Strings & Symmetries, 1991: Proceedings of the Conference, Stony Brook, May 20-25, 1991 9810207425, 9789810207427

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STRINGS &

SYMMETRIES r~91

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Proceedings of the Conference

=EEEF€== #--==-F=€F€=E=E== Stony Brook, May 20 -25,1991

Editors

N. Berkovits H. Itoyama K. Schoutens

A. Sevrin W. Siegel P. van Nieuwenhuizen J. Yamron a NATO Advanced Research Workshop Director P. van Nieuwenhuizen

World Scientific Singapore. NewJersey. Tsndon. Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd. Box 128, Faner Road, Singaporc 9128 IISA office: Suite tB, 1060 Main Street, River Edge' NI 07561

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UK ofJice: 73 Lynton Mead, Totteridge' lnndon N20 8DH

STRINGS

& SYMMETRIES I99I

Copyright O 1992 by World Sciendfic Publishing Co' Pte' Ltd'

All rights reserved. This book, or Nrts thereof, nay not be rePrduced in any lorm orby-ony**,electronicormechanical,includingphotocopying'recordingorany n\ir-itio" storage and relrieval system now known or to be invented' without written pe rmission from the

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ISBN 981-02-0742-5

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Preface

These are the proceedings of the international conference "Strings and Symmetries 1991", held at Stony Brook from May 20 through May 25, 1991. This conference is one in a series which started in 1979 with a conference on supersymmetry and supergravity at Stony Brook, then moved for several years to Trieste (Italy), where it became a spring school followed by a short conference on supergravity and strings. It then split into a European series (still annually held at Trieste) and an American series, held twice at Maryland (1987, 1988), twice at Texas A k M (1989, 1990), and now at Stony Brook. It intends to be the major string conference in the U.S.A. There was an enormous interest in this conference: 300 participants (local attendance included). During six days, a total of 68 of the top experts in their field gave seminars. The days were opened by a review lecture on an important area, followed by presentations of new results in this area. In the later afternoons, short talks on other topics of interest were given. In addition, a number of brief (5 or 10 minute) communications were presented. As main subjects of the conference, the following areas were selected (i) supergravity and string phenomenology (ii) black holes and quantum gravity (iii) d = 2 gravity and matrix models (iv) quantization methods (v) conformal field theory and related topics (vi) jy-gravity

These proceedings contain 54 talks presented at the conference.

In addition,

there were talks by Bershadsky, Brezin, Dijkgraaf, Gross, Kaplunovsky, Martinec, Nemeschansky, Polyakov, Seiberg, Sotkov, E. Verlinde, Zumino and Zwiebach. Every morning, during the coffee break, string quartets formed by graduate students from Stony Brook's music department played classical music.

In addition,

various other events took place: the conference banquet on a boat with live music which sailed along Long Island's south coast, a 5km footrace (won by E. Zaslow in 17:05 min.), trips to theatre and concerts in New York City, a reception at the home

vi of Stony Brook's president and parties at the homes of the organizers for all participants. These events gave the participants some opportunity to discuss the talks and exchange information on research. As in 1979, the conference was concluded with a soccer game, this time between "worldsheet" and "space-time". (Space-time won by 8-5). The conference was made possible by financial support from the offices of the President and Provost of Stony Brook, from DOE and an Advanced Research Workshop grant from NATO. In addition there were the conference fees, but we decided to keep these down as much as possible (S75 for faculty, S50 for post-docs, $25 for students. This fee included the banquet and these proceedings). The reason so much could be offered for a reasonable conference fee is that we decided to organize this conference as much as possible ourselves. Our secretary, Betty Gasparino, was lent to us by the Institute for Theoretical Physics, to do much administrative work. She did much more than her call of duty. Thank you again, Betty. Then, a special word of thanks goes to the 26 physics graduate students, with whom we had for several months weekly meetings on the conference. They gave us tremendous support during the conference: they served coffee and donuts, drove vans, made photo copies of the transparencies of the lectures and helped in all possible other ways. They deserve to be mentioned: A. Abada, C. Ahn, F. Bastianelli, D. Coker, C. Coriano, S. Dasmahapatra, F. EBler, M. Galassi, M. Hatsuda, C-M. Hung, R. Kedem, B.-B. Kim, E. Laenen, H.-N. Li, S.Mendoza, J. Montag, R. Parwani, B. Peeters, S. Ray, B. Razzaghe-Ashrafi, S. Riemersma, M. Sotiropoulos, J. Uralil, C. Villorente, H.M. Weiser, E. Williams. Their enthusiasm promises 26 exciting conferences in the future! The conference secretariat N. Berkovits H. ltoyama K. Schoutens A. Sevrin W. Siegel P. van Nieuwenhuizen J. Yamron

