Poisson–Lie loop groups PGL(N only their phase spaces and commuting flows, but also the discrete group action and the canonical cluster coordinates. In both constructions the spectral curve of an integrable system is given by the degeneracy condition of some matrix operator (A(x, λ) − μ · Id in the group theory approach and the Dirac operator D(A(x, λ, μ)), respectively). Though the matrices do not coincide, their determinants do; roughly, the correspondence goes as follows. The determinant of any matrix D(A) can be written as a Grassmann integral:  

SΓ (A) = det D(A) = exp D(A)bw ξb η w dξb dη w . b

w

Therefore, det D(A) can be interpreted as a partition function SΓ (A) of some lattice fermions in the background gauge field A. Cutting the torus into a cylinder corresponds to rewriting this partition function as a trace of the evolution operator from one boundary circle to another. This evolution operator is given by the matrix A(λ) acting in the external algebra of the N -dimensional space. Moreover, cutting further this cylinder into a set of smaller cylinders, one can present the evolution operator as a product of elementary steps, each depending on no more than one variable xf and exactly coinciding with elementary matrices, used to parameterise the double Bruhat cells, thus establishing the coincidence of spectral curves.

6

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

The first part of this program is establishing a correspondence between words ˆ ×W ˆ ) , enumerating cluster coordinate systems in the in the generators of (W first approach, with the bipartite graphs, enumerating coordinates in the second approach. Moreover, this correspondence should identify the letters of the word with the faces of the corresponding bipartite graphs, drawn on a torus Σ, since both sets correspond to the cluster coordinates in the corresponding cases. In order to do this we use a third combinatorial object, suggested by Dylan Thurston in the unpublished paper [29] (and already used in [11] in our context), which we call Thurston diagrams and describe in detail in Appendix 1.12. A Thurston diagram is an isotopy class of a collection of curves on a surface, either closed, or connecting two boundary points with only triple intersection points, and such that the connected components of the complement (faces) are colored in white and grey, with any two faces sharing a segment of a curve having different colors (chessboard coloring). Such diagrams admit elementary modifications called Thurston moves. As it was already observed by D. Thurston and A. Henriques, every Thurston diagram defines a cluster seed (a chart on a cluster manifold) with cluster variables attached to the white faces. Thurston moves correspond to mutations (passing from one chart to another). Having Thurston diagrams on open surfaces one can glue together boundary components respecting their coloring, and thus obtain a new surface with a Thurston diagram. In order to construct a Thurston diagram out of a reduced decomposition of  ×W  ) , we first associate a Thurston diagram on a cylinder with an element u ∈ (W a single triple point and with N grey (and white) segments on every boundary ×W  ) (except the cocentral one). Then, we glue circle to every generator of (W the cylinders together according to the order of the generators in the reduced decomposition, and finally we glue both ends of the resulting cylinder together with a twist, given by the power of the cocentral generator Λ. In order to construct a bipartite graph out of a Thurston diagram, we put a black vertex at every triple point and a white vertex at every grey face. Then we draw three edges from each black vertex inside the three grey sectors, meeting at this vertex, to the respective white vertices. It is easy to see that the set of letters of the reduced word is in a canonical bijection with the set of white faces of the Thurston diagram, and the latter are in bijection with faces of the bipartite graph. The next observation, almost as simple, is that this bijection induces a bijection between the cluster seeds, i.e., the Poisson brackets between the coordinates coincide. Finally, we need to show that the equations det (A(x, λ) − μ) = 0 and D(A(x, λ, μ)) = 0 define the same curve. With this purpose, in Section 1.5 we extend the lattice fermion partition functions on a bipartite graph to surfaces with boundary. Graphs on such surfaces are allowed to have vertices of the third type, terminating on the boundary and which can be connected to both white and black vertices, but not to each other. Denote the set of such vertices by T . The Dirac operator now acts as D(A) : CB∪T → CW ∪T and, for extra Grassmann

1.2. Integrable Systems and r-Matrices

7

variables ζ = {ζt |t ∈ T }, we define  

dξb dη w . S(A, ζ) = exp D(A)bw ξb η w + D(A)bt ξb ζ t + D(A)tw ζt η w b

w

Gluing two boundary components of Σ with a bipartite graph Γ on it in a way that terminal vertices are glued to terminal vertices, one gets a bipartite graph Γ on the glued surface Σ. The connection A on Γ induces a connection A on Γ: we just multiply the numbers of two halves of a glued edges. If SΓ (A, ζ) is a partition function for Γ, then the partition function for Γ is given by 

 dζt dζσ(t) , ζt ζσ(t) SΓ (ζ, A) = SΓ (ζ, A) exp where the index t runs over terminal edges on one side, σ is a map sending a terminal vertex to the one it is glued to, and λ is λ with the entries corresponding to glued vertices removed. On the other hand, observe that to any N × N matrix M one can associate a function of 2N Grassmann variables ξ = {ξi } and η = {ηi } given by SM (ξ, η) = exp(Mji ξi η j ). Matrix product corresponds then to convolution of the corresponding functions,  

ξi ηi SM1 M2 (ξ, η) = SM1 (ξ, η  )SM2 (ξ  , η) exp dξi dηi . Thus, if a partition function of a graph on a cylinder coincides with a partition function of a matrix, the partition function of several cylinders glued together corresponds to the product of the matrices. Therefore, in order to show the coincidence of the curves, we need to cut the torus into small cylinders and verify for each of them the coincidence of partition functions. We complete the proof of the correspondence in Section 1.6, and formulate there our main result. The discrete flows in our integrable systems are considered in Section 1.7. In Section 1.8 we discuss several examples of our integrable systems. In particular, in Section 1.8.5 we show that the discrete integrable system on the space of polygons in the projective plane discovered by R. Schwartz [26] can be realised as a particular case of the scheme described here. More particular examples of systems in this class have been already considered in [21].

1.2 Integrable Systems and r-Matrices Recall the standard construction of integrable systems related to the classical rmatrices on simple or affine Lie groups (see, e.g., [24]), which we assume to be complex and with vanishing center. The phase spaces for these integrable systems are certain Poisson submanifolds of the Poisson–Lie groups, and the mutually commuting Hamiltonians or integrals of motion are given by the conjugation invariant functions. Indeed, let G be a Lie group, g = Lie(G) be the corresponding Lie algebra and r ∈ g ⊗ g a solution of the Yang–Baxter equation

8

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

[r12 , r13 ] + [r13 , r23 ] + [r12 , r23 ] = 0. Such r-matrix defines a Poisson bracket on the group G, 1 {g ⊗, g} = − [r, g ⊗ g], (1.1) 2 and this bracket is compatible with the group structure in the sense that the group multiplication G × G → G and the inversion G → G are the Poisson maps. It is easy to see directly from (1.1) that any two Ad-invariant functions on G Poisson commute with each other. Indeed, presenting the r-matrix as r =  (1) (2) I vI ⊗vI , where all vI ∈ g, and denoting by Lv (resp. Rv ) the left (resp. right) invariant vector field corresponding to v, the Poisson bracket for any two functions   is given by {H1 , H2 } = I Lv(1) H1 Lv(2) H2 − Rv(1) H1 Rv(2) H2 . Since any AdI I I I invariant function H satisfies Lv H = −Rv H, the bracket of two such functions vanishes. Observe also that this argument is local, i.e., the bracket vanishes even if the functions are defined not on the whole G, but on any Poisson Ad-invariant subvariety of G. ˆ where We shall restrict ourselves to the case of a simple or affine Lie group G there exists a canonical Drinfeld–Jimbo solution of the Yang–Baxter equation,  1 i dα eα ⊗ eα¯ + d hi ⊗ hi , (1.2) r= 2 α∈Δ+

i∈Π

where Δ+ is the set of positive roots, dα = (α, α)/2, Π is the set of positive simple roots, α ¯ is just another notation for −α, and eα and hi constitute the standard Cartan–Weyl basis of g. To simplify the presentation we will assume in what follows that the group is simply laced, i.e., (α, α) = 2 and dα = 1 for all roots α. Obviously, on a simple group there exists rank G independent Ad-invariant functions: a possible choice of these functions is the set {Hi }, where i ∈ Π, Hi (g) = Tr πμi (g) and πμi is the i-th fundamental representation of G with the highest weight (μi , αj ) = δji dual to αi , i ∈ Π. These functions define integrable systems on Poisson submanifolds of G of rank at most 2 rankG (see, e.g., [6]). For a loop ˆ which we understand below as a group of Laurent polynomials with group G, values in a simple group G, the number of independent Ad-invariant functions is infinite since every coefficient of Tr πμi (g) is now an Ad-invariant function. Thus, a loop group gives a much larger set of integrable models. On the space G/AdH, where H is a Cartan subgroup for G, one can define, following [12], an action of a discrete birational Poisson transformation τ : G/AdH → G/AdH, preserving the double Bruhat cells and the functions Hi . Namely, let g = g+ g− be the Gauss decomposition of g ∈ G/AdH. Define τ (g) as the product g− g+ . (The Gauss decomposition is ambiguously defined on g, since the Cartan part can be equally well attached to the upper or lower triangular one, or just split between the two. But on the quotient G/AdH, the action of τ is nevertheless well defined.) In Section 1.7 we show how this transformation can be generalised to loop groups.

1.3. Cluster Parametrisation of Double Bruhat Cells: Simple Groups

9

Recall now the classification of symplectic  leaves of G (see for example [12, 19]). The group G can be decomposed as G = u∈W ×W Gu into the double Bruhat cells, enumerated by elements of the group W × W , where W is the Weyl group of G. Each double Bruhat cell is isotypic, i.e., it is birationally equivalent to the product of a symplectic manifold and a manifold with trivial Poisson bracket. The dimension of a cell Gu is given by dim Gu = l(u) + rank G, where l(u) is the length of u. One can modify the Poisson manifold G in order to make all constructions a little bit more symmetric. Namely, consider the action of the Cartan subgroup H ⊂ G on G by conjugation. Since H is a Poisson subgroup of G with trivial Poisson structure, the quotient G/AdH inherits the Poisson structure and the collection of Poisson commuting functions {Hi } as well as a decomposition into Poisson  submanifolds G/AdH = u∈W ×W Gu /AdH. The dimensions of the corresponding Poisson submanifolds are now just dim Gu /AdH = l(u).

1.3 Cluster Parametrisation of Double Bruhat Cells: Simple Groups Following [4], we describe here how to introduce the structure of a cluster variety on Gu and Gu /AdH. Namely, starting from a decomposition of u as a reduced product si1 · · · sil of standard generators of W × W , we define a cluster seed (see Appendix 1.9), a split algebraic torus provided with log-constant Poisson structure, and a Poisson embedding of it into Gu with Zariski open image. Similarly, a cluster seed for the space Gu /AdH is constructed from the same data. Finally, we show that for the group G = PGL(N ), the cluster seeds are isomorphic to those corresponding to the Thurston diagrams constructed out of the decomposition u = si1 · · · sil on a disk.

1.3.1 Cartan–Weyl Generators of a Simple Group To describe cluster coordinates on Gu and Gu /AdH, we need to introduce first a set of generators of the Lie group G analogous to the Cartan–Weyl generators of the corresponding Lie algebra g. The set of standard generators of a Weyl group W is in canonical bijection with the set Π of simple roots of G, and we shall not distinguish between these two sets. The set of generators of the second copy of W ¯ will be identified with the set of negative simple roots Π. Recall that, given a Cartan matrix Cij , the corresponding Lie algebra g is ¯ satisfying the following relations (for generated by {hi | i ∈ Π} and {ei | i ∈ Π ∪ Π}, simplicity, we denote by ¯i the root opposite to the root i, and extend h and C to the negative roots, assuming that hi = h¯i , that C¯i,¯j = Cij , and that Cij = 0 if i and j have different signs):

10

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems [hi , hj ] = 0, [hi , ej ] = sign(j)Cij ej , [ei , e¯i ] = sign(i)hi , (ad ei )1−Cij ej = 0 for i + j = 0.

(1.3)

i For the  same Liej algebra one can replace the set {hi } by the set {h }, defined by hi = j∈Π Cij h , and then the relations (1.3) take the form

[hi , hj ] = 0, [hi , ej ] = sign(j)δij ej , [ei , e¯i ] = sign(i)Cij hj , (ad ei )1−Cij ej = 0 for i + j = 0.

(1.4)

¯ introduce the group element Ei = exp(ei ) and a one-parameter For any i ∈ Π ∪ Π subgroup Hi (x) = exp(hi log x), which will be our set of generators of the group G. The commutation relations (1.4) imply the relations between Hi and Ei (we list them in Appendix 1.10 for the simply laced case). As an immediate consequence of (1.4), notice that Hi (x) commutes with Ej unless i = j, and that Ei commutes with Ej if Cij = 0 and i = ¯j.

1.3.2 Construction of the Cluster Seeds Take a decomposition u = si1 · · · sil of an element u ∈ W × W . A seed ((C× )l , ε) and a map ev : (C× )l → G/Ad H are associated to such u by (x1 , . . . , xl ) → Hi1 (x1 )Ei1 · · · Hil (xl )Eil .

(1.5)

The image of this map is Zariski open in the double Bruhat cell C u , and it is an embedding if the word u is reduced. Different reduced decompositions of the same element u ∈ W × W give rise to different parametrisations, related by a cluster transformation. To construct a seed ((C× )(l+r) , ε), where r is the rank of G, i.e., to parameterise the cells of the simple group G itself, one just needs to multiply this expression from the right by an arbitrary element of the Cartan subgroup: (x1 , . . . , xl+r ) → Hi1 (x1 )Ei1 · · · Hil (xl )Eil H1 (xl+1 ) · · · Hr (xl+r ).

(1.6)

The construction of the corresponding exchange matrix ε is given in Appendix 1.11. Using the fact that Hi (x)Ej = Ej Hi (x) unless i = j, one can rewrite expressions (1.5) and (1.6) in many different ways by moving any Hi until it meets Ei or E¯i . Therefore, every cluster variable is naturally associated to a positive simple root i, and for a given i to a minimal segment of the word si1 · · · sil delimited by si , s¯i , or ends of the word (for a cyclic word all segments are delimited just by si or s¯i ). One can check (see [4]) that cluster seeds corresponding to different decompositions of the same word u are related by a cluster transformation. In particular,

1.3. Cluster Parametrisation of Double Bruhat Cells: Simple Groups

11

their images coincide up to codimension one. For example, if Cij = Cji = −1, the coordinates corresponding to the decomposition u = Asi sj si B, where A and B are arbitrary words, are related to the coordinates corresponding to the decomposition u ˜ = Asj si sj B by a mutation in the variable, associated to the segment [si sj si ]. Similarly, for the relation between Asi s¯i B and As¯i si B we should make a mutation in the variable associated to the segment [si s¯i ]. If Cij = 0, applying the relation for Asi sj B ↔ Asj si B does not change parametrisation, since the corresponding matrices Ei and Ej commute. If a decomposition is not reduced, the maps (1.5) and (1.6) are still defined, but they are no longer embeddings. Instead, such maps are embeddings corresponding to a reduced word, pre-composed with a projection along some coordinates and some mutations. Indeed, the map corresponding to the word Asi si B is a composition of the mutation in the variable associated to the segment [si si ] with the map corresponding to Asi B. Therefore, we claim that the double Bruhat cells of G and of G/AdH are cluster varieties. In [4] it is proven that the Poisson brackets on Gu (and thus on Gu /AdH), given by the exchange matrix ε, and the one given by the Drinfeld– Jimbo r-matrix (1.2) do coincide.

1.3.3 Generators and Thurston Diagrams for the Group PGL(N ) In the case G = PGL(N ), we can make the construction much more explicit. The generators Ei and Hi (x) in the standard representation have a particularly simple form: ⎛ ⎛ ⎞ ⎞ x 0 ··· 0 1 0 ··· 0 . . ⎜0 . . ⎜0 . . 0⎟ 0⎟ ⎜ ⎜ ⎟ ⎟ 1 1 x ⎜. ⎜. ⎟ ⎟ tr Hi (x) = ⎜ . ⎟ , (1.7) ⎟ , Ei = E¯i = ⎜ . 1 1 ⎜. ⎜. ⎟ ⎟ . . ⎝ ⎝ . . 0⎠ . . 0⎠ 0 ··· 0 1 0 ··· 0 1 where the lowest row of Hi (x) with x, and the row in Ei containing the off-diagonal 1 have row number i > 0. For negative i, the corresponding matrix E−i = E¯i is just transposed to the matrix of the positive root. Let us now give an alternative description of the cluster seeds, corresponding to the decomposition of u ∈ W × W into a product of generators for the group PGL(N ), using Thurston diagrams (see Appendix 1.12). Every decomposition of u into a product of generators corresponds to a Thurston diagram, and the latter in its turn corresponds to a cluster seed. We claim that this alternative way gives the same seed. To verify this statement one needs, first, to compare the seeds corresponding to a single generator. This can be done by comparing the exchange graphs for such elementary Thurston diagrams containing only one triple point (see Fig.1.1.A) and the exchange graph (chord) described in Appendix 1.11. Then,

12

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

→

s1 s¯1 s1 s¯2 s2

A

B

Figure 1.1: Thurston diagrams. A: for an elementary generator si of the Weyl group; B: for the word s1 s¯1 s1 s¯2 s2 .

we need to verify that gluing exchange graphs corresponds to gluing Thurston diagrams. We leave these verifications as easy exercises. For the group G = PGL(N ) every word u ∈ W × W in the generators corresponds to a Thurston diagram on a disk, which we draw as an infinite vertical strip. Every diagram consists of 2N curves connecting the sides of the strip and having no vertical tangent. The curves are oriented in such a way that, for a generic vertical section, the orientations of the curves at the intersection points with the section alternate. Generators si (and s¯i ) correspond to a triple intersection of the curves with the numbers 2i − 1, 2i, 2i + 1 (and 2i, 2i + 1, 2i + 3, respectively), counted from above along a section. An example of a Thurston diagram is shown in Fig. 1.1. The properties of the correspondence between the diagrams and the words are listed in Appendix 1.12. The face variables, corresponding to the top and bottom white faces, for the simple group G = PGL(N ) are restricted to the unity. The Thurston diagrams for Gu /AdH are obtained from those for Gu by gluing together the right and the left sides, thus getting diagrams on a cylinder instead of a strip.

1.3.4 Example: Poisson submanifolds of PGL(3) In the case of G = PGL(3) there are just two simple roots, and the Cartan matrix is   (αi , αj ) 2 −1 Cij = 2 = (αi , αj ) = . (1.8) −1 2 (αi , αi ) In this case, the elementary matrices from (1.7) are

1.3. Cluster Parametrisation of Double Bruhat Cells: Simple Groups

x1

x2 x3

x6

x5

x7

x1

x8

x4

x1

x2 x3

s1 s¯1 s2 s¯2 s1 s¯1

s1 s¯1 s2 s¯2

A

B

x2 x3

x4

13

x4

s¯1 s1 s¯2 s2 C

Figure 1.2: Thurston diagrams for A: the word 1¯ 12¯ 21¯ 1; B: the cyclic word 1¯ 12¯ 2; C: ¯ ¯ the cyclic word 1122. (For B and C, the right and the left sides are glued together.) 

 1 1 0 E1 = = 0 1 0 ,  0 0 1 x 0 0 H1 (x) = 0 1 0 , 0 0 1 E¯1tr



 1 0 0 E2 = = 0 1 1 ,  0 0 1 x 0 0 H2 (x) = 0 x 0 . 0 0 1 E¯2tr

(1.9)

The big cell in G = PGL(3) is parametrised by a particular case of the expression (1.6), corresponding to a decomposition of the longest element of W ×W (here of length l = 6). For an element 1¯ 12¯ 21¯ 1 with the Thurston diagram presented in Fig. 1.2.A, the corresponding product is (x1 , . . . , x8 ) → H1 (x1 )E1 H1 (x2 )E¯1 H2 (x3 )E2 H2 (x4 )E¯2 H1 (x5 )E1 H1 (x6 )E¯1 H1 (x7 )H2 (x8 ). (1.10) 1], ]1¯ 12], [2¯ 2], The coordinates x1 , . . . , x8 are associated with the segments ]1], [1¯ [¯ 21¯ 1[, [1¯ 1], [¯ 1[, [¯ 21¯ 1[, respectively. For the Poisson submanifold of dimension l = 2 rank G = 4 in G/AdH, corresponding to the longest cyclically irreducible word 1¯ 12¯ 2 with the Thurston diagram presented in Fig. 1.2.B, the parameterisation (1.6) gives (x1 , x2 , x3 , x4 ) → H1 (x1 )E1 H1 (x2 )E¯1 H1 (x3 )E2 H1 (x4 )E¯2 ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ 1 0 0 x2 0 0 1 1 0 x1 0 0 = ⎝0 1 0⎠ ⎝ 0 1 0⎠ ⎝1 1 0⎠ ⎝ 0 1 0⎠ 0 0 1 0 0 1 0 0 1 0 0 1 ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞ 1 0 0 x3 0 0 1 0 0 x4 0 0 × ⎝0 1 1⎠ ⎝ 0 x3 0⎠ ⎝0 1 0⎠ ⎝ 0 x4 0⎠ 0 0 1 0 0 1 0 1 1 0 0 1 ⎛ ⎞ x1 x 2 x 3 x 4 + x 2 x 3 x 4 x 3 x 4 + x 4 1 x 3 x 4 + x 4 1⎠ , x 2 x3 x4 =⎝ 0 x4 1

(1.11)

14

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

where the last matrix should be understood as an element of PGL(3), i.e., modulo multiplication by a constant. The parametrisation for the word ¯ 11¯ 22 (the corresponding Thurston diagram is presented in Fig. 1.2.C) is given by (x1 , x2 , x3 , x4 ) → H1 (x1 )E¯1 H1 (x2 )E1 H1 (x3 )E¯2 H1 (x4 )E2 ⎞ ⎛     x1 x3 x4 x1 x3 x1 x 2 x 3 x 4 = ⎝x1 x2 x3 x4 x1 x3 x4 + x3 x4 x1 x3 + x3 ⎠ , x3 + 1 0 x3 x4 where the coordinates x1 , x2 , x3 , x4 are related to the coordinates x1 , x2 , x3 , x4 by mutations in the variables x2 and x4 : (x1 , x2 , x3 , x4 ) = (x1 (1 + x2 )(1 + 1/x4 )−1 , 1/x2 , x3 (1 + 1/x2 )−1 (1 + x4 ), 1/x4 ). The Poisson structures corresponding to the symplectic leaves in PGL(3), can be described by exchange graphs, see Appendix 1.11. For the big cell in PGL(3), corresponding to the word u = 1¯ 12¯ 21¯ 1 ∈ W × W , the corresponding graph is presented in Fig. 1.21, while for the Poisson submanifold in G/AdH, corresponding to u = 1¯ 12¯ 2, the exchange graph can be found in Fig. 1.22.

1.4 Cluster Parameterisation of Double Bruhat Cells: Loop Groups  ). Let us generalise this construction to loop groups, namely, present it for PGL(N For loop groups one gets infinitely many Ad-invariant functions, and they possess finite-dimensional Poisson submanifolds, thus allowing to construct a much wider class of integrable systems. The main difference with the case of simple groups, since the Cartan matrix for affine groups is non-invertible, is that relations (1.3) and (1.4) define now non-isomorphic Lie algebras. The former defines a centrally extended loop group  # (N ) = PGL(N  )  C× , while the latter corresponds to the co-extended one PGL   (N ) = PGL(N  )  C× ; see, e.g., [15]. For our purposes we shall use the group PGL 

 (N ), since it admits cluster parameterisation. The simple roots for this group PGL can be identified with the set Π = Z/N Z, with the Dynkin diagram given by a closed necklace with N vertices.