vii

Local Organizing Committee G.E. Brown V. Korepin H.T. Nieh R. Shrock J. Smith G. Sterman W. Weisberger

A.S. Goldhaber B. McCoy M. Rocek W. Siegel G.D. Sprouse P. van Nieuwenhuizen C.N. Yang

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xi Contents Preface

v

Supergravity and String Phenomenology Superstring compactification and target space duality J.H. Schwarz Sigma model anomalies, non-harmonic couplings and string theory G.L. Cardoso and B.A. Ovrut The complete action of chiral D = 10, N = 2 supergravity L. Castellani, I. Pesando Topological defects in the moduli sector of string theory M. Cvetic Homogeneous Kahler and quaternionic manifolds in N = 2 supergravity B. de Wit A duality between strings and fivebranes M.J. Duff and J.X. Lu Target-space duality, u-model anomalies, and the moduli space of (2,2) compactifications S. FerTQra Supersymmetrization of the Lorentz Chern-Simons term in D=10 P. Fre and I. Pesando Particle physics and superstrings M.K. Gaillard Lagrangian chiral coset construction of heterotic string theories in ( 1, 0) superspace S.J. Gates, Jr., S. V. Ketov, S.M. Kuzenko and O.A. Soloviev Duality invariant effective string actions D. Liist Chern-Simons theories in three dimensions from heterotic string in ten dimensions H. Nishino

3 19

24

29 36 55

82

94 113 127 128 130

Black Holes and Quantum Gravity Curved space-time strings and black holes I. Bars Physical states of the string in a. black hole background J. Distler and P. Nelson

135

146

xii Ashtekar's approach to quantum gravity G. Horowitz Non-Einsteinian gravity with torsion at d = 2 W. Kummer and D.J. Schwarz Singular solutions in string theory from nonsingular initial data A.R. Steil Target space duality and dilaton A.A. Tseytlin Two-loop quantum gravity A. van de Ven On black holes in string theory E. Witten

154 168 170 173

182 184

D = 2 Gravity and Matrix Models

Continuum approaches to 2D quantum gravity E. D'Hoker Gauge-independent analysis of 20-gravity E. Abdalla, M.C.B. Abdalla, J. Gamboa and A. Zadra The discrete integrability of matrix models of 20-gravity M.J. Bowick Multicut criticality in the Penner model and c = 1 strings S. Chaudhuri Quantum cosmology on the worldsheet A. Cooper, L. Susskind and L. Thorlacius Matrix model constraints and symmetries of integrable hierarchies K. de Vos Quantum group derivation of 20 gravity-matter couplings J.-L. Gervais Topological strings from Liouville gravity N. Ishibashi Matrix models at finite N H. Itoyama Topological gravity with minimal models K. Li 20 supergravity and supermatrix models J.L. Maiies

193

219 221

223

225 245 247 266 268 280 292

xiii Solving (1,q) KDV gravity G. Rivlis

294

Quantization Methods First quantization and supersymmetric field theories W. Siegel Twistors and the Green-Schwarz superstring N. Berkovits q-Quantum mechanics D.C. Caldi Differential regularization and renormalization: recent progress D.Z. Freedman D = 10 supersymmetric Chern-Simons gauge theory R.E. Kallosh Some applications of string field theory A. Sen Lorentz-harmonic (super)fields and (super)particles E. Sokatchev Batalin-Vilkovisky lagrangian quantisation A. Van Proeyen

301 310 325 327 341 355 373

388

CFT and Related Topics Some aspects of free field resolutions in 2D CFT with application to the quantum Drinfeld-Sokolov reduction P. Bouwknegt, J. McCarthy and K. Pilch Why are there no exact S-matrices for affine Toda theories based on nonsimply-laced Lie algebras? G. W. Delius, M. T. Grisaru and D. Zanon Extended superconformal algebras from the conformal bootstrap J.M. Figueroa-O'Farrill and S. Schrans Interacting bosonic constructions and their spectrum D. Gepner Recent developments in the Virasoro master equation M.B. Halpern

407

423 435 437 447

xiv Unitary and modular invariant strings on SU(l,l) manifolds S. Hwang Solitons in integrable, N = 2 supersymmetric Landau-Ginzburg models W. Lerche and N. Warner Kibler-Chern-Simona theory v:P. Nair Covariantly coupled chiral algebras and the associated hierarchies P. van Driel On spontaneous breaking of topological symmetry W. Zhao

462 464

479 488 491

W-gravity Classical and quantum W-gravity C.M. Hull Gauging the extended conformal algebras A. MikoviC Anomaly-free W-gravity theories C.N. Pope Induced gauge theories and W-gravity K. Schoutens, A. Sevrin and P. van Nieuwenhuizen

495 540 543 558

Erratum Erratum V. Gate~, Empty Kangaroo, M. Roachcock and W. C. Gall List of Participants

593 601

Supergravity and String Phenomenology

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3

SUPERSTRING COMPACTIFICATION AND TARGET SPACE DUALITY JOHN H. SCHWARZ Department of Physics, California Institute of Technology Pasadena., California 91125, USA ABSTRACT This review talk focusses on some of the interesting developments in the area of superstring compactification that have occurred in the last couple of years. These include the discovery that "mirror symmetric" pairs of Calabi-Yau spaces, with completely distinct geometries and topologies, correspond to a single (2,2) conformal field theory. Also, the concept of target-space duality, originally discovered for toroidal compactification, is being extended to Calabi-Yau spaces. It also associates sets of geometrically distinct manifolds to a single conformal field theory. A couple of other topics are presented very briefly. One concerns conceptual challenges in reconciling gravity and quantum mechanics. It is suggested that certain "distasteful allegations" associated with quantum gravity such as 1088 of quantum coherence, unpredictability of fundamental parameters of particle physics, and paradoxical features of black boles are likely to be circumvented by string theory. Finally there is a brief discussion of the importance of supersymmetry at the TeV scale, both from a practical point of view and as a potentially significant prediction of string theory.

1. Introduction

The conference organizers have asked me to give a review survey of significant developments in superstring compactification that have occurred in the last couple of years since the last review papers that I wrote on this subject. 1 A great deal of impressive progress has been made, and it will only be possible to survey a portion of it. The choice of topics is based mostly on what has caught my attention, and what I have been able to digest. There are undoubtedly many important developments that will not be mentioned. The two main topics to be discussed are Calabi-Yau compactification and targetspace duality. Two important developments in Calabi-Yau compactification will be stressed. The first is the existence of holomorphic prepotentials that determine the Ka.hler potentials that describe the moduli spaces M21 and Mn, associated with complex structure deformations and Kahler form deformations, respectively. The second is the remarkable mirror symmetry that associates a pair of Calabi-Yau spaces to

the same (2,2) conformal field theory. They are related to one another by interchange of the moduli spaces Mz1 and Mu.