1.4.1 The Coextended Affine Weyl Group, Wiring, and Thurston Diagrams 

 (N ) also admits a central co-extension The Weyl group W of the group PGL W  = W  Z/N Z, and can be defined by generators, corresponding to the simple

1.4. Cluster Parameterisation of Double Bruhat Cells: Loop Groups

15

roots {si |i ∈ Z/N Z}, and an additional generator Λ, with relations si si+1 si = si+1 si si+1 , Λsi = si+1 Λ, s2i = 1, ΛN = 1.

(1.12)

 ×W  )  = (W  ×W  )(Z/N Z) is generated by {si |i ∈ Π∪ Π}, ¯ Similarly, the group (W corresponding now to positive or negative simple roots, and Λ subject to (1.12), with one additional relation si sj = sj si if i > 0 and j < 0.

(1.13)

  by the ×W  ) generated by si s and Λ is isomorphic to W The subgroup of (W i obvious isomorphism si si → si . It will be called the diagonal subgroup and denoted  . by W +   as well as their decomposition into products of Elements of the group W generators can be visualised by wiring diagrams, similarly to the finite Weyl groups of type AN −1 , see, e.g, [7]. The only difference is that, for the affine case, the diagrams are drawn on cylinders instead of strips. A wiring diagram is a collection of N paths γk on the cylinder R/N Z × [0, 1], connecting bijectively the integral points on one edge of the cylinder to the integral points on the other. The paths are considered up to homotopy and up to the diffeomorphisms of the torus, preserving the boundary of the cylinder pointwise, i.e., the Dehn twists; this rule ensures the relation ΛN = 1. The group product corresponds to gluing the right side of one cylinder to the left of another. The generator si corresponds to the diagram with single crossing of the path connecting the i-th point to the (i + 1)-st, with the path connecting (i + 1)-st to the i-th, and the rest of the paths remaining horizontal. The diagram corresponding to the generator Λ connects i-th point to the (i + 1)-st for any i (see Fig. 1.3). Conversely, given a wiring diagram with only pairwise crossings and with the paths going monotonously from left to right, one can associate it with a word in   in the following way. Cut the cylinder into a rectangle by a line going from W the left to the right side of the cylinder starting on the left at some point between the points (N − 1 (mod N )) and (0 (mod N )), and such that for every face of the diagram it passes through, it enters the fact through the leftmost point and leaves through the rightmost one. Once the cylinder is cut, we can associate to every crossing point the generator si just as we did for finite diagrams on a strip, with an additional generator s0 corresponding to the intersection points occurring on the cut. The power of the coextension generator Λ is determined by the number of the segment where the cut meets the right side of the cylinder.   → Z/N Z can also be described as the The coextension homomorphism W intersection index of the diagram with a generator of the cylinder: orient all paths from left to right and count the number of intersections, taking into account the orientation, with the straight horizontal line from left to right, disjoint from the

16

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

Λ

s1

s2 s1 s2 s0

Figure 1.3: Cylindrical diagrams for the elements of the coextended affine Weyl group.

integral points. Yet another way to compute this coextension is the following. To every path one can associate an integer measuring the difference of the coordinates of the two ends of the path on the universal cover of the cylinder. The sum of such numbers divided by N modulo N is the desired homomorphism. One can easily check that such a sum is always divisible by N , and that it remains unchanged modulo N 2 under the Dehn twist. ×W  ) , it is sufficient to give just two To present elements of the group (W  wiring diagrams, one for each copy of W , with the same number of lines and the same coextension. However, to present a particular decomposition of this group as a product of generators, we will use Thurston diagrams (see Appendix 1.12) which are constructed from two wiring diagrams corresponding to two W -factors, drawn on the same cylinder in a way that the second is slightly shifted down with respect to the first one, with two additional conditions: (1) every vertical line intersects lines of both diagrams alternatively; (2) as a consequence, all intersection points must be triple; (3) the lines corresponding to the first (resp. second diagrams) are oriented from left to right (resp. from right to left), see Fig. 1.4.C. The triple intersection points are therefore of two types, namely when two lines from the first diagram intersect a line from the second and visa versa; they correspond to the generators si with i > 0 or i < 0, respectively. Below, we draw cylindrical diagrams on rectangles assuming that the bottom side of the rectangle is identified with the top one.

1.4.2 Realisations of the Coextended Loop Group 

 (N ): the infinite matrix We shall use two different realisations of the group PGL realisation and the loop realisation. In the infinite matrix realisation it can be identified with the group of infinite matrices {AJI |I, J ∈ Z}, considered up to

1.4. Cluster Parameterisation of Double Bruhat Cells: Loop Groups

s2 s1

s¯0

s0 s2

17

s¯2

A

B

s¯0 s2 s1 s¯2 s0 s2 C Figure 1.4: A: wiring diagram; B: wiring diagram for the second copy of the group W ; C: Thurston diagram.

multiplication by a constant and satisfying the following two conditions: AJI = 0 for |I − J| 0, J × AJ+N I+N = xAI for some x ∈ C .

(1.14)



 (N ) can be identified with the group In the Laurent realisation, the group PGL of expressions A(λ)Tx , where Tx is the operator of multiplicative shift by x, 

∂ Tx = exp log xλ ∂λ

 = xλ∂/∂λ ,

(1.15)

and A(λ) is a Laurent polynomial with values in N × N matrices, considered again up to a multiplicative constant. The multiplication rule of such expressions is therefore A1 (λ)Tx1 · A2 (λ)Tx2 = A1 (λ)A2 (x1 λ)Tx1 x2 .

(1.16)

18

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

The correspondence between the infinite matrix and the loop realisations is given by the isomorphism  AJI → AJ+KN λK Tx , I K∈Z

where in the right-hand side I, J ∈ 1, . . . , N and x is the quasi-periodicity factor from condition (1.14).   (N ) has one more triple of generators in addition to The loop group PGL those of PGL(N ), which we denote E0 , E¯0 , and H0 (x). In loop realisation, the matrices Ei for i = 0, ¯0 coincide with the corresponding matrices for the corresponding finite-dimensional group PGL(N ), and the matrices Hi (x) get multiplied by Tx , i.e., ⎞ ⎞ ⎛ ⎛ x 0 ··· 0 1 0 ··· 0 . . . . ⎜0 ⎜0 . . 0⎟ 0⎟ ⎟ ⎟ ⎜ ⎜ x 1 1 ⎟ ⎜. ⎟ ⎜. Tx , E i = ⎜ . Hi (x) = ⎜ . ⎟ ⎟ 1 1 ⎟ ⎜. ⎟ ⎜. . . ⎝ ⎝ . . 0⎠ . . 0⎠ 0 ··· 0 1 0 ··· 0 1 (1.17) for i > 0, where Ei = E¯itr . Additionally, for i = 0 we have H0 (x) = Tx , and ⎞ ⎛ ⎛ ⎞ 1 0 ··· 0 1 0 · · · λ−1 . . . . .. ⎟ .. ⎟ ⎜ .. . . . ⎜. ... ⎟ , E¯0 = ⎜ . ⎟. (1.18) E0 = ⎜ . . ⎝0 ⎝0 . . 0⎠ .. 0 ⎠ λ ··· 0 1 0 ··· 0 1  ), having in the infinite It is also useful to introduce the element Λ ∈ PGL(N matrix realisation the form ΛJI = δIJ+1 , or in the loop realisation ⎛ ⎞ 0 1 ··· 0 ⎜ .. . . .⎟ .. ⎜ . . .. ⎟ (1.19) Λ = ⎜. ⎟. ⎝0 · · · 0 1⎠ λ ··· 0 0 This matrix satisfies ΛEi Λ−1 = Ei+1 ,

ΛHi (z)Λ−1 = Hi+1 (z),

i ∈ Z/N Z,

(1.20)

 ). i.e., the operator Λ acts as a unit shift along the Dynkin diagram of PGL(N

1.4.3 Integrable Systems on Double Bruhat Cells The Weyl group for the affine Lie group is infinite (for example, the product of all generators has infinite order) and therefore, in contrast to the finite-dimensional

1.4. Cluster Parameterisation of Double Bruhat Cells: Loop Groups

19

→ A

B Figure 1.5: A: exchange graph, and B: Thurston diagram, both for the cell of   (2)/AdH corresponding to the word 0¯ PGL 01¯ 1. case, one can consider arbitrarily long words, corresponding to the Poisson submanifolds of arbitrarily large dimensions. The space of Ad-invariant functions   (N ) is finitely generated, but it is infinitely generated on its subgroup on PGL  ). PGL(N The cluster parameterisation of double Bruhat cells for the loop group  ) goes exactly along the lines we had for simple groups. The only difPGL(N ference on the level of exchange graphs (see Appendix 1.11) is that one should add one extra line; an example of the exchange graph for u = 0¯ 01¯ 1 is shown in ˆ u are on a cylinder with left and right Fig. 1.5.A. The Thurston diagrams for G boundaries, which replaces the infinite strip for the case of simple groups, while ˆ u /AdH are on a torus, which is obtained after gluing the Thurston diagrams for G the left with the right boundary of the cylinder (i.e., in Fig. 1.5.B one should identify top with bottom and left with right). × Let u = si1 , . . . , sin Λk be an arbitrary reduced decomposition of u ∈ (W   W ) . The corresponding double Bruhat cell can be parametrised by n variables x = (x1 , . . . , xn ) via the formula (x1 , . . . , xn ) → A(λ, x) = Hi1 (x1 )Ei1 · · · Hin (xn )Ein Λk .

(1.21)



 )/AdH and not from PGL  (N )/AdH, we In order to get an element from PGL(N need to impose in (1.21) that

xj = 1. (1.22) j

If this condition is satisfied, the matrix A(λ, x) is just a matrix with entries being Laurent polynomials, but defined up to multiplication by a constant α and up to a shift λ → βλ (which is just the result of its conjugation by H0 (β) = Tβ ).

20

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems Consider the function Ssi1 ,...,sin Λk (λ, μ, x) = det(A(λ, x) − μ)),

(1.23)

which we shall call the generating function of the integrable model. This function is ill defined due to the ambiguity described above. However, considering it defined up to a transformation Ssi1 ,...,sin Λk (λ, μ, x)  αSsi1 ,...,sin Λk (βλ, γμ, x), where α, β and γ are arbitrary functions of x1 , . . . , xn , then it becomes already well defined. Computing the determinant in (1.23), this function is expressed as a Laurent polynomial of λ and μ,  Ssi1 ,...,sin Λk (λ, μ, x) = i,j Hij (x)λi μj . (1.24) Denote by Δ the Newton polygon of (1.24), i.e., the convex hull in the plane of the points (i, j) for which Hij (x) = 0. Fix any three corners of Δ, and adjust α, β and γ to make the corresponding coefficients Hij to be equal to one; the resulting partition function will be called normalised. Theorem 1.4.1. The coefficients {Hij (x)} of the expansion (1.24) of the normalised partition functions are well-defined Ad-invariant functions on the Poisson submanˆ si1 ,...,sin Λk /AdH, and therefore they Poisson commute with each other. ifold G The proof that they indeed form a set of integrals of motion of an integrable system, and the discussion of the properties of this system will be done below, after we establish the isomorphism of these systems with the integrable systems of Goncharov and Kenyon.

1.5 Dimers Following [11], in this section we define the dimer partition function for a bipartite graph with weights attached to its edges. We show that partition functions on certain graphs can be computed as minors of certain matrices of small size (related to the number of vertices). We define then the spectral variety, and describe the GK integrable systems, constructed out of certain bipartite graphs on a torus.

1.5.1 Recollection About Dimers Let Γ be a graph, and denote by EΓ and VΓ the sets of its edges and vertices, respectively. A dimer configuration on Γ is a subset D ⊂ EΓ , such that every vertex is contained in exactly one edge of D. Denote by DΓ the set of all dimer configurations on Γ, which we assume to be nonempty. Fix a function A : e → Ae associating a complex number, called weight, to any edge e ∈ EΓ .

1.5. Dimers

21

Consider the sum over all dimer configurations 

Ae , SΓ (A) =

(1.25)

D∈DΓ e∈D

and call it the dimer partition function of the graph Γ. This partition function is a polynomial in the weight variables Ae with unit coefficients. One can slightly generalise this construction for the graphs called open, where one allows the terminal edges, having one special univalent vertex, also called terminal. Graphs without terminal edges are called closed. One can identify two terminal vertices, and then erase the resulting two-valent vertex: this procedure can be used to glue together two graphs. Conversely, one can cut an edge and declare the emerging vertices to be terminal. A dimer configuration on an open graph is a collection of edges containing every internal vertex exactly once, without any condition for the terminal edges. Denote by TΓ the set of terminal edges of the graph. For every subset T ⊂ TΓ denote by DΓ (T ) the set of dimer configurations with terminal edges from T occupied and all other terminal edges free. One can define the partition function with boundary 

SΓ (A, T ) = Ae . (1.26) D∈DΓ (T ) e∈D

All such partition functions can be put together into a single generating partition function. Fix an ordering of the terminal edges and introduce odd variables ξ = {ξt | t ∈ TΓ }. Define then 

SΓ (A, ξ) = SΓ (A, T ) ξt , T ⊂TΓ

t∈D∩T

where the product is taken respecting the fixed order of the terminal edges. This generating function is especially convenient, since it behaves in a nice way under gluing the terminal edges together. Namely, let the graph Γ be obtained from the graph Γ by gluing a terminal edge i with weight Ai to the neighbor edge j, which follows i with respect to the chosen order and carries the weight Aj . The resulting edge e carries the weight Ae = Ai Aj .  Lemma 1.5.1. SΓ (A , ξ  ) = SΓ (A, ξ)eξi ξj dξj dξi . Here, ξ  denotes the collections ξ with ξi and ξj removed, while a is the collection a with the weight Ae = Ai Aj replacing Ai and Aj . The proof of this lemma is obvious, since    ξi ξj SΓ (A, ξ)e dξj dξi = SΓ (A, ξ)dξj dξi + SΓ (A, ξ)ξi ξj dξj dξi . The first term in the right-hand side gives the sum over the dimer configurations with both terminal edges i and j occupied, which are in bijection with dimer configurations for the graph Γ with the edge e occupied. Similarly, the second term gives the sum over configurations where the edge e is empty.

22

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

1.5.2 Matrices and Dimers on a Disk Now, we are going to show that to any matrix one can associate a bipartite graph with terminal edges and with the weights such that its partition function is a generating function for the minors of the matrix. Multiplication of matrices corresponds to gluing graphs. Let M be an n × n matrix, ξ = (ξ1 , . . . , ξn ) and η = (η 1 , . . . , η n ) be two sets of the Grassmann variables. Define the minor generating function by S(M, ξ, η) = exp

 i,j

Mji ξi η j .

(1.27)

Lemma 1.5.2. The function (1.27) satisfies   (i) S(M1 M2 , ξ  , η  ) = S(M1 , ξ , η)S(M2 , ξ, η  ) i exp(−ξi η i )dξ i dηi , (ii) S(M, ξ, η) = 1 + 

n 



k=1 ii 0.

1.11 Appendix C: Exchange Graphs and Decompositions of u ∈ W × W As it is shown in [4], the Poisson brackets induced by the Drinfeld–Jimbo rmatrix (1.2) on the parameters xi are log-constant and half integer, i.e., given by the formula (1.57), where εij is a skew-symmetric matrix taking integer or half integer values. The same matrix plays the role of the exchange matrix (if the group G is simply laced) for the corresponding cluster variety. Now, we shall give the description of the matrix εij for the parametrisations of cells used in Sections 1.3 and 1.4. Instead of writing formulae, we shall give the construction of a graph with oriented edges called exchange graph 5 with vertices in bijection with the indices i 5 This graph is dual to the bipartite graph Γ on the torus Σ used to construct the dimer partition functions.

54

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

Figure 1.20: Left: example of a graph, made of an ordered sequence of chords for the word ¯ 11¯ 23 in the case of PGL(4). Right: the result of gluing it into a single graph.

(or the respective coordinate functions xi ) and εij arrows from i to j if εij  0. If the value of εij is a half integer, we shall indicate the fractional part by a grey arrow, see Fig. 1.20 and Fig. 1.21. Let u = sk1 · · · skl be a word made from the generators of W × W . To construct the graph, start with a staff: a collection of disjoint horizontal lines on the plane, enumerated by positive simple roots i ∈ Π. The points on the staff are ordered, like in music, by their projections on a horizontal axis. For each kj draw a chord – a graph Γ(kj ) with the vertices on the staff and oriented edges. Different chords are put on the staff respecting the order of the generators skj in the word, e.g., the chord Γ(kj+1 ) is located to the right from the chord Γ(kj ). A chord Γ(kj ) has a leftmost vertex Lj (kj ) and a rightmost vertex Rj (kj ) on the j-th line j ∈ Π, together with the vertices {Sl (kj ) | l = j}, located at each other line between Lj (kj ) and Rj (kj ) with arbitrary mutual order. If kj > 0, we draw on the j-th line of the staff between Lj (kj ) and Rj (kj ) a forward oriented edge, and connect Lj (kj ) and Rj (kj ) with Sl (kj ) by −Cjl /2 backward oriented edges (drawn in gray for most of the cases, indicating that often −Cjl /2 is equal to 1/2). For kj < 0, we draw the same chords, but the orientation of all arrows is opposite, see the left pictures in Figs. 1.20 and 1.21. To get the desired graph, just contract all staff lines and remove pairs of arrows connecting the same vertices with opposite orientation, as illustrated at the right in pictures on Figs. 1.20 and 1.21. The graph from Fig. 1.21 exactly corresponds to the example considered above (see formula (1.10) in Section 1.3), of the big cell in P GL(3). To construct the graphs for the cells of the quotient G/AdH, just consider the staff on a cylinder instead of a plane, so that the staff lines turn into circles (e.g., the graph for the symplectic leaf in PGL(3)/AdH, corresponding to Fig. 1.21, can be obtained by simply identifying the leftmost and rightmost points for every staff line). Similarly, the graph for the symplectic leave of the PGL(3) Toda chain (cf. with (1.11)) is constructed exactly in the same way, as shown in Fig. 1.22.

1.12. Appendix D

55

Figure 1.21: Example of a graph, constructed as in Fig. 1.20 from glued chords, for the word 1¯ 12¯ 21¯ 1 (big cell) in the group PGL(3).

Figure 1.22: Example of exchange graph for the word 1¯ 12¯ 2 for PGL(3)/AdH.

1.12 Appendix D: Thurston Diagrams Here, we briefly introduce and discuss a combinatorial object, introduced by D. Thurston in [29], and show its relations to triangulations, bipartite planar graphs, decompositions of permutations into product of generators, coverings of surfaces, and link diagrams. A Thurston diagram on an oriented surface Σ is an isotopy class of collections of oriented curves, such that all intersection points are triple and the orientation of the curves at every intersection point is alternating. An example of a Thurston diagram on a disk is shown in Fig. 1.23.A. The curves may be either closed or go from boundary to boundary. Connected components of the complement to

3

4 1

5

2

6

8 7

A

B

C

Figure 1.23: A: Thurston diagram on a disk; B: construction of the exchange matrix; C: construction of the bipartite graph.

56

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

←→

←→

A

B

←→

←→

C

D

Figure 1.24: A: Grey Thurston move; B: white Thurston move; C: grey Thurston reduction; D: white Thurston reduction. a diagram are called faces. The orientation condition at the intersection points implies that the segments of the curves binding a face are oriented either clockwise (in this case the face is called white), or counter-clockwise (in this case the face is called grey). All neighbouring faces of a white face are grey and visa versa. On the picture we do not indicate the orientations of the curves, since they can be restored from the colouring of the faces. Thurston diagrams admit four kinds of standard modifications, shown in Fig. 1.24.A,B,C,D. The first two, called Thurston moves, do not change the number of faces. The second two reduce the number of faces and are called Thurston reductions. A move can be performed each time there is a face with one or two corners, and diagrams related by Thurston moves are called equivalent. Diagrams not equivalent to those where a Thurston reduction can be applied, are called minimal. Thurston diagrams and cluster varieties. Following Thurston and Henriques [29], we define a cluster variety starting from an equivalence class of Thurston diagrams. The charts of this manifold correspond to particular diagrams in a given class, and Thurston moves correspond to transition functions called mutations in the cluster language. Cluster variables parametrising a chart are assigned to the white faces. For surfaces with boundary the faces neighbouring the boundary correspond to frozen variables. The exchange matrix εij is defined as follows. Draw three arrows connecting white faces around every triple intersection and directed counter-clockwise (as

1.12. Appendix D

57

A

B

Figure 1.25: Exchange matrix from Thurston diagram.

shown in Fig. 1.25.A). For every two segments of the boundary belonging to white faces and separated by a segment belonging to a grey face, connect them by a grey arrow pointing to the right if viewed from inside the surface (as shown in Fig. 1.25.B). Then, the value of εij is equal to the number of arrows from the white face i to the white face j minus the number of arrows in the inverse direction; the grey arrows are counted with the coefficient one half. One can easily check that the exchange matrix does not change under the mutation A, and it changes according to the cluster rule under the mutation B. An example of such collection of arrows, together with the indices enumerating white faces, is shown in Fig. 1.23.B. From this picture one can see, for example, that ε12 = −1 (one solid arrow going from the face 2 to the face 1) and ε34 = −1/2 (one solid arrow from 4 to 3 and one grey arrow in the backward direction). Thurston diagrams and bipartite graphs. For every Thurston diagram one can associate a bipartite graph. Conversely, every bipartite graph with three-valent white vertices corresponds to a Thurston diagram. To construct a bipartite graph out of a Thurston diagram, just place a white vertex inside every grey face and a terminal vertex at every grey segment of the boundary. Then, place a black vertex at every triple intersection point and connect it to the three black vertices in the three faces touching the vertex. Connect also the terminal vertices with the corresponding white ones. An example of such a graph corresponding to a Thurston diagram is shown in Fig. 1.23.C. Observe that this correspondence has the following properties: (i) a grey Thurston move corresponds to two GK moves of type A (Fig. 1.9); (ii) a white Thurston move corresponds to the GK spider move (type C, Fig. 1.9); (iii) faces of the bipartite graph correspond to white faces of the Thurston diagram; (iv) the zig-zag paths of the bipartite graph correspond to curves of the Thurston diagram.

58

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

A

B

Figure 1.26: Thurston diagrams corresponding to a triangle for the cases A: N = 2 or SL(2); B: N = 3 or SL(3).

Thurston diagrams and triangulations. To any triangulation of a surface and any integer N  2 one can associate a Thurston diagram. We will illustrate this construction for N = 2 and N = 3. Let the surface Σ be triangulated with edges belonging either entirely to the boundary, or to the interior. Replace every triangle by 3(N − 1) curves, as shown in Fig. 1.26.A for N = 2, Fig. 1.26.B for N = 3, and analogously for larger values of N . The constructed correspondence has the following properties: (i) a flip of a triangulation corresponds to a sequence of Thurston moves; for N = 2 every flip corresponds to a single Thurston move; (ii) the corresponding cluster variety is the (framed) space of SL(N ) local systems on Σ, see [5]; (iii) every closed curve of the Thurston diagram contracts to a curve surrounding one puncture; (iv) every puncture is surrounded by exactly N − 1 curves; ˜ is an N -fold cover of Σ ramified in the triple intersection (v) the dual surface Σ points. Triality of Thurston diagrams. Thurston diagrams come in triples such that each member of the triple defines the other two. Given a Thurston diagram, glue a disk by its boundary to every closed curve and by a half of its boundary for nonclosed ones. We get a CW-complex with every 1-cell belonging to three 2-cells – white, grey, and the new disks which we will paint in yellow. Removing all disks of one type one gets a smooth 2-dimensional surface with a Thurston diagram on it. It is easy to see that the yellow-grey surface with reversed orientation is the dual to the initial (white-grey) one.