4

The most famous example of target-space duality is the R--+ 1/ R symmetry associated with compactification on a circle of tadius R. As with mirror symmetry this transformation relates distinct geometries t.hat are associated with the same conformal field theory. Generalizations appropriate to toroidal compactifications have been known for some time and will be reviewed very briefly. The little bit that is known about such symmetries in the case of K3 and Calabi-Yau space compactification will also be discussed. An interesting proposal has been made to restrict the possibilities for low-energy effective actions that incorporate nonperturbative supersymmetry breaking by the duality symmetries. Recently, my interest in target-space duality was reactivated by the realization that it can be viewed as a discrete symmetry group that is a subgroup of spontaneously broken continuous gauge symmetries-what has been referred to as 'local gauge symmetry.' This led me to propose that it could have a bearing on resolving certain deep problems associated with quantum gravity. 2 After reviewing some of the disturbing allegations that are made about the inevitable consequences of reconciling · general relativity and quantum mechanics, I will discuss possible ways in which they may be circumvented in string theory. The concluding section makes a plea to demonstrate to our experimental colleagues that our work is relevant. It is suggested that a strong case could be made for supersymmetry at the TeV scale as an almost inevitable feature of any quasi-realistic string vacuum. This being so, perhaps it would not be inappropriate for us to stick our necks out a bit and call this a 'prediction of string theory.'

2. Progress in Calabi-Yau Compactification Let me begin by recalling a few basic facts. By definition, a Calabi-Yau space is a Kahler manifold of three (complex) dimensions and vanishing first Chern class. The complex (Dolbeault) cohomology groups H"·" have dimensions b11q given by the Hodge diamond shown below.

1 0 0

0

~1

1

0

bu

0

1

IJ.ll

0

bu 0

0

1 One generator of H 1•1 is the Kahler form J = ig,..pdxP II. dxP. Thus bn ~ 1. A Calabi-Yau space, with specified Kahler class, admits a unique Ricci-flat metric.

5 Also, there exists a covariantly constant spinor A, in terms of which the holomorphic three-form is 0,..,, = A-y11,,A. In the context of heterotic string compactification, the existence of A is responsible for the fact that the 4D low-energy theory has N=l supersymmetry. Altogether, the massless spectrum contains the following N=l supermultiplets: a) N=l supergravity (graviton and gravitino) b) Yang-Mills supermultiplets (adjoint vectors and spinors for Ee xEs

X ••• )

c) Various chiralsupermultiplets (Weyl spinor and a scalar) The chiral supermultiplets include matter and moduli multiplets. The matter multiplets consist of bu generations (27 of~) and ~1 antigenerations (27 of E6). Which of these one chooses to call 'generations' and 'antigenerations' is purely a matter of convention, of course. The moduli consist of an 'S field' and 'T fields.' The S field contains the dilaton t/1 and the axion 9. The vev of the dilaton gives the string coupling constant ( < t/1 >- 1/g2 ), and 9 is the 4D dual of the antisymmetric tensor field B,..,. Locally, the vevs of' and 9 parametrize the coset manifold SU(l, 1)/U(l). The T fields consist of bu Ee singlets whose vevs parametrize the moduli space of Kahler form deformations, Mn, and ~~ E6 singlets whose vevs parametrize the space of complex structure deformations, M 21· Altogether, the Calabi-Yau moduli space is the tensor product Mcy = Mu x M21· Locally, this is the same thing as the space of (2,2) conformal field theories, though the Kahler geometry of M21 differs in the two cases by the effects of world sheet ins tantons. However, as we will discuss, they differ globally by duality symmetries, so that the moduli space of (2,2) theories is given by M(2,2) = MeyfG, where G is a discrete group. This is the space that classifies inequivalent string compactifications. Remarkably, the moduli spaces Mu and M21 are themselves Kahler manifolds. Here I will simply state the salient facts without attempting to give the proofs, which can be found in the literature. 3 Let za, a = 1, 2, ... , ~~ be local complex coordinates for M21· Then linearly independent generators of H 2•1 are given by a=

1, 2, ... , ~1·

Out of Xa = Xadpdx" 1\ dxA 1\ dxP and X.p one constructs the Kahler metric for M 21 by the formula

fMxaAX.p a a Gap = - fM 0" = - aza azP log

n

From this it follows that K =-log (i fM

n A fl)

(·J ) l

M 0 "0 .

is the Kahler potential.