1.12. Appendix D

si si+1 si

59

si+1 si si+1

Figure 1.27: Thurston moves corresponding to the relation si si+1 si = si+1 si si+1 . Thurston diagrams and double permutation group. The correspondence between the Thurston diagrams and the words in double permutation groups has the following properties: (i) the relations si si+1 si = si+1 si si+1 as well as s¯i si+1 s¯i = si+1 s¯i si+1 correspond to a composition of two Thurston moves, one grey and one white (see Fig. 1.27); (ii) the relation si s¯i = s¯i si corresponds to a single white Thurston move; (iii) the relations si si−1 = si−1 si corresponds to a single grey Thurston move; (iv) all other Weyl group relations correspond just to isotopies of the diagrams; (v) product of words corresponds to gluing the strips; (vi) irreducible words correspond to minimal diagrams; (vii) cluster seed corresponding to a minimal Thurston diagram for a given word si1 · · · sin is isomorphic to the cluster seed corresponding to this word. Thurston diagrams and the coextended double affine Weyl group. For the group  ), the coextended double of the Weyl group (W × W ) , presented in SubPGL(N section 1.4.1, corresponds to a Thurston diagram on a cylinder which we shall draw horizontally. Every diagram as in the previous case consists of 2N curves without vertical tangents, going from left to right. The diagrams representing si and s¯i for i = 0 are constructed in the same way as in the finite case. The diagrams for s0 and s¯0 correspond to triple intersections of the curves 2N − 1, 2N, 1 and 2N, 1, 2, respectively. The generator Λ is represented by the diagram without intersection, but connects the point number i on the left of the cylinder to the point number i − 2 modulo N on the right. The list of properties of this correspondence reproduces that for the finite case. Thurston diagrams and link diagrams. For every knot or link diagram on a surface Σ, one can associate Thurston diagrams according to the rules shown on Figure 1.28. This can be done in two different ways, according to the orientation.

60

Chapter 1. Loop Groups, Clusters, Dimers and Integrable Systems

←−

−→

Figure 1.28: Thurston diagrams corresponding to a link diagram.

This correspondence has the following properties: (i) changing an over-crossing to an under-crossing on a diagram corresponds to a Thurston move; (ii) Thurston diagrams of the first type have white faces corresponding to segments between crossings of the link diagram; (iii) Thurston diagrams of the second type have white faces corresponding to faces and to crossings of the knot diagram; (iv) the Reidemeister-III move of the link diagram corresponds to the composition of four grey and four white Thurston moves; this move corresponds to the move si s¯i si+1 si+1 si s¯i → si+1 si+1 si s¯i si+1 si+1 ; (v) the Reidemeister-II move corresponds to one gray and one white Thurston move and two grey and two white Thurston reductions; (vi) the Reidemeister-I move corresponds to one grey and one white Thurston reduction; (vii) the bipartite graph corresponding to a link diagram on a surface of genus g has 2g − 2 more white than black vertices. An essential part of this paragraph is a translation of the constructions done by Cohen, Dasbach, and Russell in [3], into the language of Thurston diagrams.

1.13 Appendix E: Proofs of the Properties of the Minors Generating Functions First, let us prove that convolution of generating functions gives a generating function for the product of matrices. Using matrix notation and denoting a row

1.14. Appendix F

61

(respectively, a column) by ξ and ξ  (respectively, η and η  ), we get     S(M1 , ξ  , η)S(M2 , ξ, η  )e−ξη dξdη = exp ξ  M1 η + ξM2 η  − ξη dξdη    = exp −(ξ − ξ  M1 )(η − M2 η  ) + ξ  M1 M2 η  dξdη      = exp ξ  M1 M2 η  exp (−ξη) dξdη = exp ξ  M1 M2 η  , n where dξdη = j=1 dξj dη j = dξ1 · · · dξn dη n · · · dη 1 . Second, we prove that the expression exp (ξM η) is, indeed, a generating function for the minors of the matrix, exactly as stated in Lemma 1.5.2(ii): ⎞ ⎛ n   1  exp ⎝ Mji ξi η j ⎠ = Mji11 · · · Mjikk ξi1 η j1 · · · ξik η jk k! i,j i ,...,i k=0

1

k

j1 ,...,jk

=

=

n 



Mji11 · · · Mjikk ξi1 η j1 · · · ξik η jk

k=0 i1 0, are the tensor products T ⊗T e (T ∨ )⊗T (n−1) , with the

3.5. Cobordism Hypothesis in Higher Dimension

159

(n − 1)-st power (T ∨ )⊗T (n−1) := T ∨ ⊗T · · · ⊗T T ∨ of the Serre bimodule T ∨ . The description for n ≤ 0 involves the inverse bimodule of T ∨ , but an alternative presentation exploits the dualizability of T over T e to rewrite the tensor product with inverse powers as a Hom-category. Specifically, the categories for n ≤ 0 are HomT e ((T ∨ )⊗T (−n) , T ). Here, Hom stands for the category of linear, right exact functors compatible with the T e -action. Unlike the case of algebras, compatibility with the T -action carries data, not just conditions, as we will see momentarily in examples. The category HomT e (T , T ) for the 0-framed circle plays a distinguished role, as it has a natural tensor structure under composition. It is equivalent to the Drinfeld center DC(T ) of T : this is the category of pairs (x, β) where x ∈ T and β is a half brading with x, a multiplicative isomorphism between the functors x⊗ and ⊗x of left and right tensoring with x. The reader can probably guess what we mean by multiplicativity of β, based on the condition for a braiding (Section 3.3): it is the commutativity of the triangle

β(y)⊗Idz

x⊗y⊗z

y 8⊗ x ⊗ z β(y⊗z)

Idy ⊗β(z)

& / y⊗z⊗x ∼

The Drinfeld center comes with a natural braiding β(x, y) : x ⊗ y −−→ y ⊗ x, from the half braiding carried by the first object x. This braiding is usually not symmetric. Remark 3.5.7. In practice, the braiding tries to be ‘maximally non-symmetric’. For example, if T is the tensor category (Rep(F ), ⊗) of representations of a finite abelian group F under the tensor product over C, its center is the tensor category of representations of F × F ∨ , but with a Heisenberg-like, non-degenerate braiding defined from the natural pairing of F , F ∨ into C× . Specifically, calling n = #F , a distinguished class in H 4 (B 2 F × B 2 F ∨ ; μn ) is induced by the Pontrjagin pairing into μn ⊂ C× . Treating this as a k3 -invariant builds for us a space with π2 = F ×F ∨ and π3 = μn . Hence, we construct a braided tensor category as in Proposition 3.3.14. This tensor category splits into Fourier components according to the characters of μn . The summand corresponding to the standard character is isomorphic to the Drinfeld center of (Rep(F ), ⊗). More generally, any finite group G has a ‘categorified group ring,’ which is the category of vector bundles on G, with the tensor structure coming from convolution. This is the obvious categorical analogue of the functions on the group. The Drinfeld center of this is the tensor category of G-equivariant vector bundles on G, with the convolution structure and with an interesting braiding. This is the categorified analogue of class functions on G. The previous example has G = F ∨ . Remark 3.5.8. In the 3-dualizable case, the square of the Serre functor is the identity because π1 SO(3) = Z/2; so, there will be only two categories going with the

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1 1 two 3-framings on the circle, Seven and Sodd . They are DC(T ) = HomT e (T , T ) and T ⊗T T .

3.5.7 Fusion Categories There is no complete classification of 3-dimensional extended TQFTs based on tensor categories resembling the one seen in 2D, by semisimple (not necessarily commutative) algebras. There are some constraints; for instance, when T is abelian, then full dualizability requires the Drinfeld center (and its twisted form, Hochschild homology) to be semisimple. This follows by considering a special case of an important operation in TQFTs, dimensional reduction. In this case, we reduce the 3D TQFT ZT generated by T along S 1 , to produce a 2-dimensional theory S 1 \ZT , defined by S 1 \ZT (M ) := ZT (M × S 1 ); this theory must be generated by S 1 \ZT (+) = ZT (S 1 ), which implies that DC(T ) is semisimple. Semisimplicity of the center is automatic for rigid categories, but generally a strong constraint. Exercise 3.5.9. Compute the Drinfeld center of the “2 × 2 upper triangular matrix algebra over V ect”, the category V ect⊕3 , with tensor product imitating the multiplication of matrices [ a0 cb ]. However, a beautiful class of extended 3D TQFTs comes form the following result of Douglas, Schommer-Pries, and Snyder. It is based on the notion of fusion category, which has been the subject of much research (Etingof, Ocneanu et al ). The wording similarity with the 2-dimensional story is a bit deceptive, as the structure of a fusion category is substantially more involved than that of a complex semisimple algebra. Definition 3.5.10. A fusion category is a semisimple rigid tensor category with finitely many simple isomorphism classes. Examples 3.5.11. The categorified group ring of a finite group G; see Remark 3.5.7. This generates the 3D gauge theory with finite group F . The tensor structure, and the resulting TQFT, can be twisted by a cocycle τ ∈ H 3 (BF ; C× ) – a higher analogue of a central extension of the group by C× , which in 2D generates a twisted version of gauge theory. In the tensor category, the cocycle appears as an associator. The semisimplified category of representations of a quantum group at a root of unity (the category of finite-dimensional representations modded out by representations of quantum dimension zero). The associated 3-dimensional field theory computes Turaev–Viro theory, whose closed 3-manifold invariants are the square norms of the famous Chern–Simons invariants.24 24 There is at present no construction of the Chern–Simons invariants within a fully extended TQFT for general non-abelian G, although a program by Bartels, Henriques, and Douglas is nearing completion. The generating tensor category analogue is far from discrete, though, and is described in terms of von Neumann algebras. For torus groups, one can land in topological categories, and a C ∗ Hopf-like algebra appears, [11].

3.5. Cobordism Hypothesis in Higher Dimension

161

The conditions are strong enough to imply the good behavior of the tensoring operation. Theorem 3.5.12 (Etingof–Nykshich–Ostrik). For any right and left semisimple module categories M , N over a fusion category F , the tensor product M F N is semisimple. It is also exact in M and N . So, we can define a ‘small’ 3-category of fusion categories, semisimple bimodule categories, functors and natural transformations. In the setting of fusion categories, we also have a duality which allows us to interpret tensor products as functor categories. In particular, this establishes the semisimplicity of the Drinfeld center and its twisted version. Theorem 3.5.13 (Douglas–Schommer-Pries–Snyder, [8]). Every fusion category F is a fully dualizable object of the 3-category T cat. In particular, it defines a framed 1 TQFT ZF : Bordfr 3 → T cat. The category ZF (S0 ) is the Drinfeld center of F , with its natural tensor structure. The Serre functor is the double dual functor in F. Since π1 SO(3) = Z/2 (and no longer Z = π1 SO(2)), the Serre automorphism of any 3-dualizable object must square to 1. As a consequence, the authors get an enlightening proof of a recent result about fusion categories [10], in turn based on earlier work of Radford [24]. Corollary 3.5.14 (Etingof–Nykshich–Ostrik). In any fusion category, there is a canonical isomorphism of the quadruple dual functor with the identity functor.

3.5.8 The Serre Automorphism T ∨ Let us spell out its identification of Serre with the double right dual functor, in the rigid case. This does not rely on semisimplicity of T . Return to the identification T ◦ = T ∨ from Subsection 3.5.5, sending a◦ ∈ T ◦ to the functional ◦ X → HomT (a, X). I claim that we have x ⊗ a◦ ⊗ y = (y ∨ ⊗ a ⊗ ∨ x) because on a functional X → F (X), we have (x ⊗ F )(X) = F (X ⊗ x); following this, we can use the Hom-duality properties of ∨ x, y ∨ , HomT (a, y ⊗ X ⊗ x) = HomT (y ∨ ⊗ a ⊗ ∨ x, X). Now let us use the right dual to identify T with T ◦ : a → (a∨ )◦ . The relation becomes 3 4 ◦ ∨ ◦ . x ⊗ (a∨ )◦ ⊗ y = (y ∨ ⊗ a∨ ⊗ ∨ x) = (∨∨ x ⊗ a ⊗ y) The result is identifying the Serre bimodule T ∨ with T , but with the left tensor action twisted by double left dualization. This is the bimodule implementation of the double left dual functor.

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Remark 3.5.15. If S is not isomorphic to Id, then the TQFT can be defined for Spin surfaces. A rigid category is called pivotal if the double dual is isomorphic to the identity. In the fusion case, a pivotal structure allows us to pass from Spin surfaces to oriented surfaces. Going on to 3-manifolds, in order to pass from the framed to the oriented setting, we need what is called a spherical structure: a pivotal structure in which the trivialization of Serre, the double dual, squares to the canonical trivialization of the quadruple dual. The condition that Serre should be a tensor functor, rather than a bimodule category, is closely related to the existence of internal duals: a (F − F )-bimodule S is implemented by a tensor automorphism precisely when S is equivalent to F as a left and as a right F -module separately. The tensor automorphism it implements is the composition of these separate ‘straightening’ isomorphisms. In the rigid case, both straightening isomorphisms between F and F ∨ are the internal duality functors.

Bibliography [1] L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996). [2] M.F Atiyah and R. Bott, The Yang–Mills equation on Riemann surfaces, Philos. Trans. Roy. Soc. London A 308 (1983). [3] M.F. Atiyah and G.B. Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), 287–330; translation in Ukr. Math. Bull. 1 (2004), 291–334. [4] J. Baez, An Introduction to n-categories, arxiv.org/pdf/q-alg/9705009. [5] J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995). [6] B. Bakalov and A. Kirillov, Lectures on Tensor Categories and Modular Functors, University Lecture Series 21, AMS, Providence, R.I. (2001). [7] K. Costello, Topological quantum field theories and Calabi–Yau categories, Adv. Math. 210 (2007), 165–214. [8] C. Douglas, C. Schommer-Pries, and N. Snyder, Dualizable tensor categories, arXiv:1312.7188. [9] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Math. Surveys and Monographs, AMS (2015). [10] P. Etingof, D. Nikshych, and V. Ostrik, The analog of Radford’s S 4 formula for finite tensor categories, Int. Math. Res. Not. 54 (2004), 2915–2933. [11] D. Freed, M.J. Hopkins, J. Lurie, and C. Teleman, Topological quantum field theories from compact Lie groups, in A Celebration of the Mathematical Legacy of Raoul Bott, CRM Procs. Lecture Notes 60 (2010), 367–403. [12] D. Freed, M.J. Hopkins, and C. Teleman, Twisted equivariant K-theory with complex coefficients, J. Topol. 1 (2008), 16–44. [13] D. Freed, M.J. Hopkins, and C. Teleman, Consistent orientations on moduli spaces, in The Many Facets of Geometry, Oxford (2010), 395–419. 163

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[14] D. Freed, M.J. Hopkins, and C. Teleman, Loop groups and twisted K-theory I, J. Topol. 4 (2011), 737–798. [15] E. Getzler and M. Kapranov, Modular operads, Compositio Math. 110 (1998). [16] J. Greenough, Monoidal 2-structure of bimodule categories, J. Algebra 324 (2010), 1818–1859. [17] S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545–574. [18] M. Kontsevich and Y. Soibelman, Notes on A∞ -algebras, A∞ -categories and non-commutative geometry, in Homological Mirror Symmetry, Lecture Notes in Phys. 757 (2009), 153–219. [19] J. Lurie, On the classification of topological field theories, in Current Developments in Mathematics (2008), 129–280. [20] Y. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, American Mathematical Society Colloquium Publications, 47 (1999). [21] G. Moore, N. Nekrasov, and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000), 97–121. [22] G. Moore and G.B. Segal, D-branes and K-theory in topological field theory, preprint, arXiv: hep-th/0609042. [23] S. Morrison and K. Walker, Blob homology, Geom. Topol. 16 (2012), 1481– 1607. [24] D.E. Radford, The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976), 333–355. [25] N. Reshetikhin and V.G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991). [26] C. Rezk, A Cartesian presentation of weak n-categories, Geom. Topol. 14 (2010). [27] C. Teleman, K-theory and the moduli space of bundles on a surface and deformations of the Verlinde algebra, in Topology, Geometry and Quantum Field Theory, LMS. Lecture Note Ser. 308, 358–378. [28] C. Teleman and C.T. Woodward, The index formula on the moduli of Gbundles, Ann. Math. 170 (2009), 495–527. [29] C. Teleman, The structure of 2D semisimple field theories, Invent. Math. 188 (2012), 525–588. [30] E. Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988). [31] E. Witten, Two-dimensional gauge theories revisited, in On Quantum Gauge Theories in Two Dimensions, Comm. Math. Phys. 141 (1991); arxiv.org/abs/hep-th/9204083.

Chapter 4

Higgs Bundles and Local Systems on Riemann Surfaces Richard A. Wentworth 1 4.1 Preface These notes are based on lectures given at the Third International School on Geometry and Physics at the Centre de Recerca Matem`atica in Barcelona, March 26–30, 2012. The aim of the School’s four lecture series was to give a rapid introduction to Higgs bundles, representation varieties, and mathematical physics. While the scope of these subjects is very broad, that of these notes is far more modest. The main topics covered here are: (i) the Hitchin–Kobayashi–Simpson correspondence for Higgs bundles on Riemann surfaces; (ii) the Corlette–Donaldson theorem relating the moduli spaces of Higgs bundles and semisimple representations of the fundamental group; (iii) a description of the oper moduli space and its relationship to systems of holomorphic differential equations, Higgs bundles, and the Eichler–Shimura isomorphism. These topics have been treated extensively in the literature. I have tried to condense the key ideas into a presentation that requires as little background as 1 R.W. supported in part by NSF grants DMS–1037094 and DMS–1406513. The author also acknowleges support from NSF grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network).

© Springer International Publishing Switzerland 2016 V. Fock et al., Geometry and Quantization of Moduli Spaces, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-319-33578-0_4

165

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possible. With regard to the first item, I give a complete proof of the Hitchin– Simpson theorem (Theorem 4.2.17) that combines techniques that have emerged since Hitchin’s seminal paper [35]. In the case of Riemann surfaces a direct proof for arbitrary rank which avoids introduction of the Donaldson functional can be modeled on Donaldson’s proof of the Narasimhan–Seshadri theorem in [18] (such a proof was suggested in [56]). Moreover, the Yang–Mills–Higgs flow can be used to extract minimizing sequences with desirable properties. A similar idea is used in the Corlette–Donaldson proof of the existence of equivariant harmonic maps (Theorem 4.3.14). Indeed, I have sought in these notes to exhibit the parallel structure of the proofs of these two fundamental results. Continuity of the two flows is the key to the relationship between the equivariant cohomology of the moduli space of semistable Higgs bundles on the one hand, and the moduli space of representations on the other. On first sight the last item in the list above is a rather different topic from the others, but it is nevertheless deeply related in ways that are perhaps still not completely understood. Opers [3] plays an important role in the literature on the Geometric Langlands program [24]. My intention here is to give fairly complete proofs of the basic facts about opers and their relationship to differential equations and Higgs bundles (see also [62]). Due to the limitation of time and space, I have necessarily omitted many important aspects of this subject. Two in particular are worth mentioning. First, I deal only with vector bundles and do not consider principal bundles with more general structure groups. For example, there is no discussion of representations into the various real forms of a complex Lie group. Since some of the other lectures at this introductory school will treat this topic in great detail I hope this omission will not be serious. Second, I deal only with closed Riemann surfaces and do not consider extra “parabolic” structures at marked points. In some sense this ignores an important aspect at the heart of the classical literature on holomorphic differential equations (cf., [7, 58]). Nevertheless, for the purposes of introducing the global structure of moduli spaces, I feel it is better to first treat the case of closed surfaces. While much of the current research in the field is directed towards the two generalizations above, these topics are left for further reading. I have tried to give references to essential results in these notes. Any omissions or incorrect attributions are due solely to my own ignorance of the extremely rich and vast literature, and for these I extend my sincere apologies. Also, there is no claim to originality of the proofs given here. A perusal of Carlos Simpson’s foundational contributions to this subject is highly recommended for anyone wishing to learn about Higgs bundles (see [56, 57, 59, 60, 61]). In addition, the original articles Corlette [11], Donaldson [18, 20] and, of course, Hitchin [35, 36, 38] are indispensable. Finally, I also mention more recent survey articles [9, 10, 27] which treat especially the case of representations to general Lie groups. ´ I am grateful to the organizers, Luis Alvarez-C´ onsul, Peter Gothen, and Ignasi Mundet i Riera, for inviting me to give these lectures, and to the CRM for its hospitality. Additional thanks to Bill Goldman, Fran¸cois Labourie, Andy Sanders, and Graeme Wilkin for discussions related to the topics presented here, and to

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167

Benoˆıt Cadorel for catching several typos. The anonymous referee also made very useful suggestions, for which I owe my gratitude. Notation • X = a compact Riemann surface of genus g ≥ 2; • π = π1 (X, p) = the fundamental group of X; • H = the upper half-plane in C; • O = OX = the sheaf of germs of holomorphic functions on X; • K = KX = the canonical sheaf of X; • E = a complex vector bundle on X; • H = a hermitian metric on E; • ∇ = a connection on E; • A (or dA ) = a unitary connection on (E, H); • CE = the space of connections on a rank n bundle E; • AE = the space of unitary connections on E; • BE = the space of Higgs bundles; • Bss E = the space of semistable Higgs bundles; • GE (resp., GC E ) = the unitary (resp., complex) gauge group; • ∂¯E = a Dolbeault operator on E, which is equivalent to a holomorphic structure; • (∂¯E , H) = the Chern connection; • E = sheaf of germs of holomorphic sections of a holomorphic bundle (E, ∂¯E ); • gE = the bundle of skew-hermitian endomorphisms of E; • End E = gC E the endomorphism bundle of E; • V = a local system on X; • Vρ = the local system associated to a representation ρ : π → GLn (C); • R = the locally constant sheaf modeled on a ring R; • Lpk = the Sobolev space of functions/sections with k derivatives in Lp ; • C k,α = the space of functions/sections with k derivatives being H¨ older continuous with exponent α.

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4.2 The Dolbeault Moduli Space 4.2.1 Higgs Bundles Holomorphic Bundles and Stability Throughout these notes, X will denote a closed Riemann surface of genus g ≥ 2 and E → X a complex vector bundle. We begin with a discussion of the basic differential geometry of complex vector bundles. Good references for this material are Kobayashi’s book [44] and Griffiths–Harris [26]. A holomorphic structure on ¯ E is equivalent to a choice of ∂-operator, i.e., a C-linear map ∂¯E : Ω0 (X, E) −→ Ω0,1 (X, E) ¯ ⊗s+f ∂¯E s, for a function f and a section s satisfying the Leibniz rule: ∂¯E (f s) = ∂f of E. Indeed, if {si } is a local holomorphic frame of a holomorphic bundle, then the ¯ Leibniz rule uniquely determines the ∂-operator on the underlying complex vector bundle. Conversely, since there is no integrability condition on Riemann surfaces, ¯ given a ∂-operator as defined above one can always find local holomorphic frames (cf., [2, §5]). When we want to specify the holomorphic structure, we write (E, ∂¯E ). We also introduce the notation E for a sheaf of germs of holomorphic sections of (E, ∂¯E ). We will sometimes confuse the terminology and call E a holomorphic bundle. If S ⊂ E is a holomorphic subbundle with quotient Q, then a smooth splitting ¯ E = S ⊕ Q allows us to represent the ∂-operators as   ∂¯S β ∂¯E = , (4.1) 0 ∂¯Q where β ∈ Ω0,1 (X, Hom(Q, S)) is called the second fundamental form. A hermitian metric H on E gives an orthogonal splitting. In this case the subbundle S is determined by its orthogonal projection operator π, which is an endomorphism of E satisfying: (i) π 2 = π; (ii) π ∗ = π; and (iii) tr π is constant. The statement that S ⊂ E be holomorphic is equivalent to the further condition (iv) (I − π)∂¯E π = 0. Notice that (i) and (iv) imply (iii), and that β = −∂¯E π. Hence, there is a oneto-one correspondence between holomorphic subbundles of E and endomorphisms π of the hermitian bundle E satisfying conditions (i), (ii), and (iv). I should point out that the generalization of this description of holomorphic subsheaves to higher dimensions is a key idea of Uhlenbeck–Yau [65]. A connection ∇ on E is a C-linear map ∇ : Ω0 (X, E) −→ Ω1 (X, E), satisfying the Leibniz rule: ∇(f s) = df ⊗ s + f ∇s, for a function f and a section s. Given a hermitian metric H, we call a connection unitary (and we will always then denote it by A or dA ) if it preserves H, i.e., d s1 , s2 H = dA s1 , s2 H + s1 , dA s2 H .

(4.2)

4.2. The Dolbeault Moduli Space

169

The curvature of a connection ∇ is F∇ = ∇2 (perhaps more precise notation: ∇ ∧ ∇). If gE denotes the bundle of skew-hermitian endomorphisms of E and gC E its complexification, then FA ∈ Ω2 (X, gE ) for a unitary connection, and F∇ ∈ Ω2 (X, gC E ) in general. Remark 4.2.1. We will mostly be dealing with connections on bundles that induce a fixed connection on the determinant bundle. These will correspond, for example, to representations into SLn as opposed to GLn . In this case, the bundles gE and gC E should be taken to consist of traceless endomorphisms. ¯ Finally, note that a connection always induces a ∂-operator by taking its ¯ (0, 1) part. Conversely, a ∂-operator gives a unique unitary connection, called the Chern connection, which we will sometimes denote by dA = (∂¯E , H). The complex structure on X splits Ω1 (X) into (1, 0) and (0, 1) parts and, hence, also splits the connections. We denote these by, for example, dA and dA , respectively. So for dA = (∂¯E , H), dA = ∂¯E , and dA is determined by ∂ s1 , s2 H = dA s1 , s2 H , for any pair of holomorphic sections s1 , s2 . Henceforth, I will mostly omit H from the notation if there is no chance of confusion. Example 4.2.2. Let L be a holomorphic line bundle with hermitian metric H. For ¯ log Hs , and the a local holomorphic frame s, write Hs = |s|2 . Then F(∂¯L ,H) = ∂∂ right-hand side is independent from the choice of frame. The transition functions of a collection of local trivializations of a holomorphic line bundle on the open sets of a covering of X give a 1-cocycle with values in the sheaf O∗ of germs of nowhere vanishing holomorphic functions. The set of isomorphism classes of line bundles is then H 1 (X, O∗ ). Recall that on a compact Riemann surface, every holomorphic line bundle has a meromorphic section. This gives an equivalence between the categories of holomorphic line bundles under tensor products and linear equivalence classes of divisors D = x∈X mx x with their additive structure (here, mx ∈ Z is zero for all but finitely many x ∈ X). We shall denote by O(D) theline bundle thus associated to D. Furthermore, a divisor has a degree, deg D = x∈X mx . We define this to be the degree of O(D). Alternatively, from the exponential sequence f →e2πif

0 −→ Z −→ O −−−−−−−−→ O∗ −→ 0, we have the long exact sequence in cohomology: c1

0 −→ H 1 (X, Z) −→ H 1 (X, O) −→ H 1 (X, O∗ ) −−−−→ H 2 (X, Z) −→ 0. The fundamental class of X identifies H 2 (X, Z) ∼ = Z, and it is a standard exercise to show that, under this identification, deg (D) = c1 (O(D)). For a holomorphic vector bundle E, we declare the degree deg E := deg det E. Notice that the degree is topological, i.e., it does not depend on the holomorphic structure, just on the

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underlying complex bundle E. By the Chern–Weil theory, for any hermitian metric H on E we have √  √  −1 −1 c1 (E) = tr F(∂¯E ,H) = F(∂¯det E ,det H) . (4.3) 2π 2π Complex vector bundles on Riemann surfaces are classified topologically by their rank and degree. We will also make use of the slope (or normalized degree) of a bundle, which is defined by the ratio μ(E) = deg E/rank E. If a line bundle L = O(D) has a nonzero holomorphic section then, since D is linearly equivalent to an effective divisor (i.e., one with mx ≥ 0 for all x), deg L ≥ 0. It follows that, if E is a holomorphic vector bundle with a subsheaf S ⊂ E and rank S = rank E, then deg S ≤ deg E. Indeed, the assumption implies that det E ⊗ (det S)∗ has a nonzero holomorphic section. We will use this fact later on. Notice that in the case above, Q = E/S is a torsion sheaf. In general, for any subsheaf S ⊂ E of a holomorphic vector bundle, S is contained in a uniquely defined holomorphic subbundle S of E called the saturation of S. It is obtained by taking the kernel of the induced map E → Q/Tor(Q) → 0. From this discussion we conclude that deg S is no greater than the degree deg S of its saturation. Let ω be the K¨ahler form associated to a choice of conformal metric on X. This will be fixed throughout and, for convenience, we normalize so that  ω = 2π. X

The contraction: Λ : Ω2 (X) → Ω0 (X), is defined by setting Λ(f ω) = f for any function f . For a holomorphic subbundle S of a hermitian holomorphic bundle E with projection operator π we have the following useful formula, which follows easily from direct calculation using (4.3):   √ 1 1 tr (π −1ΛF(∂¯E ,H) ) ω − |β|2 ω. (4.4) deg S = 2π X 2π X Definition 4.2.3. We say that E is stable (resp., semistable) if for all holomorphic subbundles S ⊂ E, 0 < rank S < rank E, we have μ(S) < μ(E) (resp., μ(S) ≤ μ(E)). We call E polystable if it is a direct sum of stable bundles of the same slope. Remark 4.2.4. Line bundles are trivially stable. If E is (semi)stable and L is a line bundle, then E ⊗ L is also (semi)stable. Before giving an example, recall the notion of an extension 0 −→ S −→ E −→ Q −→ 0.

(4.5)

The extension class is the image of the identity endomorphism under the coboundary map of the long exact sequence associated to (4.5), H 0 (X, Q ⊗ Q∗ ) −→ H 1 (X, S ⊗ Q∗ ).

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Notice that the isomorphism class of the bundle E is unchanged under scaling, so the extension class (if not zero) should be regarded as an element of the projective space P(H 1 (X, S ⊗ Q∗ )). It is then an exercise to see that, in terms of the second fundamental form β, the extension class coincides (projectively) with the corre∗ sponding Dolbeault cohomology class [β] ∈ H∂0,1 ¯ (X, S ⊗ Q ). We say that (4.5) is split if the extension class is zero. Clearly, this occurs if and only if there is an injection Q → E lifting the projection. Example 4.2.5. Suppose g ≥ 1. Consider extensions of the type 0 −→ O −→ E −→ O(p) −→ 0. These are parametrized by H 1 (X, O(−p)) ∼ = H 0 (X, K(p))∗ ∼ = H 0 (X, K)∗ , which has dimension g. Any non-split extension of this type is stable. Indeed, if L → E is a destabilizing line subbundle, then deg L ≥ 1. The induced map L → O(p) cannot be zero, since then by the inclusion L → E it would lift to a nonzero map L → O, which is impossible. Hence, L → O(p) must be an isomorphism. Such an L would therefore split the extension. A√connection is flat if its curvature vanishes. We say that ∇ is projectively flat if −1ΛF∇ = μ, where μ is a constant (multiple of the identity). Note that by our normalization of the area, μ = μ(E). In Section 4.4, we will prove Weil’s criterion for when a holomorphic bundle E admits a flat connection (i.e., ∇ = ∂¯E , F∇ = 0). Demanding that the connection be unitary imposes stronger conditions. This is the famous result of Narasimhan–Seshadri. Theorem 4.2.6 (Narasimhan–Seshadri, [49]). A holomorphic bundle E → X admits a projectively flat unitary connection if and only if E is polystable. In Subsection 4.2.3 we will prove Theorem 4.2.6 as a special case of the more general result on Higgs bundles (see Theorem 4.2.17). Higgs Fields A Higgs bundle is a pair (E, Φ) where E is a holomorphic bundle and Φ is a holomorphic section of K ⊗ End E. We will sometimes regard Φ as a section of ¯ Ω1,0 (X, gC E ) satisfying ∂E Φ = 0. Definition 4.2.7. We say that a pair (E, Φ) is stable (resp., semistable) if for all Φ-invariant holomorphic subbundles S ⊂ E, 0 < rank S < rank E, we have μ(S) < μ(E) (resp., μ(S) ≤ μ(E)). It is polystable if it is a direct sum of Higgs bundles of the same slope. The following is a simple but useful consequence of the definition and the additive properties of the slope on exact sequences.

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Lemma 4.2.8. Let f : (E1 , Φ1 ) → (E2 , Φ2 ) be a holomorphic homomorphism of Higgs bundles, Φ2 f = f Φ1 . Suppose (Ei , Φi ) is semistable, i = 1, 2, and μ(E1 ) > μ(E2 ). Then f ≡ 0. If μ(E1 ) = μ(E2 ) and one of the two is stable, then either f ≡ 0, or f is an isomorphism. Proof. Consider the first statement. Then if f ≡ 0, the assumption Φ2 f = f Φ1 implies that the image of f is Φ2 -invariant so, by the condition on slopes, f must have a kernel. But then ker f is Φ1 -invariant. So μ(ker f ) ≤ μ(E1 ) ≤ μ(coker f ) ≤  μ(E2 ), a contradiction. The second statement follows similarly. A Higgs subbundle of (E, Φ) is, by definition, a Φ-invariant holomorphic subbundle S ⊂ E. The restriction ΦS of Φ to S then makes (S, ΦS ) a Higgs bundle, where now the inclusion S → E gives a map of Higgs bundles. Similarly, Φ induces a Higgs bundle structure on the quotient Q = E/S. Given an arbitrary Higgs bundle, the Harder–Narasimhan filtration of (E, Φ) is a filtration by Higgs subbundles 0 = (E0 , Φ0 ) ⊂ (E1 , Φ1 ) ⊂ · · · ⊂ (E , Φ ) = (E, Φ), such that the quotients (Qi , ΦQi ) = (Ei , Φi )/(Ei−1 , Φi−1 ) are semistable (cf., [31]). The filtration is also required to satisfy μ(Qi ) > μ(Qi+1 ), and one can show that the 0 associated graded object GrHN (E, Φ) = i=1 (Qi , ΦQi ) is uniquely determined by the isomorphism class of (E, Φ). The collection of slopes μi = μ(Qi ) is an important invariant of the isomorphism class of the Higgs bundle. Remark 4.2.9. By construction, μi is the maximal slope of a Higgs subbundle of E/Ei−1 with its induced Higgs field. We can also interpret μi as the minimal slope of a Higgs quotient of (Ei , Φi ). Indeed, (E1 , Φ1 ) is semistable, so this is trivially true if i = 1. Suppose Ei → Q → 0 is a Higgs quotient for i ≥ 2 and μ(Q) ≤ μi . If Q is the minimal such quotient, then it is semistable with respect to the induced Higgs field. It follows from Lemma 4.2.8 that the induced map E1 → Q must vanish. Hence, the quotient passes to E/E1 → Q → 0. Now, by the same argument, E2 /E1 → Q vanishes if i ≥ 3. Continuing in this way, we obtain a quotient Qi → Q → 0. And now, since (Qi , ΦQi ) is semistable and the quotient is nonzero, applying Lemma 4.2.8 once again, we conclude that μi ≤ μ(Q). Consider the n-tuple of numbers μ  (E, Φ) = (μ1 , . . . , μn ) obtained from the Harder–Narasimhan filtration by repeating each of the μi ’s according to the ranks  (E, Φ), called the Harder–Narasimhan type of of the Qi ’s. We then get a vector μ (E, Φ). There is a natural partial ordering on vectors of this type that is key to the stratification wedesire. Fora pair μ  , λ of n-tuples satisfying μ1 ≥ · · · ≥ μn , n n λ1 ≥ · · · ≥ λn , and i=1 μi = i=1 λi , we define   λ ≤ μ λj ≤ μj for all k = 1, . . . , n.  ⇐⇒ j≤k

j≤k

The importance of this ordering is that it defines a stratification of the space of Higgs bundles. In particular, the Harder–Narasimhan type is upper semicontin-

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uous. This is the direct analog of the Atiyah–Bott stratification for holomorphic bundles [2, §7]. There is a similar filtration of a semistable Higgs bundle (E, Φ), where the successive quotients are stable, all with slope = μ(E). This is called the Seshadri filtration, see [54], and its associated graded object GrS (E, Φ) is therefore polystable. When Φ ≡ 0, we recover the usual Harder–Narasimhan and Seshadri filtrations of holomorphic bundles E. We will denote these by GrHN (E) and GrS (E). Example 4.2.10. Consider an extension (4.5) where rank S = rank Q = 1 and deg S > deg Q. Then the Harder–Narasimhan filtration of E is given by 0 ⊂ S ⊂ E.

4.2.2 The Moduli Space Gauge Transformations Let AE denote the space of unitary connections on a rank n hermitian vector bundle E. If gE denotes the associated bundle of skew-hermitian endomorphisms of E, then one observes from the Leibniz rule that AE is an infinite-dimensional affine space modeled on Ω1 (X, gE ). By the construction of the Chern connection discussed in Subsection 4.2.1, we also see that AE can be identified with the space of holomorphic structures on E. We will most often be interested in the case of fixed determinant, i.e., where the induced holomorphic structure on det E is fixed. The gauge group is defined by GE = {g ∈ Ω0 (X, End E) : gg ∗ = I}. In the fixed determinant case we also impose the condition that det g = 1 (see Remark 4.2.1). The gauge group acts on AE by pulling back connections: dg(A) = g ◦ dA ◦ g −1 . On the other hand, because of the identification with holomorphic structures, we see that the complexification GC E , the complex gauge group, also acts on AE . Explicitly, if ∂¯E = dA , then g(A) is the Chern connection of g ◦ ∂¯E ◦ g −1 . The space of Higgs bundles is  BE = {(A, Φ) ∈ AE × Ω0 (X, K ⊗ gC E ) : dA Φ = 0}.

Let Bss E ⊂ BE denote the subset of semistable Higgs bundles. Definition 4.2.11. The moduli space of rank n semistable Higgs bundles (with ## C (n) fixed determinant) on X is MD = Bss E GE , where the double slash means that the orbits of (E, Φ) and GrS (E, Φ) are identified. (n)

We have not been careful about topologies. In fact, MD can be given the structure of a (possibly nonreduced) complex analytic space using the Kuranishi map (cf., [44]). An algebraic construction using geometric invariant theory is given in [60]. A second comment is that GC E /GE may be identified with the space of hermitian metrics on E. This leads to an important interpretation when studying the behavior of functionals along GC E orbits in AE /GE : we may either think of varying the complex structure g(∂¯E ) with a fixed hermitian metric, or we may keep ∂¯E fixed and vary the metric H by s1 , s2 g(H) = gs1 , gs2 H .

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Deformations of Higgs Bundles Let D = dA + Φ, D = dA + Φ∗ . The metric ω on X and the hermitian metric on E define L2 -inner products on forms with values in E and End E. We have the K¨ ahler identities √ (D )∗ = − −1 [Λ, D ], (4.6) √ (D )∗ = −1 [Λ, D ] (see [26, p. 111] for the case Φ = 0; the case Φ = 0 follows by direct computation). The infinitesimal structure of the moduli space is governed by a deformation complex C(A, Φ), which is obtained by differentiating the condition dA Φ = 0 and the action of the gauge group: D 

D 

1,0 0,1 1,1 (X, gC (X, gC (X, gC C(A, Φ) : 0 −→ Ω0 (X, gC E ) −−→ Ω E )⊕Ω E ) −−→ Ω E ) → 0. (4.7) Note that the holomorphicity condition on Φ guarantees that (D )2 = 0. Serre’s duality gives an isomorphism H 0 (C(A, Φ))  H 2 (C(A, Φ)). We call a Higgs bundle simple if H 0 (C(A, Φ))  C (or {0} in the fixed determinant case). Remark 4.2.12. A stable Higgs bundle is necessarily simple. Indeed, if ϕ ∈ ker D , then ϕ is a holomorphic endomorphism of E commuting with Φ. In particular, det ϕ is a holomorphic function and is therefore constant. Also, ker ϕ is Φ-invariant. If ϕ is nonzero, but not an isomorphism, then

0 −→ ker ϕ −→ E −→ E/ ker ϕ −→ 0. Since E/ ker ϕ is also a subsheaf of E, stability implies both μ(ker ϕ) and μ(E/ ker ϕ) are less than μ(E), which is a contradiction. Hence, ϕ is either zero or an isomorphism. But applying the same argument to ϕ − λ for any scalar λ, we conclude that ϕ is a multiple of the identity. (n)

Proposition 4.2.13. At a simple Higgs bundle [A, Φ], MD is smooth of complex dimension (n2 − 1)(2g − 2), and the tangent space may be identified with √  (4.8) H 1 (C(A, Φ))  (ϕ, β) : dA ϕ = −[Φ, β], (dA )∗ β = −1Λ[Φ∗ , ϕ] . Example 4.2.14 (cf., [35, 38]). We now give important examples of stable Higgs bundles; namely, the Fuchsian ones. First, for rank 2. Fix a choice of square root K1/2 of the canonical bundle, and let E = K1/2 ⊕ K−1/2 . Then the part of the endomorphism bundle that sends K1/2 → K−1/2 is isomorphic to K−1 . Tensoring by K, it becomes trivial. Hence,   0 0 Φ= 1 0 makes sense as a Higgs field, and it is clearly holomorphic. While E is unstable as a holomorphic vector bundle, the Higgs bundle (E, Φ) is stable since the only

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175

Φ-invariant sub-line bundle is K−1/2 , which has negative degree. Let us remark in passing, that if we consider a different holomorphic structure V on E given by the ¯ ∂-operator   ∂¯K1/2 ω ∂¯E + Φ∗ = , 0 ∂¯K−1/2 then V is the unique (up to isomorphism) non-split extension 0 −→ K1/2 −→ V −→ K−1/2 −→ 0. (2)

We now compute the tangent space MD at [(E, Φ)]. Write     b b1 ϕ ϕ1 β= , ϕ= , b2 −b ϕ2 −ϕ and compute



[Φ, β] =

−b1 2b

0 b1



√

,

−1Λ[Φ∗ , ϕ] =



ϕ2 0

 −2ϕ . −ϕ2

Then the conditions (4.8) that (β, ϕ) define a tangent vector are     b1 ϕ2 −2ϕ 0 ∗ β= , ∂¯E . ∂¯E ϕ = −2b −b1 0 −ϕ2 ∗ 0 2 ∗ In particular, ϕ1 ∈ H 0 (X, K2 ) and b2 ∈ H∂0,1 ¯ (X, K )  H (X, K ) . I claim ¯ that the other entries vanish. Indeed, the equations for ϕ and b1 are ∂ϕ = b1 , and ∂¯∗ b1 = −2ϕ. But this implies (∂¯∗ ∂¯ + 2)ϕ = 0. Hence ϕ, and therefore also b1 , must vanish. The same argument works for ϕ2 and b. We therefore have an isomorphism

T[EF ,ΦF ] MD  H 0 (X, K2 ) ⊕ (H 0 (X, K2 ))∗ . (2)

For n ≥ 2, there is a similar argument. Here we take EF = K(n−1)/2 ⊕ K(n−3)/2 ⊕ · · · ⊕ K−(n−1)/2 and

⎛

0

⎜ ⎜1 ⎜ ⎜ ΦF = ⎜0 ⎜ ⎜. ⎝ .. 0

0

0

···

0

0

···

1

0 .. . 0

··· .. . 1

···

⎞ 0 .. ⎟ .⎟ ⎟ .. ⎟ . .⎟ ⎟ .. ⎟ .⎠ 0

Notice that, with respect to this splitting, the (i, j)-th entry of ϕ is a section of Kj−i+1 , and the (i, j)-th entry of β is in Ω0,1 (X, K j−i ). We obtain the following equations on the entries of a tangent vector (β, ϕ): ∂¯E ϕi,j = βi−1,j − βi,j+1 , ∗ ∂¯E βi,j = ϕi,j−1 − ϕi+1,j ,

(4.9)

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where it is understood that terms with indices ≤ 0 or ≥ n + 1 are set to zero. Upon further differentiation as in the n = 2 case, we find (L − δi,1 − δj,n )ϕi,j = ϕi+1,j+1 + ϕi−1,j−1 ,  − δi,n − δj,1 )βi,j = βi+1,j+1 + βi−1,j−1 , (L

(4.10)

∗ ¯  = ∂¯E ∂¯∗ + 2. I claim that ϕi,j = 0 (resp., βi,j = 0) where L = ∂¯E ∂E + 2 and L E for i ≥ j (resp., i ≤ j). For example, by (4.10), Lϕn,1 = 0, and since L is a positive operator, ϕn,1 vanishes. More generally, fix 0 ≤ p ≤ n − 2. Then for 0 ≤  ≤ n − p − 1, there are polynomials P such that

ϕp++1,+1 = P (L)ϕp+1,1 .

(4.11)

Indeed, let P0 (L) = 1, P1 (L) = L if p = 0, and P1 (L) = L − 1 if p = 0. Suppose Pk (L) has been defined for 0 ≤ k ≤ , where 0 <  < n − p − 1. Use (4.10) and (4.11) to find Lϕp++1,+1 = ϕp++2,+2 + ϕp+, , LP (L)ϕp+1,1 = ϕp++2,+2 + P−1 (L)ϕp+1,1 . Hence, we let P+1 (L) = LP (L) − P−1 (L). Since L ≥ 2, we see from the recursive definition that P+1 (L) ≥ P (L) and hence, for all  ≥ 1, P (L) ≥ P1 (L) ≥ 1, and ≥ 2 if p = 0. Taking  = n − p − 1 in (4.11), we have ϕn,n−p = Pn−p−1 (L)ϕp+1,1 .

(4.12)

On the other hand, a similar argument implies ϕp+1,1 = Pn−p−1 (L)ϕn,n−p , from which we obtain 2 0 = (Pn−p−1 (L) − 1)ϕp+1,1 = (Pn−p−1 (L) + 1)(Pn−p−1 (L) − 1)ϕp+1,1 .

Hence, ϕp+1,1 is in the kernel of Pn−p−1 (L) − 1. But then, by the remark above, for p ≥ 1, ϕp+1,1 must vanish. Since p ≥ 1 is arbitrary, this implies by (4.11) that ϕi,j = 0 for all i > j. In the case p = 0, notice that for all  ≥ 1, P (L) is a ∗ ¯ polynomial of positive degree in ∂¯E ∂E with nonnegative coefficients and constant term equal to 1. Indeed, by the definition ∗ ¯ P+1 (L) − P (L) = (∂¯E ∂E )P (L) + P (L) − P−1 (L),

and so, by induction, P+1 (L) − P (L) has nonnegative coefficients and zero constant term. In this case, (Pn−1 (L) − 1)ϕ1,1 = 0 implies that ϕ1,1 is holomorphic. Using (4.11) again, ϕ+1,+1 = P (L)ϕ1,1 = (P (L) − 1)ϕ1,1 + ϕ1,1 = ϕ1,1 , for all  = 0, . . . , n − 1. But since (ϕi,j ) is traceless, it follows that, in fact, ϕi,i = 0 for all i. The proof for βi,j is similar.

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177

Going back to (4.9), we see that ϕi,j (resp., βj,i ) is holomorphic (resp., harmonic) if i < j. Moreover, for p ≥ 1, (4.10) becomes (2 − δi,1 − δi,n−p )ϕi,i+p = ϕi+1,i+1+p + ϕi−1,i−1+p .

(4.13)

If i = 1 this implies ϕ1,p+1 = ϕ2,p+2 . Suppose by induction that ϕk,k+p = ϕ1,p+1 for all k ≤ i. Then if i + p = n, (4.13) implies 2ϕi,i+p = ϕi+1,i+1+p + ϕi−1,i−1+p

=⇒

ϕ1,p+1 = ϕi+1,i+1+p .