6 Moreover, M21 is a Kahler manifold with a holomorphic prepotential. (Such Kahler manifolds are sometimes said to be of 'restricted type.') The formula is given most succinctly using projective coordinates (in ~~) z 11 , a = 1, 2, ... , ~~ + 1, defined with respect to a canonical homology basis of Ha. The basic cycles A 11 and B& are arranged to intersect in a manner analogous to the A and B cycles of a Riemann surface. The coordinates z 11 are given by z11 = fA• n and the derivatives of the holomorphic prepotential 9(z 11 ) are given by {;og = f 8 • fl. In terms of 9 and its complex conjugate C(z11 ) the Kahler potential is given by e-K

{)Q- z{)Q =-•· ( z -) 8z 8z 11

-11

11

11

.

The prepotential also encodes the Yukawa couplings of the antigenerations, which are given by its third derivatives. (I am confident that more details of this construction will be presented by other speakers.) The moduli space Mu has a very similar description. In terms of local complex coordinates wA, with A = 1, 2, ... , bu. Its metric is given in terms of a Kahler potential by GAB = ~pK, as usual. The Kahler potential is given by K = -log K.(J, J, J), where K.(J, J, J) fM J 1\ J 1\ J and J is the Kahler form. In terms of projective coordinates wi, i = 1, 2, ... , bn + 1, there is again a holomorphic prepotential F( wi) with

=f

e

-K

{)j: = -z· ( w i EJwi -

-i

{)J=') •

w EJwi

As before, the third derivatives of F give the Yukawa couplings of the generations. The significant asymmetry between :F and g in the case of the geometric limit is that F is a cubic function so that the Yukawa couplings are constants, whereas g is a complicated nonpolynomial expression, so that those Yukawa couplings depend on the complex structure deformation moduli. , A&alabi-Yau space corresponds to a geometric limit of a (2,2) superconformal field theory. This geometric limit includes effects to every order in o' in the associated sigma model, but it does not include the nonperturbative effects associated with world-sheet instantons. It turns out that these world-sheet instantons contribute to the moduli space of Kahler form deformations M u but not to the moduli space of complex structure deformations Mzt. As a result the prepotential 9 can be computed exactly in the geometric limit, but the prepotential :F cannot. The earlier assertion that F is a cubic function referred to the geometric limit. When the instanton contributions are included it is no longer cubic. It is the latter expression that is relevant to string theory. This modification of Calabi-Yau geometry implied by the corresponding conformal field theory is sometimes referred to as "quantum geometry" in the recent literature.

7

We now turn to the mirror symmetry conjecture. The similarity in the description of the two moduli spaces Mn and M21 suggests that to every Calabi-Yau space M (with b:n > 0) there is a mirror partner Calabi-Yau space !If such that the two moduli spaces are interchanged:

Mn = M21

and

M21 = Mn.