If i + p = n, we immediately get ϕi,n = ϕi−1,n−1 = ϕ1,p+1 . Hence, all differentials ϕi,j , j−i = p, are equal. The same argument applies to βi,j . From this, we conclude that the map (ϕ, β) → (ϕ1,2 , . . . , ϕ1,n , β2,1 , . . . , βn,1 ) gives an isomorphism (n)

T[EF ,ΦF ] MD 

n 1

H 0 (X, Kj ) ⊕ (H 0 (X, Kj ))∗ .

(4.14)

j=2

¯ The rank n holomorphic vector bundle V whose ∂-operator is ∂¯E + Φ∗F is unstable and has a Harder–Narasimhan filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V, Vj+1 /Vj = K−j+(n−1)/2 , such that 0 −→ Vj −→ Vj+1 −→ K−j+(n−1)/2 −→ 0. is the (unique) non-split extension. This is an example of an oper. Opers will be discussed in Subsection 4.4.3. The Hitchin Map Given a Higgs bundle (E, Φ), the coefficient of λn−i in the expansion of det(λ + Φ) is a holomorphic section of Ki , i = 1, . . . , n. In the case of fixed determinant that we will mostly be considering, tr Φ = 0, so the sections start with i = 2. These pluricanonical sections are clearly invariant under the action (by conjugation) of GC E , so we have a well-defined map, called the Hitchin map, (n)

h : MD −→

n 1

H 0 (X, Ki ).

(4.15)

i=2

The structure of this map and its fibers turns out be extremely rich (cf., [36]). In these notes, however, I will only discuss the following important fact which will be proven in the next section using Uhlenbeck compactness (for algebraic proofs, see [50, 59]). Theorem 4.2.15. The Hitchin map is proper.

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4.2.3 The Hitchin–Kobayashi Correspondence Stability and Critical Metrics Hitchin’s equations for Higgs bundles on a trivial bundle are FA + [Φ, Φ∗ ] = 0.

(4.16)

Here, Φ is regarded as an endomorphism valued (1, 0)-form. It will also be convenient to consider the case of bundles of nonzero degree. In this case, the equations become √ f(A,Φ) := −1Λ(FA + [Φ, Φ∗ ]) = μ. (4.17) Here, we recall the normalization vol(X) = 2π and then, on the right hand side, the scalar multiple of the identity endomorphism necessarily satisfies μ = μ(E). There are two ways of thinking of (4.17): for a Higgs bundle (E, Φ) a choice of hermitian metric gives a Chern connection A = (∂¯E , H). Hence, we may either view (4.17) as an equation for a hermitian metric H, or alternatively (and equivalently) we may fix H and consider f(A,Φ) for all (A, Φ) in a complex gauge orbit. We will often go back and forth between these equivalent points of view. The solutions to the equations (4.17) may be regarded as the absolute minimum for the Yang–Mills–Higgs functional on the space of holomorphic pairs, defined as  2 |FA + [Φ, Φ∗ ]| ω. (4.18) YMH(A, Φ) = X

The Euler–Lagrange equations for YMH are dA f(A,Φ) = 0 ,

[Φ, f(A,Φ) ] = 0.

(4.19)

We call a metric critical if (4.19) is satisfied. In this case, it is easy to see the bundle (E, Φ) splits holomorphically and isometrically as a direct sum of Higgs bundles that are solutions to (4.17) with possibly different slopes. Proposition 4.2.16. If a Higgs bundle (E, Φ) admits a metric satisfying (4.17), then (E, Φ) is polystable. Proof. Let S ⊂ E be a proper Φ-invariant subbundle. Let π denote the orthogonal projection to S and β = −∂¯E π the second fundamental form. Then, since S is invariant, (I − π)Φπ = 0, or Φπ = πΦπ, πΦ∗ = πΦ∗ π. In particular, this implies tr (π[Φ, Φ∗ ]) = tr (πΦΦ∗ ) + tr (πΦ∗ Φ) = tr (πΦΦ∗ ) − tr (ΦπΦ∗ ) = tr (πΦΦ∗ π) − tr (ΦπΦ∗ π) = tr (πΦΦ∗ π) − tr (πΦπΦ∗ π) = tr (πΦ(I − π)Φ∗ π) = tr (πΦ(I − π)(I − π)Φ∗ π) = tr {(πΦ(I − π))(πΦ(I − π))∗ } ,

4.2. The Dolbeault Moduli Space and

179

√ tr (π −1Λ[Φ, Φ∗ ]) = |πΦ(I − π)|2 .

(4.20)

Plugging (4.17) into (4.4), and using (4.20), we have deg S = rank (S)μ(E) −

1 (#πΦ(I − π)#2 + #β#2 ). 2π

This proves μ(S) ≤ μ(E). Moreover, equality holds if and only if the two terms on the right-hand side above vanish; i.e., the holomorphic structure and Higgs field split.  The main result we prove in this section is the converse to Proposition 4.2.16. Theorem 4.2.17 (Hitchin [35], Simpson [57]). If (E, Φ) is polystable, then it admits a metric satisfying (4.17). Remark 4.2.18. The result is straightforward in the case of line bundles L. Indeed, in rank 1 the term [Φ, Φ∗ ] vanishes, so (4.17) amounts to finding a constant curvature metric on L. If H is any metric, let Hϕ = eϕ H for a function ϕ. Then ¯ and the problem is solved if we can find ϕ such that F(∂¯L ,Hϕ ) = F(∂¯L ,H) + ∂∂ϕ, √ Δϕ = 2 −1Λ(F(∂¯L ,H) ) − 2deg (L). By Hodge’s theorem, the only condition to finding a solution to this equation is that the integral of the right-hand side vanish (cf., [26, p. 84]), which it does by (4.3). In order to prove Theorem 4.2.17 in higher rank, it will be important to construct approximate critical metrics. Let 0 ⊂ (E1 , Φ1 ) ⊂ · · · ⊂ (E , Φ ) = (E, Φ) be the Harder–Narasimhan filtration of the Higgs bundle (E, Φ). 0 We let Qi = Ei /Ei−1 and μi = μ(Qi ). Then, there is a smooth splitting E = i Qi and, given a hermitian metric H, we can make this splitting orthogonal. Hence, there is a well-defined endomorphism ⎞ ⎛ μ1 ⎟ ⎜ .. μ(Gr(E,Φ),H) = ⎝ (4.21) ⎠, . μ where the block μi has dimension rank Qi . Definition 4.2.19. We say that a metric on (E, Φ) is ε-approximate critical if 5 5 5 5 sup 5f((∂¯E ,H),Φ) − μ(Gr(E,Φ),H) 5 < ε. ¯ Note that the ∂-operator for E may be written in an upper triangular form with respect to this splitting, and the strictly upper triangular piece is determined by the extension classes. By acting with a complex gauge transformation that is

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Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

block diagonal, the extension classes may be made arbitrarily small. If, moreover, the bundles Qi with their induced Higgs fields admit Hermitian–Yang–Mills–Higgs connections, then we can sum these up and obtain the following (for more details, see [14]). Lemma 4.2.20. Let (E, Φ) be an unstable Higgs bundle of rank n, and suppose that Theorem 4.2.17 has been proven for Higgs bundles of rank less than n. Then for any ε > 0 there is an ε-approximate critical metric on (E, Φ). Preliminary Estimates Recall the map (4.15). A crucial point is the following a priori estimate. Proposition 4.2.21. Let (E, Φ) be a Higgs bundle. There are constants C1 , C2 > 0 depending only on the metrics on X and E, and on #h[E, Φ]#, such that 5√ 5 sup |Φ|2 ≤ C1 + C2 sup 5 −1Λ(FA + [Φ, Φ∗ ])5 . We need the following Lemma. Lemma 4.2.22 (cf., [59, p. 27]). For a matrix P there are constants C1 , C2 > 0 depending only on the eigenvalues of P such that |[P, P ∗ ]|2 ≥ C1 |P |4 − C2 (1 + |P |2 ). Proof. Choose a unitary basis such that P = S + N , where S is diagonal and N is strictly upper triangular. By assumption, |S| is bounded. It is easy to see that it then suffices to show there is C > 0 such that for all strictly upper triangular N , |[N, N ∗ ]| ≥ C|N |2 . Suppose not. Then, by scaling, we can find a sequence Nj , |Nj | = 1, and [Nj , Nj∗ ] → 0. After passing to a subsequence, we may assume Nj → N , with [N, N ∗ ] = 0, |N | = 1. But this is a contradiction. Indeed, if a1 , . . . , an and b1 , . . . , bn are the rows and columns of N , then reading off the diagonal of N N ∗ = N ∗ N implies |ai |2 = |bi |2 for i = 1, . . . , n. But b1 = 0, which from this equality implies a1 = 0. This in turn implies b2 = 0, and hence a2 = 0. Continuing in this way, we conclude N = 0, a contradiction.  We will also need the following computation: [[P, P ∗ ], P ] = (P P ∗ − P ∗ P )P − P (P P ∗ − P ∗ P ) = 2P P ∗ P − P ∗ P 2 − P 2 P ∗ , [[P, P ∗ ], P ], P = tr ([[P, P ∗ ], P ]P ∗ ) = tr ((2P P ∗ P − P ∗ P 2 − P 2 P ∗ )P ∗ ) = 2tr (P P ∗ )2 − 2tr (P 2 (P ∗ )2 ), ad([P, P ∗ ])P, P = |[P, P ∗ ]|2 .

(4.22)

Proof of Proposition 4.2.21. Regard Φ as a holomorphic section of K ⊗ End E. We ! also make use of three easy facts. First, if H is a hermitian metric on E and H

4.2. The Dolbeault Moduli Space

181

is the induced metric on End E, then F(End E,H)  = ad F(E,H) , where the adjoint ! h indicates that the curvature endomorphism acts by commutation. Second, if H, are hermitian metrics on End E and K, respectively, then F(K⊗End E,h⊗H)  = F(End E,H)  + F(K,h) · I.

(4.23)

Third, if s is a holomorphic section of a vector bundle with unitary connection A and curvature FA , then we have √ √ ¯ 2 = 2 −1Λ∂ ∂|s| ¯ 2 = 2 −1Λ∂ s, d s Δ|s|2 = −2∂¯∗ ∂|s| A √ √ = 2 −1Λ dA s, dA s + 2 −1Λ s, dA dA s √ = 2|dA s|2 + 2 −1Λ s, FA s √ = 2|dA s|2 − 2 −1ΛFA s, s , giving the Weitzenb¨ock formula √ Δ|s|2 = 2|dA s|2 − 2 −1ΛFA s, s .

(4.24)

Now using eqs. (4.22), (4.23), and (4.24), along with Lemma 4.2.22, we have √ Δ|Φ|2 ≥ −2 −1ΛF(K⊗End E,h⊗H)  Φ, Φ √ 2 ≥ −2 −1ΛF(End E,H)  Φ, Φ − C3 |Φ| √ = −2 ad( −1ΛF(E,H) )Φ, Φ − C3 |Φ|2 √ √ = 2 ad( −1Λ[Φ, Φ∗ ])Φ, Φ − 2 ad( −1Λ(F(E,H) + [Φ, Φ∗ ])Φ, Φ − C3 |Φ|2 5√ 5 ≥ C1 |Φ|4 − C2 (1 + |Φ|2 ) − C4 sup 5 −1Λ(F(E,H) + [Φ, Φ∗ ]5 |Φ|2 . Finally, at a maximum of |Φ|2 the left-hand side is nonpositive. Since C1 > 0, the proposition follows immediately.  Remark 4.2.23. Notice that the sign in (4.16) is crucial for this argument (cf., [37]). Finally, the existence proof will be based on Donaldson’s elegant argument in [18]. This requires the introduction of the functional J = J(A, Φ), defined as follows. For a hermitian endomorphism ϕ of E, let ν(ϕ) =

n  i=1

 |λi | ,

N 2 (ϕ) =

ν 2 (ϕ) X

ω , 2π

where the λi are the (pointwise) eigenvalues of ϕ. Then we define J(A, Φ) = N (f(A,Φ) − μ(E)).

(4.25)

We next prove the following two results by Donaldson (see [18, Lemmas 2 & 3]), adapted here to the case of Higgs bundles.

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Lemma 4.2.24. Let (A, Φ) be a Higgs bundle with underlying bundle E. Suppose it fits into an extension of Higgs bundles 0 → M → E → N → 0, and that μ(N) ≤ μ(E) ≤ μ(M). Then, (rank M)(μ(M) − μ(E)) + (rank N)(μ(E) − μ(N)) ≤ J(A, Φ). Proof. With respect to the orthogonal splitting E = M ⊕ N , and letting FE , FM , and FN denote the curvature and induced curvatures of the Chern connection for (E, H), we have √  √ −1ΛFM + bM √ −(dA )∗ β −1ΛFE = , −1ΛFN + bN −((dA )∗ β)∗ √ ∗ where √ β is∗ the second fundamental form, and bM 2 = − −1Λ(β ∧ β ), bN = −  −1Λ(β ∧ β). Notice that tr bM = −tr bN = |β| . Similarly, if we write Φ = ϕ ΦM , then 0 ΦN   [ΦM , Φ∗M ] + ϕ ∧ ϕ∗ ϕ ∧ Φ∗N + Φ∗M ∧ ϕ . [Φ, Φ∗ ] = ΦN ∧ ϕ∗ + ϕ∗ ∧ ΦM [ΦN , Φ∗N ] + ϕ∗ ∧ ϕ It follows that



f(A,Φ) =

√ fM + bM + −1Λϕ ∧ ϕ∗ ...

 ·√ ·· . fN + bN + −1Λϕ∗ ∧ ϕ

Hence, (cf., [18, p. 271])

5 √ 5 ν(f(A,Φ) − μ(E)) ≥ 5tr ( −1ΛFM ) − (rank M)μ(E) + |β|2 + |ϕ|2 5 5 √ 5 + 5tr ( −1ΛFN ) − (rank N)μ(E) − |β|2 − |ϕ|2 5

and, therefore,



J(A, Φ) ≥

ν(f(A,Φ) − μ(E))

ω 2π

5 5X 5  √  ω5 5 tr ( −1ΛFM ) − (rank M)μ(E) + |β|2 + |ϕ|2 ≥ 55 2π 5 X 5 5 5  √  ω5 5 tr ( −1ΛFN ) − (rank N)μ(E) − |β|2 − |ϕ|2 + 55 2π 5 X

≥ (rank M)(μ(M) − μ(E)) + (rank N)(μ(E) − μ(N)).  Lemma 4.2.25. Let (A0 , Φ0 ) be a stable Higgs bundle of rank n that fits into an extension of Higgs bundles 0 → S → E → Q → 0. Assume Theorem 4.2.17 has been proven for Higgs bundles of rank less than n. Then, we can choose a point (A, Φ) in the complex gauge orbit of (A0 , Φ0 ) such that J(A, Φ) < (rank S)(μ(E) − μ(S)) + (rank Q)(μ(Q) − μ(E)).

4.2. The Dolbeault Moduli Space

183

Proof. First, consider the Harder–Narasimhan filtrations of (S, ΦS ) and (Q, ΦQ ). By applying Lemma 4.2.20, we may assume for any ε > 0 that there is a metric on S such that 5 5 5 5 sup 5f((∂¯S ,HS ),ΦS ) − μ(Gr(S,ΦS ),HS ) 5 < ε, and similarly for Q. We endow E = S ⊕ Q with the sum of these two metrics. This is equivalent to a pair (A, Φ) in the orbit of (A0 , Φ0 ). Next, since (A0 , Φ0 ) (and hence also (A, Φ)) is simple, we may further assume that √   ∗ β + −1Λ ϕ ∧ Φ∗Q + Φ∗S ∧ ϕ = 0; −∂¯A 0 see (4.8). This   is accomplished via a complex gauge transformation of the form 1 ϕ ¯ g= . In particular, the ∂-operators on S and Q remain unchanged, and so 0 1 the approximate critical structure still holds. With this understood, we perform a further gauge transformation so that (A, Φ) coincides with (A0 , Φ0 ) but with β and ϕ scaled by t. Then, f(A,Φ) − μ(E) is block diagonal with entries √   fS − μ(Gr(S,ΦS ),HS ) + μ(Gr(S,ΦS ),HS ) − μ(E) + t2 bS + −1Λϕ ∧ ϕ∗ , √   (4.26) fQ − μ(Gr(Q,ΦQ ),HQ ) + μ(Gr(Q,ΦQ ),HQ ) − μ(E) + t2 bQ + −1Λϕ∗ ∧ ϕ . Since (E, Φ) is stable, μ(E) is strictly bigger than the maximal slope of a subsheaf of S, and strictly smaller than the minimal slope of a quotient of Q. This says that for t and ε chosen sufficiently small, the first line in (4.26) is negative definite and the second is positive definite. It follows that ν(f(A,Φ) − μ(E)) ≤ (rank S)(μ(E) − μ(S)) + (rank Q)(μ(Q) − μ(E))   − 2t2 |β|2 + |ϕ|2 + O(ε). Without loss of generality, assume that #β#2 + #ϕ#2 = 1. By the argument in [18], we may also assume |β|, |ϕ| are bounded uniformly in ε. The result now follows by fixing t and choosing ε sufficiently small.  The Existence Theorem We will prove the following in the next section, where the Yang–Mills–Higgs flow will be introduced. Lemma 4.2.26. In any complex gauge orbit there exists a sequence (Ai , Φi ) satisfying the following conditions: (i) (Ai , Φi ) is minimizing for J; √ (ii) if f(Aj ,Φj ) = −1Λ(FAj + [Φj , Φ∗j ]), then sup |f(Aj ,Φj ) | is bounded uniformly in j; (iii) #dAj f(Aj ,Φj ) #L2 → 0 and #[f(Aj ,Φj ) , Φi ]#L2 → 0.

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Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Next, we will need one of the most fundamental results in gauge theory, stated here for the case of Riemann surfaces. Proposition 4.2.27 (Uhlenbeck, [64]). Fix p ≥ 2. Let {Aj } be a sequence of Lp1 connections with #FAj #Lp uniformly bounded. Then there exist a sequence of unitary gauge transformations gj ∈ Lp2 and a smooth unitary connection A∞ such that, after passing to a certain subsequence, gj (Aj ) → A∞ weakly in Lp1 and strongly in Lp . Assuming these results, we now prove the existence theorem. Proof of Theorem 4.2.17. It clearly suffices to assume (E, Φ) is stable. Furthermore, by Remark 4.2.18, we may proceed by induction. Assume that the result has been proven for all bundles of rank < n = rank E. Step 1. The limiting bundle (E∞ , Φ∞ ). Choose a minimizing sequence for J as in Lemma 4.2.26. Since the sequence lies in a single complex gauge orbit, the image of the Hitchin map h[Ai , Φi ] is unchanged. Hence, by Proposition 4.2.21, the Φi are uniformly bounded. By Lemma 4.2.26 (ii), this in turn implies that #FAj #Lp is bounded for any p. We therefore may assume, by Proposition 4.2.27, that there is a smooth connection A∞ so that if we write ∂¯Aj = ∂¯A∞ + aj , then aj → 0 weakly in Lp1 . By the Sobolev embedding theorem, we may assume in particular that aj → 0 in some C α . It follows that FAj → FA∞ weakly in Lp . From the holomorphicity condition, 0 = ∂¯Aj Φj = ∂¯A∞ Φj + [aj , Φj ]. Elliptic regularity for ∂¯A∞ implies a bound #Φj #L21 ≤ C#Φj #L2 , say. Differentiating the previous equation gives ∗ ¯ ∗ [aj , Φj ] = 0. ∂ Φj + ∂¯A ∂¯A ∞ A∞ ∞

(4.27)

By the Cauchy–Schwarz inequality and the previous estimate, we have ∗ [aj , Φj ]#L2 ≤ C1 #aj #L41 #Φj #L4 + C2 #Φj #L2 . #∂¯A ∞

(4.28)

Now we may assume {aj } is bounded in L41 and, using elliptic regularity for the ∗ ¯ along with the inclusions L21 → L4 , L22 → C α , by (4.27) Laplacian ∂¯A ∂ ∞ A∞ and (4.28) we have the estimate #Φj #C α ≤ C#Φj #L2 . Since the Φj ’s are uniformly bounded, their L2 -norms are bounded so, we may assume that Φj converges in C α to some Φ∞ . Moreover, by holomorphicity of the Φj , we can write ∂¯A∞ Φ∞ = ∂¯A∞ (Φ∞ − Φj ) − [aj , Φj ] and, since [aj , Φj ] → 0 in C α , we see that ∂¯A∞ Φ∞ = 0 weakly. Hence, by Weyl’s lemma, Φ∞ is actually holomorphic, and thus (E∞ , Φ∞ ) is a Higgs bundle.

4.2. The Dolbeault Moduli Space

185

Step 2. Construction of a nonzero map E → E∞ . Let gj be complex gauge transformations such that gj (A) = Aj . Holomorphicity of gj implies ∂¯A∞ gj +[aj , gj ] = 0. By the exact same argument as in Step 1, we have the estimate #gj #C α ≤ C#gj #L2 . Now rescale gj so that #gj #L2 = 1. The C α -estimate above still holds for the rescaled map so, by compactness, we may assume there is a continuous map g∞ : E → E∞ such that gj → g∞ in C α . Because of the normalization, we know that g∞ ≡ 0. Moreover, it follows as in Step 1 that g∞ is holomorphic. Finally, by the C α -convergence of gj and Φj and the fact gj Φ = Φj gj , we have g∞ Φ = Φ∞ g∞ . Step 3. The map g∞ is an isomorphism. Suppose the contrary, and let S = ker g∞ and Q = E/S. Then Q is a subsheaf of E∞ . Let M denote its saturation and N = E∞ /M. Since Φ∞ g∞ = g∞ Φ, the subbundle S is Φ-invariant. Similarly, M is Φ∞ -invariant. Also, from the discussion in Subsection 4.2.1, we have μ(Q) − μ(E) ≤ μ(M) − μ(E),

(4.29)

μ(E) − μ(S) ≤ μ(E) − μ(N). Then we have the following extensions of Higgs bundles (see [18]): 0

/S No

0o

/E 

g∞

E∞ o

/Q

/0 (4.30)

 Mo

0.

Applying Lemma 4.2.24 to the bottom row of (4.30) and Lemma 4.2.25 to the top row, we get (rank M)(μ(M) − μ(E)) + (rank N)(μ(E) − μ(N)) ≤ J(A∞ , Φ∞ ) ≤ lim J(Aj , Φj ) j→∞

= inf J(A, Φ) < (rank S)(μ(E) − μ(S)) + (rank Q)(μ(Q) − μ(E)), where we can use either the lower semicontinuity of J (see [18]) or the argument in [14, Corollary 2.12 and Lemma 2.17]. Since rank M = rank Q and rank S = rank N, this contradicts (4.29). Step 4. Solution to Hitchin’s equations. Finally, I claim that the Higgs bundle (A∞ , Φ∞ ) is a solution to (4.16). Indeed, by the remark following equation (4.19), this follows if we can show dA∞ f(A∞ ,Φ∞ ) = 0 and [f(A∞ ,Φ∞ ) , Φ∞ ] = 0. The second fact holds, since [f(Aj ,Φj ) , Φj ] → 0 in L2 by assumption, and f(Aj ,Φj ) (resp., Φj ) converges weakly in Lp (resp., C α ). For the first claim, let B be a test form. Then, dA∞ f(A∞ ,Φ∞ ) , B L2 = f(A∞ ,Φ∞ ) , d∗A∞ B L2 = lim f(Aj ,Φj ) , d∗Aj B L2 + lim j→∞

j→∞

 X

= lim dAj f(Aj ,Φj ) , B L2 + lim j→∞

j→∞

X

 tr f(Aj ,Φj ) [aj , B ∗ ]  tr f(Aj ,Φj ) [aj , B ∗ ] .

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Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

The first term vanishes since #dAj f(Aj ,Φj ) #L2 → 0, and the second term vanishes since fj is bounded and aj → 0 in C α . Since B is arbitrary, dA∞ f(A∞ ,Φ∞ ) = 0, and this completes the proof.  Proof of Theorem 4.2.15. Let [Aj , Φj ] be a sequence of polystable Higgs bundles with h[Aj , Φj ] bounded. By Theorem 4.2.17, we may assume (Aj , Φj ) satisfies (4.16). Since h[Aj , Φj ] is bounded, the pointwise spectrum of Φj is uniformly bounded. Therefore, Proposition 4.2.21 provides uniform upper bounds on |Φj |. Again using (4.16), we have uniform bounds on |FAj |. Now Uhlenbeck compactness can be used to extract a convergent subsequence which also satisfies (4.16), as in the proof of the existence theorem above.  The Yang–Mills–Higgs Flow We define the Yang–Mills–Higgs flow for a pair (A, Φ) by the equations ∂A = −d∗A (FA + [Φ, Φ∗ ]), ∂t (4.31) √ ∂Φ = [Φ, −1Λ(FA + [Φ, Φ∗ ])]. ∂t In the above, we only consider initial conditions where Φ is dA -holomorphic. Notice then that this holomorphicity condition is preserved along a solution to (4.31). Indeed, as in Donaldson [19], the flow is tangent to the complex gauge orbit and exists for all 0 ≤ t < +∞. The flow equations may be regarded as the L2 -gradient flow of the YMH functional. They generalize the Yang–Mills flow equations. For more on this we refer to [39, 69] and the references therein. Here, we limit ourselves to a discussion of a few key properties. In particular, we justify the assumptions in the previous section. √ As in (4.17), set f(A,Φ) = −1Λ(FA + [Φ, Φ∗ ]). Lemma 4.2.28. For all t ≥ 0,

d dt

YMH(A, Φ) = −2#dA f(A,Φ) #2L2 − 4#[Φ, f(A,Φ) #2L2 .

Proof. We have

 d YMH(A, Φ) = 2 tr (f(A,Φ) f˙(A,Φ) )ω. dt X Now, using dots to denote time derivatives,  √ ˙ Φ∗ ] + [Φ, Φ˙ ∗ ] f˙(A,Φ) = −1Λ dA A˙ + [Φ, √ = −1Λ (−dA d∗A (FA + [Φ, Φ∗ ]) + [[Φ, f ], Φ∗ ] + [Φ, [Φ, f ]∗ ]) √  = −d∗A dA f(A,Φ) + −1Λ [Φ, f(A,Φ) ]Φ∗ + Φ∗ [Φ, f(A,Φ) ] + Φ[Φ, f(A,Φ) ]∗  +[Φ, f(A,Φ) ]∗ Φ .

Taking traces we get

√   tr (f(A,Φ) f˙(A,Φ) ) = −tr (f(A,Φ) d∗A dA f(A,Φ) ) − 2 −1 Λ tr [Φ, f(A,Φ) ][Φ, f(A,Φ) ]∗ , (4.32)

4.2. The Dolbeault Moduli Space and the result follows by integration by parts.

187 

As a consequence of Lemma 4.2.28, YMH decreases along the flow. Moreover, we have the inequality  ∞  dt 2#dA f(A,Φ) #2L2 + 4#[Φ, f(A,Φ) #2L2 ≤ YMH(A0 , Φ0 ). 0

It follows that if (Aj , Φj ) is a sequence with YMH(Aj , Φj ) uniformly bounded,  j ) with YMH(A j , Φ  j ) also j , Φ then we may replace it with another sequence (A and [Φ , f ] converge to 0 in uniformly bounded, but such that dAj f(Aj ,Φ j (A j ) j ,Φ j ) 2 L . Now let’s compute 5 5 52 52 2 Δ 5f(A,Φ) 5 = −d∗ d 5f(A,Φ) 5 = ∗d ∗ dtr f(A,Φ) = 2 ∗ d ∗ tr (f(A,Φ) dA f(A,Φ) ) = 2 ∗ tr (df(A,Φ) ∧ ∗dA f(A,Φ) ) − 2tr (f(A,Φ) d∗A dA f(A,Φ) ) 5 52 52 5 52 ∂ 55 = 2 5df(A,Φ) 5 + 4 5[Φ, f(A,Φ) ]5 + f(A,Φ) 5 , ∂t from (4.32). We have shown Lemma 4.2.29. For all t ≥ 0, 5 5 5 52 52 52 52 ∂ 55 f(A,Φ) 5 − Δ 5f(A,Φ) 5 = −2 5dA f(A,Φ) 5 − 4 5[Φ, f(A,Φ) ]5 . ∂t 5 5 5 5 In particular, 5f(A,Φ) 5 is a subsolution of the heat equation and so, sup 5f(A,Φ) 5 is nonincreasing. In fact, one can 5 use an5 explicit argument with the heat kernel to show that for t ≥ 1, say, sup 5f(At ,Φt ) 5 ≤ C YMH(A0 , Φ0 ) for a fixed constant C. In particular, if (Aj , Φj ) is a sequence with YMH(Aj , Φj ) uniformly bounded,  j ) with f   uniformly j , Φ then we may replace it with another sequence (A (Aj ,Φj ) bounded. Proof of Lemma 4.2.26. Choose a minimizing sequence (Aj , Φj ) for J in the complex gauge orbit of (A, Φ). Note that YMH(Aj , Φj ) is then uniformly bounded. In addition, by an argument similar to the one above (see [14]), J is also decreasing along the YMH-flow. Hence, replacing each (Aj , Φj ) with a point along the YMH-flow with initial condition (Aj , Φj ) also gives a J-minimizing sequence. On the other hand, by the discussion in this section, we can choose points along the flow where items (ii) and (iii) are also satisfied. This completes the proof.  Let Bmin be the set of all Higgs bundles satisfying Hitchin equations (4.17). E The YMH-flow sets up an infinite-dimensional, singular Morse theory problem, is the minimum of the functional, and Higgs bundles not in Bmin where Bmin E E , but satisfying (4.19) play the role of higher critical points. This Morse theory picture can actually be shown to be more than just an analogy. In particular, we have the following

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Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Theorem 4.2.30 (Wilkin, [69]). The YMH-flow gives a GE -equivariant deformation min retraction of Bss E onto BE .

4.3 The Betti Moduli Space 4.3.1 Representation Varieties Fix a base point p ∈ X and set π = π1 (X, p). Let Hom(π, SLn (C)) denote the set of homomorphisms from π to SLn (C). This has the structure of an affine algebraic variety. Let ## (n) MB = Hom(π, SLn (C)) SLn (C) denote the representation variety, where the double slash indicates the invariant (n) theoretic quotient by overall conjugation of SLn (C). Then, MB is an irreducible 2 affine variety of complex dimension (n − 1)(2g − 2). There is a surjective alge(n) braic quotient map Hom(π, SLn (C)) → MB , and this is a geometric quotient (n) on the open set of irreducible (or simple) representations. Points of MB are in one-to-one correspondence with conjugacy classes of semisimple (or reductive) representations, and every SLn (C)-orbit in Hom(π, SLn (C)) contains a semisimple representation in its closure (for these results, see [48]). Following Simpson [60, 61] (n) I will refer to MB as the Betti moduli space of rank n. Let E → X be a trivial rank n complex vector bundle. A flat connection ∇ on E gives rise to a representation of π as follows. Recall that we have fixed a base point p ∈ X. We also fix a frame {ei } of Ep . For each loop γ based at p, parallel translation of the frame {ei } defines an element of GLn (C). Since the connection is flat, this is independent of the choice of path in the homotopy class. In this way we have defined an element hol(∇) ∈ Hom(π, GLn (C)). If ∇ induces the trivial connection on det E, the holonomy lies in SLn (C), and we will assume this from now on. Conversely, given a representation ρ : π1 (X, p) → SLn (C), we obtain a  × Cn /π, holomorphic bundle Vρ with a flat connection ∇ by the quotient Vρ = X  where X is the universal cover of X, and the quotient identifies (x, v) ∼ (xγ, vρ(γ)). Let CE denote the space of connections on E, and Cflat E ⊂ CE the flat connections. (p) denote the space of complex gauge transformations that are the identity Let GC E at p, acting on CE by conjugation (warning: this is a different action of GC E from the one on the space of unitary connections in Subsection 4.2.2). Proposition 4.3.1. The holonomy map gives an SLn (C)-equivariant homeomor## C ∼ (n) C flat GE  MB . phism hol : Cflat E /GE (p) −→ Hom(π, SLn (C)). In particular, CE

4.3.2 Local Systems and Holomorphic Connections Definition 4.3.2. A complex n-dimensional local system on X is a sheaf of abelian groups that is locally isomorphic to the constant sheaf Cn .

4.3. The Betti Moduli Space

189

Here, C denotes the locally constant sheaf modeled on C. Clearly, a local system V is a sheaf of modules over C. Definition 4.3.3. Let V → X be a holomorphic bundle. A holomorphic connection on V is a C-linear operator ∇ : V → K ⊗ V satisfying the Leibniz rule ∇(f s) = df ⊗ s + f ∇s,

(4.33)

for local sections f ∈ O, s ∈ V. For a local system V let V be the holomorphic bundle V = O ⊗C V. Then V inherits a holomorphic connection as follows: choose a local parallel frame {vi } n for V. Any local section of V may be written uniquely as s = i=1 fi ⊗ vi , with n fi ∈ O. Then define ∇s = i=1 dfi ⊗ vi . Since the transition functions for V are constant, this is well defined independently from the choice of frame, and ∇ also immediately satisfies the Leibniz rule. Conversely, a holomorphic connection defines a flat connection on the underlying complex vector bundle, since in a local holomorphic frame the curvature F∇ is necessarily of type (2, 0), and on a Riemann surface there are no (2, 0)-forms. In particular, the C-subsheaf V ⊂ V of locally parallel sections ∇s = 0 defines a local system. This gives a categorical equivalence between local systems and holomorphic bundles with a holomorphic connection (see [16, Th´eor`eme 2.17]). A local system has a monodromy representation ρ : π → GLn (C), obtained by developing local parallel frames. Conversely, given ρ we construct a local system as in the previous section. We will sometimes denote these Vρ and Vρ . For simplicity, in these notes I will almost always assume the monodromy lies in SLn (C), or in other words, det Vρ  O and the induced connection on det Vρ is trivial. Not every holomorphic bundle V admits a holomorphic connection. In particular, such a connection is flat and so, by (4.3), a necessary condition is that deg V = 0. In fact, one can say more about the Harder–Narasimhan type of a bundle with a holomorphic connection. Proposition 4.3.4 (cf., [8, 23]). Suppose V is an unstable bundle with an irreducible holomorphic connection, and let μ1 > μ2 > · · · > μ be the Harder–Narasimhan type. Then, for each i = 1, . . . ,  − 1, μi − μi+1 ≤ 2g − 2. Proof. Let 0 ⊂ V1 ⊂ · · · ⊂ V = V be the Harder–Narasimhan filtration of V. Then, since the connection is irreducible, the O-linear map ∇

Vi −−→ V/Vi ⊗ K is nonzero for each i = 1, . . . ,  − 1. Let j ≤ i be the smallest integer such that Vj → V/Vi ⊗ K is nonzero. Then it follows from the sequence 0 −→ Vj−1 −→ Vj −→ Qj −→ 0

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Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

that there is a nonzero map Qj → V/Vi ⊗ K. With this fixed j, let k ≥ i be the largest integer such that Qj → V/Vk ⊗ K is nonzero. It follows from 0 −→ Qk+1 −→ V/Vk −→ V/Vk+1 −→ 0 that Qj → Qk+1 ⊗ K is nonzero. Since the Qi are all semistable, we have by Lemma 4.2.8 that μj = μ(Qj ) ≤ μ(Qk+1 ⊗ K) = μk+1 + 2g − 2, and the result follows, since μi − μi+1 ≤ μj − μk+1 .



The Weil–Atiyah Theorem. The goal now is to prove the following theorem. Theorem 4.3.5 (Weil [66], Atiyah [1]). A holomorphic bundle V → X admits a holomorphic connection if and only if each indecomposable factor of V has degree zero. The proof I give here follows Atiyah. The following construction will be useful (see [1, p. 193]). Any holomorphic bundle V → X gives rise to a counterpart D(V) as follows. First, as a smooth bundle, D(V) = (V ⊗ K) ⊕ V . With respect to this splitting, define the O-module structure by f (ϕ, s) = (f ϕ + s ⊗ df, f s) ,

f ∈ O, ϕ ∈ V ⊗ K, s ∈ V.

One checks that this gives D(V) the structure of a locally free sheaf over O. Then we have a compatible inclusion ϕ → (ϕ, 0) and projection (ϕ, s) → s making D(V) into an extension (4.34) 0 −→ V ⊗ K −→ D(V) −→ V −→ 0. Observe that (4.34) splits if and only if V admits a holomorphic connection. Indeed, such a ∇ gives a splitting by s → (∇s, s), and if (4.34) splits, then there is a Clinear map V → V ⊗ K satisfying (4.33). Remark 4.3.6. The construction is functorial with respect to subbundles. If 0 = V0 ⊂ V1 ⊂ · · · ⊂ V = V is a filtration of V by holomorphic subbundles, then there is a filtration 0 = D(V0 ) ⊂ D(V1 ) ⊂ · · · ⊂ D(V ) = D(V). Lemma 4.3.7. Given a holomorphic bundle V → X, let [β] ∈ H 1 (X, (V ⊗ K) ⊗ V∗ )  H∂1,1 ¯ (X, End V ) √ denote the extension class. Then, [tr β] = −2π −1 c1 (V ). Proof. Choose s(i) local holomorphic frames for V on Ui , and let ψij denote the transition functions, s(i) = s(j) ψi,j . We can define local splittings of (4.34) by s(i) f (i) → s(i) ⊗df (i) , for f (i) a vector of holomorphic functions on Ui . In particular, f (j) = ψi,j f (i) ,

−1 ∂f (j) = ψi,j (ψi,j ∂ψi,j f (i) + ∂f (i) ).

4.3. The Betti Moduli Space

191

Since the extension class is given by the image of I under the map H 0 (X, End V) → H 1 (X, End V ⊗ K), it follows from the local splitting above that [β] is represented by the cocycle −1 dψi,j ]. Hence, [tr β] = [d log det ψ]. On the other hand, if h is a hermitian [ψi,j (i)

(i)

(j)

(j)

metric on det V, then hi |s1 ∧· · ·∧sn |2 = hj |s1 ∧· · ·∧sn |2 , so hi | det ψi,j |2 = hj . This implies d log det ψi,j = ∂ log hj − ∂ log hi . By the √ Dolbeault isomorphism, [β] ¯ log hi ] = [F ¯ is represented by [∂∂ (∂det V ,h) ] = −2π −1 c1 (V ) (see Example 4.2.2 and equation (4.3)).  Lemma 4.3.8. If V → X is an indecomposable holomorphic bundle and ϕ ∈ H 0 (X, End V), then there is λ ∈ C such that ϕ − λI is nilpotent. Proof. Since det(ϕ−λI) is holomorphic and X is closed, the eigenvalues of ϕ must be constant. So, without loss of generality, assume ker ϕ = {0}, V, and consider the sequence 0 −→ S = ker ϕ −→ V −→ Q = coker ϕ −→ 0. (4.35) 

Write ∂¯E =

∂¯S 0

β ∂¯Q



 ,

ϕ=

 0 ϕ1 . 0 ϕ2

We wish to show ϕ2 = 0. First note that   0 ∂¯E ϕ1 + βϕ2 0 = ∂¯E ϕ = . 0 ∂¯Q ϕ2 So, ϕ2 is holomorphic as an endomorphism of Q. If ϕ2 = 0, then it is an isomorphism. This is so because, again, the eigenvalues of ϕ2 are constant, and by assumption 0 is not an eigenvalue. Hence, we can rewrite the upper right entry in the matrix equation above as, ∂¯E (ϕ1 ϕ−1 2 ) + β = 0. But then the Dolbeault class of β vanishes and (4.35) splits, contradicting the assumption that V be indecomposable.  Proof of Theorem 4.3.5. Suppose V has a holomorphic connection. Then, by Remark 4.3.6, D(V) splits. Moreover, since D(V) is natural with respect to subbundles, D(Vi ) splits for each indecomposable factor of V. But then, by Lemma 4.3.7, deg (Vi ) = 0 for all i. Conversely, suppose V is indecomposable and deg (V) = 0. It suffices to show that D(V) splits. Now by Serre duality the extension class is  ∗ [β] ∈ H 1 (X, End (V) ⊗ K)  H 0 (X, End (V))  and the perfect pairing is (β, ϕ) = tr (βϕ). By Lemma 4.3.8, we may express X

ϕ = λI + ϕ0 , where ϕ0 is nilpotent. Then, by Lemma 4.3.7,  tr β (β, ϕ) =(β, ϕ0 ) + λ(β, I) = (β, ϕ0 ) + λ X √ =(β, ϕ0 ) − 2π −1λ deg (E) = (β, ϕ0 ).

(4.36)

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Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Set V = V, and recursively define Vi−1 to be the saturation of ϕ0 (Vi ). Note that Vi−1 is a proper subbundle of Vi since, otherwise, the restriction of ϕ0 would be almost everywhere an isomorphism. Eventually, the process terminates. Adjust  so that V0 = {0}, V1 = {0}. By Remark 4.3.6, β preserves the filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ V = V. Choose a hermitian metric on V and let πi be the orthogonal projection to Vi . Note that I=

 

(πi − πi−1 ) =

i=1

 

(πi − πi πi−1 ) =

i=1

 

πi (I − πi−1 ),

i=1

and (I − πi )βπi = (I − πi−1 )ϕπi = 0. Then, tr (βϕ0 ) = tr (ϕ0 β) =

 

tr (ϕ0 βπi (I − πi−1 ))

i=1

=

 

tr ((I − πi−1 )ϕ0 βπi ) =

i=1

 

tr ((I − πi−1 )ϕ0 πi βπi ) = 0.

i=1

So, (β, ϕ0 ) = 0 and, by (4.36), we conclude [β] = 0. The proof is complete.



4.3.3 The Corlette–Donaldson Theorem Hermitian Metrics and Equivariant Maps Let D = SUn \SLn (C) and ρ : π → SLn (C). Then π acts on the right on D via the representation ρ. Following Donaldson, we give a concrete description of D with its SLn (C)-action. Set D to be the set of positive hermitian n × n matrices M with det M = 1. Then, the right SLn action is given by (M, g) → g −1 M (g −1 )∗ . Note that the space D may be interpreted as the space of hermitian inner products on Cn which induce a fixed one on det Cn . The invariant metric on D is given by |M −1 dM |2 = tr (M −1 dM )2 .  → D is ρ-equivariant if u(xγ) = u(x)ρ(γ) for all Definition 4.3.9. A map u : X x ∈ X, γ ∈ π.  × Cn /π. We now claim that a hermitian metric on the bundle Let E = X E is equivalent to a choice of ρ-equivariant map, up to the choice of basepoints.  → D is ρ-equivariant. By definition, a section of E is a Indeed, suppose u : X  → Cn such that σ(xγ) = σ(x)ρ(γ). Hence, if we define #σ#2 (x) = map σ : X σ(x), σ(x)u(x) Cn then, #σ#2 (xγ) = σ(x)ρ(γ), σ(x)u(x)(ρ(γ)−1 )∗ Cn = #σ#2 (x) and so, this is a well-defined metric on E. In the other direction, given a metric H, if σi are sections, then write σi , σj H (x) = σi (x), σj (x)u(x) Cn , for a hermitian

4.3. The Betti Moduli Space

193

matrix valued function u(x). Then, σi (x), σj (x)u(x) Cn = σi , σj H (x) = σi , σj H (xγ) = σi (x)ρ(γ), σj (x)ρ(γ)u(xγ) Cn = σi (x), σj (x)ρ(γ)u(xγ)ρ(γ)∗ Cn for all sections. Hence, ρ(γ)u(xγ)ρ(γ)∗ = u(x), and u is ρ-equivariant. Harmonic Metrics  → D is a continuously differentiable ρ-equivariant map, we define its If u : X  ⊗ u∗ (T D). We have fixed energy as follows. The derivative du is a section of T ∗ X 2 an invariant metric on D, so the norm is eu (x) = |du| (x). In fact, by equivariance, eu (x) is invariant under π, so it gives a well-defined function on X which is called the energy density. The energy of u is then by definition  Eρ (u) = eu (x) ω. (4.37) X

Note that the energy only depends on the conformal structure on X, and not on the full metric. The Euler–Lagrange equations for Eρ are easy to write down. Define τ (u) = d∗∇ du.

(4.38)

In the above, we note that the bundle u∗ (T D) has a connection ∇: the pull-back of the Levi–Civita connection on D. It is with respect to this connection that d∇ is defined. The tensor τ (u) is called the tension field. It is a section of u∗ (T D). Definition 4.3.10. A C 2 ρ-equivariant map u is called harmonic if it satisfies τ (u) = 0.