Clearly, in view of the remarks made above, this is only possible if we include the instanton corrections in the description of the moduli spaces. Thus the precise statement of the conjecture is that there is a a mirror partner Calabi-Yau space such that the two spaces correspond to distinct geometric limits of the same (2,2) conformal field theory. This a remarkable conjecture from a mathematical point of view, since the CY spaces have completely different geometries and topologies. It is also of some practical importance from the physical point of view, since computations of the prepotentials are significantly easier to carry out in the geometric limit (using topological formulas) than for the exact CFT in general. By computing moduli spaces of complex-structure deformations for a pair of mirror CY spaces one deduces (by the equations above) the exact geometry of the moduli spaces of Kahler form deformations, including the effects of world-sheet instantons. A considerable amount of evidence in support of the mirror symmetry conjecture has been amassed. I am not completely sure of the history, but I believe the idea originated when Dixon noticed the symmetrical way in which generations and antigenerations are treated in Calabi-Yau compactification of the heterotic string a Ia Gepner, and when Lerche, Vafa, and Warner noticed the symmetrical appearance of the (c,c) and (a,c) rings in N=2 Landau-Ginzburg theory. 4 Candelas, Lynker, and Schimmrigk computed the Hodge numbers bu and ~1 for several thousand CY spaces that can be described as intersections of weighted complex projective spaces and observed that there was an almost perfect correspondence between pairs of CY spaces with these numbers interchanged. 5 Some unpaired examples are to be expected in their list, since it is certainly not complete. Given that fact, it is remarkable how few of them there are. Also, one class of spaces is certainly special. There exist Calabi-Yau spaces with ~1 = 0. The mirror of such a space should have b11 = 0, but such a space cannot be a Calabi-Yau space, since they always have the Kii.hler form itself as at least one generator of H 1•1 . The significance of this class of exceptions is being investigated by the experts. It may be of some practical importance if we hope to eventually find an example that gives three generations and no antigenerations, so as to avoid the problem of understanding how extra generations and anti-generations pair up to acquire a large mass. Such a space with bu = 3 and ~~ = 0 might not exist, however. Further evidence in support of the mirror symmetry conjecture and insight into various structural details have been obtained from a variety of additional studies. These include analysis of explicit examples based on (2,2) orbifolds6 and additional

8 studies of the Landau-Ginzburg connection. 7 Another approach utilizes Gepner's correspondence between Calabi-Yau compactification and compactification of Type II superstring theories. The constructions based on Type IIA and Type liB theories turn out to be related by mirror symmetry. This fact provides a rather powerful tool for detailed studies, since it benefits from the restrictions implied by the additional supersymmetry of the Type II theories. 8 Finally, I should mention the detailed investigation of a specific mirror pair of CY spaces by Candelas et al. 9 They compute the prepotential :F for the one-dimensional moduli space of the Kibler class for the CY space given by a quintic polynomial in P• and compare it to the prepotential g for the mirror space (given by orbifolding the original space by a suitable discrete symmetry group). By comparing the two expressions they are able to explicitly identify the instanton contributions and infer the number of "holomorphic curves" of various sorts-deep results in algebraic geometry.

3. Target Space Duality The notion of target space duality is becoming a more and more prevalent theme in string theory, with a wide range of applications and implications. Target-space dualities are discrete symmetries of compactified string theories, whose existence suggests a breakdown of geometric concepts at the Planck scale. The simplest example is given by closed bosonic strings with one dimension of space taken to form a circle of radius R. In this case the momentum component of a string corresponding to this dimension is quantized: p = n/ R, n E Z. This is a general consequence of quantum mechanics and is not special to strings, of course. What is special for strings is the existence of winding modes. A closed string can wrap m times around the circular dimension. A string state with momentum and winding quantum numbers n and m, respectively, receives a zero-mode contribution to its mass-squared given by 10

where c/ is the usual Regge slope parameter. It is evident that the simultaneous interchanges R +-+ ol/ R and m +-+ n leaves the mass formula invariant. 11 In fact, the entire physics of the interacting theory is left unaltered provided that one simultaneously rescales the dilaton field (whose expectation value controls the string coupling constant) according to t/1 --+ t/1- ln(R/../Cl). 12 The basic idea is that the radii R and a'/R both correspond to the same (c = 1) conformal field theory, and it is the conformal field theory that determines the physics. This is the simplest example of the general phenomenon called "target-space duality." The circular compactification described above has been generalized to the case of ad-dimensional torus, characterized by Jl constants Ga6 and Ba6.U The symmetric matrix G is the metric of the torus, while B is an antisymmetric matrix. These parameters describe the moduli space of toroidal compactification and can be interpreted as the vacuum expectation values of Jl massless scalar fields. The dynamics