(4.39)

Equation (4.39) is a second-order elliptic nonlinear partial differential equation in u. This statement is slightly misleading because u is a mapping and not a collection of functions. This annoying fact makes defining weak solutions a little tricky. In the case of maps between compact manifolds (the non-equivariant problem), one way to circumvent this issue is to use a Nash isometric embedding of the target into an euclidean space, and rewrite the equations in terms of coordinate functions (cf., [53]). A more sophisticated technique, better suited to the equivariant problem, is to define the Sobolev space theory intrinsically (cf., [41, 45, 46]). On the other hand, if we assume u is Lipschitz continuous, then we can introduce local coordinates {y a } on D and write (4.39) locally. By Rademacher’s theorem, the pull-backs sa = u∗ (∂/∂y a ) give a local frame for u∗ (T D) almost everywhere, and the connection forms for ∇ in this frame are Γcab (u)dua ⊗ sc , where Γcab (u) are the Christoffel symbols on D evaluated along u. Writing u = (u1 , . . . , uN ) in

194

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

terms of the coordinates on {y a }, it is easy to see that the local expression of (4.39) becomes (4.40) −τ (u)a = Δua + Γabc (u)∇ub · ∇uc = 0. To be clear, the dot product in the second term refers to the metric on X, and Δ is the Laplace operator on X. Notice that this equation is conformally invariant with respect to the metric on X, a manifestation of the fact that the energy functional itself is conformally invariant. In light of the previous section, ρ-equivariant maps are equivalent to choices of hermitian metrics. Given a flat connection ∇ and hermitian metric on E, we can construct the equivariant map in a more intrinsic way. First, lift ∇ and E  We will to obtain a flat connection on a trivial bundle on the universal cover X. use the same notation to denote this lifted bundle and connection. If we choose a base point pˆ covering the base point p for π1 (X, p), and we choose a unitary frame p)} for the fiber Epˆ, let {ei (x)} be given by parallel transport with respect to {ei (ˆ  → D is given by x → ei , ej (x). It is ρ-equivariant and ∇. Then the map u : X uniquely determined up to the choice of pˆ and the base point in D.  → D is any ρ-equivariant map such that u(ˆ Conversely, if u : X p) = I, then u defines a hermitian metric for which it is the equivariant map constructed above. Notice that there is an equivalence of the type we saw for Higgs bundles. If g ∈ GC E (p), then the corresponding ρ-equivariant map obtained from the pair (g(∇), H) is the same as that for (∇, Hg). Finally, if we act by a constant g ∈ SLn (C), the same is true, but now the map is (ρ · g)-equivariant. The moral of the story is that finding a harmonic metric is equivalent to finding a harmonic equivariant map in the GC E orbit of ∇. dA is a Given the data (∇, H), we may uniquely write ∇ = dA + Ψ, where √ unitary connection on (E, H), and Ψ is a 1-form with values in the bundle −1gE of hermitian endomorphisms. We can explicitly define Ψ with respect to a local frame {si } by Ψsi , sj =

1 { ∇si , sj + si , ∇sj − d si , sj } . 2

(4.41)

Lemma 4.3.11 (cf., [20]). The energy of the above map is given by Eρ (u) = 4#Ψ#2 . Proof. From the definition above and the fact that dA is unitary, we obtain dui,j = dA ei , ej + ei , dA ej . On the other hand, the ei are parallel with respect to ∇,  so dA ej = −Ψej . Hence, u−1 du = −2Ψ. Definition 4.3.12. We say that H is a harmonic metric if the map u defined above is a harmonic map. Proposition 4.3.13 (Corlette, [11]). If ρ admits a harmonic metric, then it is semisimple. Proof. Suppose that H is a critical metric but that ∇ is reducible. Let V1 ⊂ V be a subbundle invariant with respect to the connection ∇. Let V2 be the orthogonal

4.3. The Betti Moduli Space

195

complement of V1 , and H1 , H2 the induced metrics. We can express     ∇1 β β d A1 + Ψ 1 ∇= = , 0 ∇2 0 d A2 + Ψ 2 where β ∈ Ω1 (X, Hom(V2 , V1 )). It suffices to show that the connection splits, or in other words that β ≡ 0. The proposition then follows by induction. Now using (4.41) it follows that if s1 , s2 are local sections of V1 , then Ψs1 , s2 = Ψ1 s1 , s2 . Similarly, Ψs1 , s2 = Ψ1 s1 , s2 for local sections of V2 . On the other hand, if si ∈ Vi , then Ψs1 , s2 = 21 s1 , βs2 . It follows that   Ψ1 12 β Ψ= 1 ∗ . Ψ2 2β We now deform the metric H to a family Ht as follows: scale H1 → e−(rank V2 )t H1 , and H2 → e+(rank V1 )t H2 . This, of course, preserves the orthogonal splitting and the condition det Ht = 1. But Ht is a geodesic homotopy of ρ-equivariant maps and so, by a result of Hartman, the energy Eρ (ut ) is convex [32]. On the other hand, by Lemma 4.3.11, 1 Eρ (ut ) = #Ψ1 #2H1 + #Ψ2 #2H2 + #β#2H e−(rank V )t/2 . 4 In particular, Eρ (ut ) is bounded as t → ∞. The only way Eρ (ut ) could have a critical point at t = 0 is if Eρ (ut ) is constant, which implies β ≡ 0. This completes the proof.  The Corlette–Donaldson Theorem In this section we prove the following theorem. Theorem 4.3.14 (Corlette [11], Donaldson [20], Jost–Yau [42], Labourie [47]). Let ρ : π → SLn (C) be semisimple. Then, there exists a ρ-equivariant harmonic map  → D. u: X The following result can be compared to Lemma 4.2.26. It will be proven when we discuss the harmonic map flow below. Lemma 4.3.15. For any ρ : π → SLn (C) there is a sequence uj of ρ-equivariant  → D satisfying the following conditions: maps uj : X (i) uj is energy minimizing; (ii) the uj have a uniformly bounded Lipschitz constant; (iii) τ (uj ) → 0 in L2 .  → D be a sequence Lemma 4.3.16. Let ρ : π → SLn (C) be irreducible, and let uj : X p) is bounded. of ρ-equivariant maps with a uniform Lipschitz constant. Then, uj (ˆ

196

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Proof. Suppose not. Set hj = uj (ˆ p) and choose εj → 0 such that (perhaps after passing to a subsequence) εj hj → h∞ = 0. Notice that det h∞ = 0 so, V = ker h∞ is a proper subspace of Cn . We claim that ρ(π) fixes V . Indeed, if ρ(γ) = g −1 and p), uj (ˆ p) · g −1 ) is uniformly bounded, we have v ∈ V then, since d(uj (ˆ | w, vhj Cn − w, vghj g ∗ Cn | ≤ B, for a constant B independent from j, and all w ∈ Cn . It follows that | w, vεj hj Cn − wg, vgεj hj Cn | −→ 0 and, since vh∞ = 0, we conclude that wg, vgh∞ Cn = 0. Since w was arbitrary, vg ∈ V .  Proof of Theorem 4.3.14. By induction, it suffices to prove the result for irreducible representations. Let uj be a minimizing sequence as in Lemma 4.3.16, the existence of which is guaranteed by Lemma 4.3.15. It follows from Ascoli’s theorem that there is a uniformly convergent subsequence, also denoted uj , with the limit uj → u∞ a Lipschitz ρ-equivariant map. I claim that we may arrange for u∞ to be a harmonic map. Indeed, since the convergence is uniform, we may choose local coordinates and write ua . Then, since |dua | is uniformly bounded, we may assume further that uj → u∞ weakly in L21,loc . By the condition in Lemma 4.3.15 (iii), the coordinates ua∞ are in L21,loc and form a weak solution of (4.40). Since u∞ is Lipschitz, elliptic regularity of the Laplace operator implies u∞ ∈ L22,loc . By the remark following (4.40), we may assume that the local metric on X is euclidean. Now, differentiate to obtain Δ(∇ua∞ ) + ∇(Γabc (u∞ )∇ub∞ · ∇uc∞ ) = 0, Δ(∇2 ua∞ ) + ∇2 (Γabc (u∞ )∇ub∞ · ∇uc∞ ) = 0. Notice that, since u∞ is Lipschitz, the second term in the first equation is in L2 . It then follows that ua∞ ∈ L23,loc . Because of the inclusion L23 → L42 , the second term of the second equation above is then in L2 . This in turn implies ua∞ ∈ L24,loc . Finally, L24 ⊂ C 2,α and so, u∞ is a strong solution to the harmonic map equations (4.39). This completes the proof.  The Harmonic Map Flow The harmonic map flow is defined by u˙ = −τ (u).

(4.42)

Here, ut is a family of ρ-equivariant maps. Since D has non-positive curvature, the flow is very well-behaved. Long time existence is proven in [21, 30].

4.3. The Betti Moduli Space

197

The variation of the energy along the flow is given by    d ∗ E(ut ) = 2 du, du ˙ = 2 d∇ du, u ω ˙ = −2 |τ (u)|2 ω. dt X X X In particular, energy decreases along the flow. Moreover,  ∞  2 dt |τ (ut )|2 ω ≤ E(u0 ). 0

(4.43)

X

We are now ready to prove Lemma 4.3.15. Proof of Lemma 4.3.15. The proof is based on the famous Eells–Sampson–Bochner formula for the change of the energy density along the harmonic map flow [21]. Let u = u(t, x) be a solution to (4.42), and e = eu (t, x). Then, −

∂e + Δe = |∇du|2 + RicX (du, du) − RiemD (du, du, du, du). ∂t

Now since RiemD ≤ 0 and RicX is bounded below a negative constant, we have ∂e − Δe ≤ C · e. ∂t Using an explicit argument with the heat kernel, this inequality along with the fact that energy is decreasing imply an estimate of the following type sup eut ≤ C · Eu0 ,

(4.44)

for t ≥ 1, say, where C depends only on the geometry of X and D. Now, let u(j) be an energy minimizing sequence of ρ-equivariant maps. Let (j) ut be the corresponding maps after the time t flow of (4.42). Then, since energy is (j) decreasing along the flow, utj is also energy minimizing for any choice of sequence tj . On the other hand, the right hand of (4.44) is uniformly bounded, so if we (j) choose each tj ≥ 1, say, then utj is also uniformly Lipschitz. Finally, for each fixed initial condition u0 , (4.43) implies τ (utj ) → 0 in L2 along some sequence. (j) By a diagonalization argument we can arrange for utj to satisfy this property as well. 

4.3.4 Hyperk¨ahler Reduction The Moduli Spaces are Real Isomorphic Using (4.41) and given a hermitian metric, we may identify the space of all connections √  CE = (A, Ψ) ∈ AE × Ω1 (M, −1gE ) .

198

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Then, CE is a hyperk¨ ahler manifold, and the action of the gauge group G has associated moment maps μ1 (A, Ψ) = FA + 21 [Ψ, Ψ],

μ2 (A, Ψ) = dA Ψ,

μ3 (A, Ψ) = dA (∗Ψ).

(4.45)

Let m = (μ1 , μ2 , μ3 ). The hyperk¨ahler quotient is, by definition, # # −1 −1 m−1 (0) G = μ−1 1 (0) ∩ μ2 (0) ∩ μ3 (0) GE . The two pictures we have been discussing above are equivalent to a reduction of CE in steps, but in two different ways. The first is the point of view of Hitchin and Simpson described in Subsection 4.2.3. Namely, the space of Higgs bundles is given by −1 BE = μ−1 2 (0) ∩ μ3 (0) ⊂ CE , where the relationship between Ψ and Φ is Ψ = Φ + Φ∗ and, conversely, Φ is the (1, 0) part of Ψ. Just like for functions on surfaces, Ψ is harmonic if and only if Φ is holomorphic. Now Theorem 4.2.17 guarantees that the orbit of every polystable ss Higgs bundle intersects locus μ−1 1 (0) in B . Hence, we have ## C # # (n) −1 −1 −1 (0) GE = μ−1 MD = Bss E GE = m 1 (0) ∩ μ2 (0) ∩ μ3 (0) GE . The second point of view (e.g., Corlette and Donaldson, see Subsection 4.3.3) comes from the observation that the space of flat connections is −1 −1 Cflat E = μ1 (0) ∩ μ2 (0) ⊂ CE .

Given ∇ ∈ Cflat E , the condition that the associated hol(∇)-equivariant map be harmonic is precisely that ∇ ∈ μ−1 3 (0). Indeed, suppose δ∇ is a variation of ∇. It follows from (4.41) that δΨ = δ∇ + (δ∇)∗ . In the case of a complex gauge transformation with g −1 δg = ϕ, we have δ∇ = ∇ϕ, and δΨ = dA (ϕ + ϕ∗ ) + [Ψ, ϕ − ϕ∗ ]. It is easy to see that the second term will not contribute in the variation tr (δΨ ∧ ∗Ψ) + tr (Ψ ∧ ∗δΨ) (by direct computation, and also from the fact that unitary gauge transformations do not vary the associated equivariant map). So, from Lemma 4.3.11 we have  tr (δΨ ∧ ∗Ψ) + tr (Ψ ∧ ∗δΨ) δE(u) = 4 X tr (dA (ϕ + ϕ∗ ) ∧ ∗Ψ) + tr (Ψ ∧ ∗dA (ϕ + ϕ∗ )) =4 X  tr ((ϕ + ϕ∗ )dA (∗Ψ)). = −8 X

Since Ψ is hermitian and ϕ is arbitrary, Ψ is a critical point for the energy if and only if dA (∗Ψ) = 0. Now Theorem 4.3.14 guarantees that the orbit of every semisimple representation contains a harmonic map. It therefore follows that the holonomy map ## C # (n) −1 −1 GE  μ−1 gives a homeomorphism MB  Cflat 1 (0) ∩ μ2 (0) ∩ μ3 (0) GE . So, E the Dolbeault and Betti moduli spaces do coincide!

4.3. The Betti Moduli Space

199

Theorem 4.3.17 (Simpson, [60, 61]). The identification described above gives a (n) (n) homeomorphism MD  MB . Equivariant Cohomology As in the case of the YMH flow, the harmonic map flow actually has continuity properties as t → ∞. To describe this, let GE (p) ⊂ GE denote the subgroup of gauge transformations that are the identity at the point p. Now the holonomy map gives a proper embedding hol : m−1 (0)/GE (p) → Hom(π, SLn (C)),

(4.46)

which is SUn -equivariant. Theorem 4.3.18 (cf., [15]). The inclusion (4.46) is an SUn -equivariant deformation retract. An explicit retraction is defined using the harmonic map flow to define a flow  of p. Given ρ ∈ Hom(π, SLn (C)), on the space of representations. Fix a lift p˜ ∈ X flat choose ∇ ∈ CE with hol(∇) = ρ. The hermitian metric gives a unique ρ-equiva → D with u(˜ riant lift u : X p) = I. Let ut , t ≥ 0, denote the solution to (4.42) with initial condition u. There is a unique continuous family ht ∈ SLn (C), h∗t = ht , such  that h0 = I, and ht ut (˜ p) = z. Notice that a different choice of flat connection ∇  with hol(∇) = ρ will be related to ∇ by a based gauge transformation g. The flow ˜ t = ht . Hence, ht is well is u corresponding to ∇ ˜t = g · ut , and since g(˜ p) = I, h defined by ρ. The flow on Hom(π, SLn (C)) is then defined by ρt = ht ρh−1 t . The  result states that this flow defines a continuous retraction to hol m−1 (0)/GE (p) . When ρ is not semisimple, the flow converges to a semisimplification. This result has consequences for computing the equivariant cohomology of moduli space [2, 12, 43]. In particular, Theorem 4.3.18 implies ∗ ∗ (m−1 (0)/GE (p))  HSU (Hom(π, SLn (C)). HSU n n

Note that, since SLn (C)/SUn is contractible, on the right-hand side we may take equivariant cohomology with respect to SLn (C). On the other hand, Theorem 4.2.30 implies ∗ ∗ ss (m−1 (0)/GE (p)) = HG∗ E (Bmin HSU E )  HGE (BE ). n

It follows that the equivariant cohomology of the space of representations may be computed by studying the equivariant Morse theory of YMH on BE , in the spirit of [2]. This is complicated since BE is singular. Some progress has been made using this approach; see [13, 67]. Figure 4.1 gives a cartoon of CE with the subspaces Cflat E and BE , and the flows that have been defined.

200

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Figure 4.1: Higgs bundles and flat connections

4.4 Differential Equations 4.4.1 Uniformization For more on the discussion in this section I refer to the classic text Gunning [28]. Definition 4.4.1. The Schwarzian derivative of a univalent holomorphic function f (z) defined on a domain in C is given by f  3 S(f ) = {f, z} =  − f 2



f  f

2 .

By straightforward calculations, one shows the following two facts: (i) S(f ◦ g) = (S(f ) ◦ g)(g  )2 + S(g); (ii) S(f ) = 0 ⇐⇒ f is the restriction of a M¨obius transformation. A particular consequence of (i) and (ii) is then (iii) S(f ) = S(g) =⇒ f = ϕ ◦ g, where ϕ is a M¨obius transformation. The Schwarzian derivative gives a link between uniformization and the monodromy of differential equations, as I briefly explain here. Let Q(z), y(z) be locally defined holomorphic functions, and consider the ODE y  (z) + Q(z)y(z) = 0.

(4.47)

4.4. Differential Equations

201

If y1 , y2 are independent solutions of (4.47) and y2 = 0, then a calculation shows that f = y1 /y2 satisfies S(f ) = 2Q. Note that, for a univalent function f , S = S(f ) is not quite a tensor: rather, by (i), it transforms with respect to local coordinate changes as S(w)(w )2 = S(z) − {w, z},

(4.48)

so S nearly transforms as a quadratic differential. A collection {S(z)} of local holomorphic functions on X transforming as in (4.48) is called a projective connection. The space of projective connections on X is an affine space modeled on the space H 0 (X, K2 ) of holomorphic quadratic differentials. Next, consider the transformation properties of the solutions y to (4.47), where 2Q = S is an arbitrary projective connection (cf., [33]). If we assume y is a local holomorphic section of K−1/2 , then we have y(z) = y(w)(w )−1/2 , y  (z) = y  (w)(w )3/2 − 21 y(z){w, z}, and so y  (z) + 12 S(z)y(z) = (y  (w) + 12 S(w)y(w))(w )3/2 . We deduce that Dy = y  + 12 Sy gives a well-defined map of C-modules D : K−1/2 → K3/2 . Therefore, given a projective connection S, we have a rank 2 local system V defined by the solution space to (4.47), 2Q = S. Moreover, there is an exact sequence of C-modules 0 −→ V −→ K−1/2 −→ K3/2 −→ 0. Now assume X has a uniformization as a hyperbolic surface. So ρF : π → PSL2 (R) is a discrete and faithful representation such that X is biholomorphic to H/ρF (π). Let u be a (multi-valued) inverse of the quotient map H → X. In other  → H that is equivariant with respect to ρF . words, u is a univalent function u : X Set SF (z) = S(u)(z). Then by items (i) and (ii) above, for any γ ∈ π, SF (γz) = S(u)(γz) = S(ρF (γ) ◦ u)(z) = S(u)(z) = SF (z). So SF is a well-defined projective connection on X. Now the key point is the following: if y1 , y2 are linearly independent solutions to (4.47) where 2Q = SF , then S(y1 /y2 ) = S(u) and so, by (iii) above, there is a M¨obius transformation ϕ such that y1 /y2 = ϕ ◦ u. It follows that the (projective) monodromy of the local system associated to (4.47) in the case 2Q = SF is conjugate to ρF . If S is any fixed choice of projective connection, one may ask for the holomorphic quadratic differential Q such that SF = S + Q. This is the famous problem of accessory parameters (cf., [52]).

202

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Remark 4.4.2. I want to clarify the following issue: the bundle K1/2 involves a choice of square root of the canonical bundle (i.e., a spin structure), of which there are 22g possibilities. This choice is precisely equivalent to a lift of the corresponding monodromy ρ from PSL2 (C) to SL2 (C). To see this, let Vρ = O ⊗C Vρ , and notice that Vρ fits into an exact sequence (now of O-modules) −→ 0, 0 −→ Kρ1/2 −→ Vρ −→ K−1/2 ρ

(4.49)

where now we also label the choice of spin structure by ρ. Since Vρ has a holomorphic connection, by Theorem 4.3.5, (4.49) cannot split. On the other hand, the extensions are parametrized by the projective space of H 1 (X, K)  (H 0 (X, O))∗ = C. So, all the bundles V obtained in this way as ρ varies are isomorphic modulo −1/2 has a nonzero the choice of K1/2 . Equation (4.49) also implies that V∗ρ ⊗ Kρ holomorphic section. Moreover, if we have such an exact sequence for one spin structure, then (4.49) cannot hold for any other choice K1/2 . Indeed, the induced −1/2 map K1/2 → Kρ would necessarily vanish, and so the inclusion K1/2 → Vρ 1/2 would lift to give an isomorphism K1/2  Kρ . So K−1/2 is determined by ρ. Changing the lift of the projective monodromy ρ to SL2 (C) amounts to ρ → ρ ⊗ χ for some character χ : π → Z/2. This corresponds to tensoring Vρ by a flat line −1/2 bundle Lχ whose square is trivial. From the condition H 0 (X, V∗ρ⊗χ ⊗Kρ⊗χ ) = {0}, 1/2

1/2

and the argument given above, it follows that Kρ⊗χ = Kρ

⊗ Lχ .

4.4.2 Higher Order Equations Invariance Properties The structure outlined in the previous subsection for equations of the type (4.47) extends to higher order equations. We consider n-th order differential equations on H: (4.50) y (n) + Q2 y (n−2) + · · · + Qn y = 0. We would like an appropriate invariance property under coordinate changes, in order to have solutions that are intrinsic to X. Motivated by the example of projective connections, we attempt to realize local solutions of (4.50) in the sheaf K1−q , where n = 2q − 1 and we have chosen a spin structure if q is a half integer. D

Solutions to (4.50) are given by the kernel of an operator K1−q −−−−→ Kq . Theorem 4.4.3 (cf., [17]; see also [34, 68]). Let D : K1−q → Kq be C-linear and locally of the form Dy = y (n) + Q2 y (n−2) + · · · + Qn y. Then, 12Q2 /n(n2 − 1) is a projective connection and, for k ≥ 3, there exist wk , linear combinations of Qj , j = 2, . . . , k and derivatives, with coefficients polynomials in Q2 , such that wk transform as a k-differentials. Conversely, given one such operator and k differentials wk , k = 2, . . . , n, these conditions uniquely determine an operator D.

4.4. Differential Equations

203

The expressions for the wk are quite complicated. For example, we reproduce some from [17, Table 1]: w 2 = Q2 , n−2  Q2 , 2 n−3  (n − 2)(n − 3)  (n − 2)(n − 3)(5n + 7) 2 Q3 + Q2 − Q2 . w 4 = Q4 − 2 10 10n(n2 − 1) w 3 = Q3 −

(4.51)

It follows from Theorem04.4.3 that the space of all such D is an affine space n modeled on the Hitchin base j=2 H 0 (X, Kj ). The map D : K1−q → Kq is clearly locally surjective. Moreover, the Wronskian of any fundamental set of solutions Dyi = 0 is constant. We therefore obtain a local system V and an exact sequence of sheaves over C, ϕ

D

0 −→ V −−−−→ K1−q −−−−→ Kq −→ 0.

(4.52)

In this situation, we say that the local system V is realized in K . Remark 4.4.4. If we tensor by a line bundle with a holomorphic connection and replace derivatives y (j) with derivatives in a local parallel frame of the line bundle, then we can consider local systems realized in L: 1−q

ϕ

D

0 −→ V −−−−→ L −−−−→ L ⊗ Kn −→ 0,

(4.53)

where deg L = −(n − 1)(g − 1). The Riemann–Hilbert Correspondence The goal of this section is to characterize which local systems can be realized as the monodromy of solutions to differential equations. To motivate the following, if V is a local system realized in L, and V = O ⊗C V, notice that in (4.53) there is a surjective sheaf map V → L given by f ⊗ v → f ϕ(v), for f ∈ O, v ∈ V. In particular, V∗ ⊗ L has a nonzero holomorphic section. Theorem 4.4.5. A representation ρ : π → SLn (C) can be realized in L if and only if ρ is irreducible, H 0 (X, V∗ρ ⊗ L) = {0}, and Ln = K−n(n−1)/2 . Proof. According to Hejhal [34, Theorem 3], the monodromy representation arising from a differential operator D is necessarily irreducible. I shall give a proof of this fact below (see Proposition 4.4.8). Accepting this point for the time being, from the discussion above we also have a nonzero section of V∗ρ ⊗ L. Moreover, if y1 , . . . , yn is an independent set of solutions Dyi = 0 on H, then the Wronskian ⎛ ⎞ y1 ··· yn ⎜ y1 ··· yn ⎟ ⎜ ⎟ W (y1 , . . . , yn ) = det ⎜ .. .. ⎟ ⎝ . . ⎠ (n−1)

y1

···

(n−1)

yn

204

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

is a well-defined nowhere vanishing global holomorphic section of Ln ⊗Kn(n−1)/2 on X. The latter is therefore trivial. This proves the necessity part of the assertion. For the converse, we follow a classical argument using the Wronskian (cf., [51]). Assume we have a nonzero holomorphic section ϕ of V∗ρ ⊗ L. This induces a map Vρ → L, also denoted by ϕ. Since ρ is irreducible, ϕ is injective. And since Ln = K−n(n−1)/2 , we can write L = L0 ⊗ K−(n−1)/2 , where L0 has a flat connection. If we express a section of L as l ⊗ w, where l is a parallel section of L0 , then we define y  = l ⊗ w . With this understood, choose a local frame {vi } for Vρ , and set ⎛

ϕ(v1 ) ϕ(v1 ) .. .

⎜ ⎜ Dy = det ⎜ ⎝ ϕ(v1 )(n)

··· ···

ϕ(vn ) ϕ(vn ) .. .

y y .. .

···

ϕ(vn )(n)

y (n)

⎞ ⎟ ⎟ ⎟. ⎠

Then, if y is a local holomorphic section of L, Dy is a well-defined local holomorphic section of Ln+1 ⊗ Kn(n+1)/2 = L ⊗ Kn . Clearly, the kernel of D is precisely Vρ . Moreover, since the monodromy of Vρ is in SLn (C), it is easy to see that Dy is actually globally defined on X. Finally, Ln = K−n(n−1)/2 , so ⎛ ⎜ ⎜ det ⎜ ⎝

ϕ(v1 ) ϕ(v1 ) .. .

··· ···

ϕ(vn ) ϕ(vn ) .. .

ϕ(v1 )(n−1)

···

ϕ(vn )(n−1)

⎞ ⎟ ⎟ ⎟ ⎠

is a nonzero holomorphic function on X, which may therefore be set equal to 1. Hence, Dy has the form (4.52). This completes the proof.  Example 4.4.6. The lift of the monodromy of a projective connection defines a representation into SL2 (C) which, via the irreducible embedding SL2 → SLn , gives a representation into SLn (C). It is straightforward, if somewhat tedious, to calculate the differential equations associated to the local systems arising in this way. Below are some examples, where we let 2Q be a projective connection on X: (i) n = 2 : y  + Qy = 0; (ii) n = 3 : y  + 4Qy  + 2Q y = 0; (iii) n = 4 : y (4) + 10Qy  + 10Q y  + (9Q2 + 3Q )y = 0; (iv) n = 5 : y (5) + 20Qy  + 30Q y  + (64Q2 + 18Q )y  + (64QQ + 4Q )y = 0; (v) n = 6 : y (6) + 35Qy (4) + 70Q y  + (63Q + 259Q2 )y  + (28Q + 518QQ )y  +(130Q2 + 155QQ + 5Q(4) + 225Q3 )y = 0. Note that w3 , w4 in (4.51) vanish for these examples.

4.4. Differential Equations

205

4.4.3 Opers Oper Structures In this section we introduce opers. For more details consult [3, 4, 5, 6, 40, 62]. Definition 4.4.7 (Beilinson–Drinfeld, [3]). An SLn -oper is a holomorphic bundle V → X, a holomorphic connection ∇ inducing the trivial connection on det V, and a filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V satisfying (i) ∇Vi ⊂ Vi+1 ⊗ K; ∇

(ii) the induced O-linear map Vi /Vi−1 −−−→ Vi+1 /Vi ⊗ K is an isomorphism for 1 ≤ i ≤ n − 1. There is an action of GC on the space of opers which pulls back connections and filtrations. Let Opn denote the space of gauge equivalence classes of SLn -opers on X. Given a holomorphic connection on a bundle V, we shall call a filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V satisfying (i) and (ii) an oper structure. Not every holomorphic connection admits an oper structure. For example, we have the following important result. Proposition 4.4.8. The holonomy representation of an oper is irreducible. First we have Lemma 4.4.9. For any SLn -oper, det Vj  Lj ⊗ Knj−j(j+1)/2 , where L  V/Vn−1 , and Ln  K−n(n−1)/2 . Proof. To simplify notation, set vi = det Vi , κ = K, and use additive notation for line bundle tensor products. Then Definition 4.4.7 (ii) gives vi −vi−1 = vi+1 −vi +κ, and so j j   vj = (vi − vi−1 ) = (vi+1 − vi + k) = vj+1 − v1 + jκ, i=1

i=1

and vj+1 − vj = v1 − jκ. Now, summing again, vi − v1 =

i−1 

(vj+1 − vj ) = (i − 1)v1 −

j=1

i(i − 1) κ, 2

i(i − 1) κ, 2 n(n − 1) κ. 0 = vn = nv1 − 2

vi = iv1 −

Set L = v1 − (n − 1)κ, and this completes the proof.



Proof of Proposition 4.4.8. (cf., [40]) Suppose that (V, ∇) has an oper structure and 0 = W ⊂ V is ∇-invariant. Let Wi = W ∩ Vi . I claim that the induced map Wi /Wi−1 −→ Wi+1 /Wi ⊗ K

206

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

is an inclusion of sheaves for all i = 1, . . . , n − 1. Indeed, consider the commutative diagram of O-modules: Wi /Wi−1

/ Wi+1 /Wi ⊗ K

 Vi /Vi−1

 / Vi+1 /Vi ⊗ K.

The vertical arrows are inclusions and the lower horizontal arrow is an isomorphism. This proves the claim. Set ri = rank (Wi /Wi−1 ). By the claim, if ri = 0, then rj = 0 for j ≤ i. Let 1 ≤  ≤ n be the smallest integer for which r = 0. It follows that ri = 1 if and only if  ≤ i ≤ n. Applying the inclusions recursively and using Lemma 4.4.9, we find Wi /Wi−1 → V/Vn−1 ⊗ Kn−i ∼ = K(n−2i+1)/2 . In particular (see Section 4.2.1), deg (Wi /Wi−1 ) ≤ (n − 2i + 1)(g − 1), and so deg W =

n 

deg (Wi /Wi−1 ) ≤

i=

n 

(n − 2i + 1)(g − 1) = −(n −  + 1)( − 1)(g − 1).

i=

The right-hand side is strictly negative unless  = 1. But, since W has a holomorphic connection induced by ∇, deg W = 0. Hence, the only possibility is  = 1, which implies W = V. This completes the proof.  We now show that if a holomorphic connection admits an oper structure, then that structure is unique up to gauge equivalence. For the next part of the discussion, it will be useful to have the following diagram in mind (cf., Lemma 4.4.9): 0  L ⊗ Kn−j

0  / Vj−1

/V

0

 / Vj

 /V

 L ⊗ Kn−j  0

∼

0

 / Rj−1  / Rj  0

/0 (4.54) /0

4.4. Differential Equations

207 %

Lemma 4.4.10. H 1 (X, (L ⊗ Kn−j ) ⊗ R∗i ) =

0, H 1 (X, K),

if i ≥ j + 1, if i = j.

Proof. Fix j and do induction on i. If i = n − 1, then Rn−1 = L and % 0, if n − j − 1 > 0, 1 n−j ∗ 1 n−j H (X, (L ⊗ K ) ⊗ Rn−1 ) = H (X, K )= 1 H (X, K), if n = j + 1. Now the exact sequence 0 → R∗i → R∗i−1 → L∗ ⊗ Ki−n → 0 gives the following: H 1 (X, (L ⊗ Kn−j ) ⊗ R∗i ) −→ H 1 (X, (L ⊗ Kn−j ) ⊗ R∗i−1 ) −→ H 1 (X, Ki−j ) −→ 0. By induction, the first term vanishes and the last two terms are isomorphic. This proves the lemma.  % 0, if i ≥ j + 1, Lemma 4.4.11. H 1 (X, Vj ⊗ R∗i ) = 1 H (X, K), if i = j. Proof. Fix i and induct on j. Now V1  L⊗Kn−1 so, the result in this case follows from Lemma 4.4.10. Next, consider the exact sequence H 1 (X, Vj−1 ⊗ R∗i ) −→ H 1 (X, Vj ⊗ R∗i ) −→ H 1 (X, L ⊗ Kn−j ⊗ R∗i ) −→ 0. By induction, the first term vanishes and so the second and third terms are isomorphic. Again, the result follows from Lemma 4.4.10.  Corollary 4.4.12. H 1 (X, Vj−1 ⊗ (L ⊗ Kn−j )∗ ) = H 1 (X, K). Proof. Consider the exact sequence H 1 (X, Vj−1 ⊗ R∗j ) → H 1 (X, Vj−1 ⊗ R∗j−1 ) → H 1 (X, Vj−1 ⊗ (Lj−1 ⊗ Kn−j )∗ ) → 0. By Lemma 4.4.11, the first term vanishes and the second one is isomorphic to H 1 (X, K).  Lemma 4.4.13. The extension 0 → Vj−1 → Vj → L ⊗ Kn−j → 0 is non-split. Proof. Consider the diagram H 1 (X, Vj−1 ⊗ R∗j )

H 0 (X, Rj−1 ⊗ R∗j−1 )

I→[β]

 / H 1 (X, Vj−1 ⊗ R∗ ) j−1 g

 / H 1 (X, Vj−1 ⊗ (L ⊗ Kn−j )∗ ) 3

 H 0 (X, Rj−1 ⊗ (L ⊗ Kn−j )∗ ) O I→[α]

H 0 (X, (L ⊗ Kn−j ) ⊗ (L ⊗ Kn−j )∗ )

 0

(4.55)

208

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

By the comment following (4.5), [α] is the extension class of 0 → Vj−1 → Vj → L ⊗ Kn−j → 0, and [β] is the extension class of 0 → Vj−1 → V → Rj−1 → 0. By Lemma 4.4.11, g is injective. By tracing through the definition of coboundary, one has [α] = g[β]. Finally, since V has a holomorphic connection and deg Vj−1 = 0 by Lemma 4.4.9, it follows from Theorem 4.3.5 that [β] = 0.  Finally, we can state the result on the uniqueness of the underlying holomorphic structures. Proposition 4.4.14. Let (V, ∇) be an SLn -oper. Then the oper structure on V is uniquely determined by L = V/Vn−1 . In particular, the isomorphism class of the bundle V is fixed on every connected component of Opn . Proof. By Lemma 4.4.9, V1 = L ⊗ Kn−1 and so it is determined. By Corollary 4.4.12 and Lemma 4.4.13, each Vj is successively determined by Vj−1 as the unique nonsplit extension of the sequence appearing in Lemma 4.4.13. Continuing in this way until j = n, this proves the first statement. The second statement follows as well, since by Lemma 4.4.9 we also have Ln  K−n(n−1)/2 , and therefore the set of possible L’s is discrete.  Corollary 4.4.15. The map sending an oper to its monodromy representation gives (n) an embedding Opn → MB . Proof. Fix a representation ρ : π → SLn (C) and suppose that, up to conjugation, ρ is the monodromy of opers (Vρ , ∇1 ) and (Vρ , ∇2 ). In light of Proposition 4.4.14, it suffices to show that the line bundle L is uniquely determined by ρ. Let L and M be line bundles of degree −(n − 1)(g − 1) such that H 0 (X, V∗ρ ⊗ L) = {0} and H 0 (X, V∗ρ ⊗ M) = {0}. Let {Vi } be the oper structure for (Vρ , ∇1 ), and assume Vρ /Vn−1 = L. If L and M are not isomorphic, it follows from 0 −→ L∗ ⊗ M −→ V∗ρ ⊗ M −→ V∗n−1 ⊗ M −→ 0 that H 0 (X, V∗n−1 ⊗ M) = {0}. Now, for j ≤ n − 1, deg R∗j ⊗ M < 0 so, by applying this argument successively, we conclude that H 0 (X, V∗1 ⊗ M) = {0}. But V∗1 ⊗ M = L∗ ⊗ M ⊗ K1−n also has negative degree, so we get a contradiction.  Remark 4.4.16. There are precisely n2g possibilities for the line bundle L in Proposition 4.4.14. These choices label the components of Opn . As in Remark 4.4.2, these correspond precisely to the n2g ways of lifting a monodromy representation in PSLn (C) to SLn (C). For simplicity, from now on we will always take L = K−(n−1)/2 where if n is even we assume a fixed choice of K1/2 . Opers and Differential Equations We first show how to obtain an oper from a local system that is realized in K1−q , n = 2q − 1. So, assume we are given the exact sequence (4.52), and set V = Vn =

4.4. Differential Equations

209

O ⊗C V. For k = 1, . . . , n − 1, define Vn−k =

" n

fi ⊗ v i :

i=1

n 

(j) fi ϕ(vi )

6 = 0, j = 0, . . . , k − 1 .

i=1

Then Vn−k ⊂ V is a coherent subsheaf and we have exact sequences 0 −→ Vn−k−1 −→ Vn−k −→ K1−q+k −→ 0.

(4.56)

Property (i) in Definition 4.4.7 is clearly satisfied. Furthermore, in view of (4.56), the connection ∇ induces an O-linear map Vn−k−1 → Vn−k /Vn−k−1 ⊗K  K2−q+k by n n   (k+1) fi ⊗ vi → fi ϕ(vi ), i=1

i=1

and this is an isomorphism of sheaves. So property (ii) holds as well. Conversely, suppose that V is a rank n holomorphic bundle with holomorphic connection ∇ that admits an oper structure. By Lemma 4.4.9, we have V/Vn−1  K1−q . It follows that for any SLn -oper (we continue to assume L = K−(n−1)/2 ), H 0 (X, V∗ ⊗K1−q ) = {0}. Since the monodromy of an oper is irreducible by Proposition 4.4.8, the hypotheses of Theorem 4.4.5 are satisfied, and (V, ∇) is realized in K1−q . Theorem 4.4.17 (Beilinson–Drinfeld, [3]). The embedding above gives an isomorphism the connected components of Opn and the (affine) Hitchin base 0n between 0 j H (X, K ). j=2 Corollary 4.4.18 (Teleman, [63]). The monodromy of a differential equation (4.52) (or (4.53)) is never unitary. Proof. If ρ is the monodromy then, by the correspondence above and Lemma 4.4.9, we see that Vρ is an unstable bundle. But then, from the easy direction of Theorem 4.2.6 (see Proposition 4.2.16), Vρ cannot admit a flat unitary connection.  Opers and Moduli Space The main goal of this section is to prove the following (n)

Theorem 4.4.19. The map Opn → MB is a proper embedding. By the upper semicontinuity of the Harder–Narasimhan type, this theorem is a direct consequence of the following one (see Subsection 4.2.1). Proposition 4.4.20. Among those bundles with holomorphic connections, opers have strictly maximal Harder–Narasimhan type. We begin with

210

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Lemma 4.4.21. The Harder–Narasimhan filtration of a bundle V with an oper structure is given by the oper filtration itself. Proof. It suffices to show that, for each j = 0, . . . , n − 1, Vj+1 /Vj is the maximally destabilizing subsheaf of V/Vj . In order to do this, let μmax (V/Vj ) denote the maximal slope of a subsheaf of F ⊂ V/Vj , 0 < rank F < rank (V/Vj ). We make the inductive hypothesis that μmax (V/Vj ) = μ(Vj+1 /Vj ) = (n − 1)(g − 1) − j(2g − 2). Note that this is trivially satisfied for j = n − 1. Now suppose j ≤ n − 2 and let F → V/Vj be the maximally destabilizing subsheaf. Then F is semistable and, from the sequence 0 −→ Vj+1 /Vj −→ V/Vj −→ V/Vj+1 −→ 0 and the inductive hypothesis, we have μ(F) ≥ μ(Vj+1 /Vj ) > μ(Vj+2 /Vj+1 ) = μmax (V/Vj+1 ). It follows that the induced map F → V/Vj+1 must vanish. Therefore, F  Vj+1 /Vj and the inductive hypothesis is satisfied for j. This concludes the proof.  Proof of Proposition 4.4.20. (cf., [40, Theorem 5.3.1]). Let (V, ∇) be an unstable bundle with holomorphic connection. I claim that it suffices to assume that ∇ is irreducible. Indeed, in the case of rank 1 there is nothing to prove. Suppose the result has been proven for rank less than n and suppose (V, ∇) is reducible. Since the Harder–Narasimhan type is upper semicontinuous, we may assume there is a splitting (V, ∇) = (V1 , ∇1 ) ⊕ (V2 , ∇2 ) with ni = rank Vi ≥ 1. Then, by the induction hypothesis, it suffices to assume the Vi have the Harder–Narasimhan types of rank ni opers. Indeed, if not, then we can change the Harder–Narasimhan types of Vi without changing the ordering of the slopes for V, so that V has a larger Harder–Narasimhan type. Let (n)

μi = μ i

= μ(Kq−i ) = (n + 1 − 2i)(g − 1)

(4.57)

be the Harder–Narasimhan type of a rank n oper (see Lemmas 4.4.21 and 4.4.9). (n ) (n ) If λi is a reordering of the slopes {μi 1 , μj 2 }, we need to show k  i=1

λi ≤

k 

(n)

μi ,

(4.58)

i=1

for all k = 1, . . . , n, with strict inequality for some k. Assume n1 ≥ n2 . Without changing the ordering of the slopes, we can sequentially subtract even integers (n) (n ) from the leading entries μi , λi = μi 1 for 2i ≤ n1 − n2 , and add the integers to last entries where n1 + n2 + 2 ≤ 2i. Notice that the multiplicities of the resulting

4.4. Differential Equations

211

first and last slopes in {μi } and {λi } are equal and will cancel in the sums. This reduces the problem to n1 = n2 or n1 = n2 + 1, where it is easy to verify (4.58). With this understood, we may assume that (V, ∇) is irreducible. The Harder– Narasimhan type of an oper is given by (4.57). Let Vi−1 ⊂ Vi , i = 1, . . . , , be the Harder–Narasimhan filtration of V, and λi = μ(Vi /Vi−1 ). Let ni = rank (Vi /Vi−1 ) and di = ni λi . Then it suffices to show j 



rank (Vj )

ni λ i ≤

i=1

(4.59)

μi ,

i=1

for j = 1, . . . , . The left hand side is just deg Vj while the right hand side is 

rank (Vj )

(n + 1 − 2i)(g − 1) = (g − 1)rank (Vj )(n − rank (Vj )).

i=1

Hence, (4.59) is equivalent to deg Vj ≤ (g − 1)

j 

ni



n−

i=1

j 

 ni .

(4.60)

i=1

Apply repeatedly Proposition 4.3.4 to find λj ≤ λj+1 +2g−2, λj ≤ λj+2 +2(2g−2), λj ≤ λj+i + i(2g − 2), and λj ≤ λ + ( − j)(2g − 2), for any i ≤  − j. This implies nj+1 dj ≤ dj+1 + (2g − 2)nj+1 , nj nj+i dj ≤ dj+i + i(2g − 2)nj+i , nj n dj ≤ d + ( − j)(2g − 2)nj+1 , nj from which we have −j 

ni+j

i=1

−j −j    dj ≤ di+j + (2g − 2) ini+j . nj i=1 i=1

Consider first the case j = 1. Then (4.61) becomes   i=2

ni

     d1 ≤ di + (2g − 2) (i − 1)ni , n1 i=2 i=2

(n − n1 )

  d1 ≤ −d1 + (2g − 2) (i − 1)ni , n1 i=2

(4.61)

212

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces d1 ≤

We claim that

  n1 (2g − 2) (i − 1)ni . n i=2

(4.62)

 2 (i − 1)ni ≤ n − n1 = ni . n i=2 i=2 



(4.63)

Note that this combined with (4.62) proves (4.60) in the case j = 1. To prove the claim, let ri = ni − 1 ≥ 0. Then (4.63) becomes 2

 

(i − 1)(ri + 1) ≤ n

i=2

2

 

7

(i − 1)ri + ( − 1) ≤

i=2

(ri + 1),

i=2  

82

(ri + 1)

+ n1

i=2

which holds if 2

 

 

7 (i − 1)ri + ( − 1) ≤

i=2

 

 

(ri + 1),

i=2

82 ri + ( − 1)

+ ( − 1),

i=2

which, in turn, after canceling like terms from both sides, holds if  

(i − 1)ri ≤

i=2

 

( − 1)ri .

i=2

This latter inequality is clearly true, since ri ≥ 0. Hence, (4.63) holds. We now proceed by induction. Suppose that (4.60) holds for j. We show that it also holds for j + 1. Adding (4.60) for j, and (4.61) for j + 1, we have deg Vj+1 = deg Vj + dj+1 ≤ (g − 1)

j  i=1

−j i=1

ni+j

i=2

ni+j

−j

ni

−j 

−j−1 j+1   nj+1 nj+1 ni+j − −j di + −j (2g − 2) ini+j+1 , i=2 ni+j i=1 i=2 ni+j i=1 i=1

deg Vj+1 ≤ (g − 1)

j  i=1

ni

−j 

−j−1  nj+1 ni+j + −j (2g − 2) ini+j+1 , i=2 ni+j i=1 i=1

j 

−j−1  nj+1 deg Vj+1 ≤ (g − 1) ni + −j (2g − 2) ini+j+1 , j+1 −j n − i=1 ni i=1 ni+j i=2 ni+j i=1 i=1

where in going from the first inequality to the second we have used the fact that j+1 deg Vj+1 = i=1 di . Hence, it suffices to show 2

−j−1  i=1

ini+j+1 ≤

−j  i=1

ni+j

−j  i=2

ni+j .

4.4. Differential Equations

213

In terms of the ri defined above, this becomes 2

−j 

(i − 1)(ri+j + 1) ≤ (rj+1 + 1)

i=2

2

−j 

−j 

(ri+j + 1) +

i=2

+

2 (ri+j + 1)

,

i=2

(i − 1)ri+j + ( − j)( − j − 1) ≤ (rj+1 + 1)

i=2

 −j 

 −j 

−j 

ri+j + (rj+1 + 1)( − j − 1)

i=2

ri+j + ( − j − 1)

2 .

i=2

But this is a consequence of 2

−j  i=2

(i − 1)ri+j ≤ 2

−j 

( − j − 1)ri+j ,

i=2

which obviously holds. This completes the proof of the maximality of the Harder– Narasimhan type. We now show that if the Harder–Narasimhan type of (V, ∇) is maximal, then the filtration {Vi } is an oper structure. Indeed, consider the Olinear map ∇ : Vi → V/Vi+1 ⊗K. By Remark 4.2.9, the minimal slope of a quotient of Vi is μi = μ(Vi /Vi−1 ), whereas the maximal slope of a subsheaf of V/Vi+1 ⊗ K is μ(Vi+2 /Vi+1 ⊗ K) = μi+2 + 2g − 2 = μi+1 = μi − (2g − 2) < μi . Hence, the map above must be zero, and ∇Vi ⊂ Vi+1 ⊗ K. By irreducibility of the connection, Vi /Vi−1 → Vi+1 /Vi ⊗ K is nonzero. Since these are line bundles with the same degree, this map is an isomorphism. Therefore, conditions (i) and (ii) in Definition 4.4.7 are satisfied. This completes the proof. 

4.4.4 The Eichler–Shimura Isomorphism Let us return in more detail to Example 4.4.6. For q ∈ 21 Z, let Vq denote the (2q − 1)-dimensional irreducible representation of SL2 (C). Let ρ : π → SL2 (C) be the (lift of the) monodromy of a projective connection on X. We can realize the local system Vρ in K−1/2 for some choice of Spin structure. For q ≥ 3/2, let Vq denote the local system obtained by composing ρ with the representation Vq , ρ(n) : π −→ SL2 (C) −→ SL(Vq ). Then Vq is realized in K1−q , and we have D

0 −→ Vq −→ K1−q −−−−→ Kq −→ 0.

(4.64)

214

Chapter 4. Higgs Bundles and Local Systems on Riemann Surfaces

Since q ≥ 3/2, H 0 (X, K1−q )  H 1 (X, Kq )∗ = {0}. This implies H 0 (X, Vq ) = H 2 (X, Vq ) = {0}, and the long exact sequence associated to (4.64) becomes δ

0 −→ H 0 (X, Kq ) −−→ H 1 (X, Vq ) −→ H 1 (X, K1−q ) −→ 0. The coboundary map δ is called Eichler integration. The reason for the terminology is the following: if ω is a global holomorphic section of Kq , then on sufficiently 5 small open sets Ui , we can solve the inhomogeneous equation Dyi = ω 5Ui . If we set vi,j = yi − yj , then {vi,j } is a 1-cocycle with values in Vq , representing δω. In any case, it follows that we have an isomorphism (cf., [22, 29, 55]) H 1 (X, Vq )  H 0 (X, Kq ) ⊕ (H 0 (X, Kq ))∗ .

(4.65)

Equation (4.65) can be used to describe the tangent space to the Betti moduli space at [ρ(n) ] (this was explained to me by Bill Goldman [25]). From Weil’s description of the tangent space, (n)

T[ρ(n) ] MB  H 1 (X, End Vq ).

(4.66)

Now representations of SL2 (C) are self-dual: Vq∗  Vq . By the Clebsch–Gordon rule for decomposition of tensor product representations, we have End Vq = (Vq ⊗ Vq∗ )tr =0  (Vq ⊗ Vq )tr =0 =

2q−1 1

Vj

j=2

j∈Z

(note that the trivial representation V3/2 is eliminated by the traceless condition). This decomposition translates into one for the local system. It follows that H 1 (X, End Vq ) =

2q−1 1

H 1 (X, Vj ).

j=2

j∈Z

Combining this with equations (4.65) and (4.66), we obtain (n)

T[ρ(n) ] MB 

n 1

H 0 (X, Kj ) ⊕ (H 0 (X, Kj ))∗ .

j=2

This should be compared with (4.14)!

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