9

of the toroidal string coordinates

xa is described by the world sheet action

In this case we can introduce d-component vectors of integers rna and na to describe the winding modes and discrete momenta, respectively. A straightforward calculation then gives the zero-mode contributions

where Gab represents the inverse of the matrix Gab· The one-dimensional case is recovered by setting Gu = R 2 /a' and B = 0. The generalization of the duality symmetry becomes G + B -+ (G + B)- 1 and f/J -+ f/J- !lndet(G +B). The B term in the world-sheet action is topological. The parameters Bab are analogous to the 8 parameter in QCD, and the quantum theory is invariant under integer shifts Bab-+ Bab +Nab· Combined with the inversion symmetry, these generate the infinite discrete group 0( d, d; Z). 14 (The analogous group in the case of the heterotic string is O(d + 16, d; Z).) Thus, whereas the moduli Gab and Bab locally parametrize the coset space O(d,d)/[O(d) x O(d)), points in this space related by O(d, d; Z) transformations correspond to the same conformal field theory and should be identified. It is conjectured that when nonperturbative effects break the flatness of the effective potential, so that the scalar fields that correspond to the moduli can acquire mass, the discrete duality symmetries of the theory are preserved. 15 The 0( d, d; Z) target space duality symmetries are discrete remnants of spontaneously broken gauge symmetries. 16 Specifically, for values of the moduli corresponding to a fixed point of a subgroup of this discrete group, the corresponding string background has enhanced gauge symmetry. (This is most easily demonstrated by showing that there are additional massless vector string states in the spectrum.) At such a point some of the duality symmetry transformations coincide with finite gauge transformations. By considering all possible such fixed points it is possible to identify an infinite number of distinct gauge symmetries, with all but a finite number of them spontaneously broken for any particular choice of the moduli. One may wonder whether the occurrence of target-space dualities is special to toroidal compactification or whether it occurs generically for curved compactification spaces such as Calabi-Yau manifolds. The four-dimensional analog, namely K3 compactification, has been analyzed in some detailY In that case (applied to the heterotic string) the moduli space is 0(20,4)/[0(20) x 0(4)), parametrized by 80 massless scalar fields and the duality group is 0(20, 4; Z). Remarkably, this is exactly the same manifold and duality group that arises in the case of toroidal compactification of the heterotic string to six dimensions. One might be tempted to speculate that

10

the two compactifications are equivalent, but that cannot be correct since K3 compactification breaks half of the supersymmetry while toroidal compactification does not break any. 18 By modding out certain symmetries of the torus, it is possible to form an orbifold for which half the supersymmetry is broken and the moduli space is still essentially the same. This orbifold seems likely to correspond (at least locally) to the same conformal field theory as K3. Having spaces of distinct topology correspond to identical conformal field theories goes beyond what we learned from tori. (There various different geometries all having the same topology were identified.) However, as we have seen, such identifications do exist for Calabi-Yau spaces, which occur in mirror pairs of opposite Euler number. The moduli space of a Calabi-Yau compactification factorizes into the manifold

M21 that describes complex-structure deformations times the manifold Mu that describes Kahler form deformations. There are duality symmetry transformations that act on each of these spaces separately. The two classes of transformations would seem to have very different interpretations from a geometrical point of view. However, the two factors are interchanged for the mirror manifold, so if one associates them with a mirror pair of Calabi-Yau spaces, then they appear on an equivalent footing. The problem is to determine the target-space duality group that acts on each of these moduli spaces. A natural action of the discrete group Sp(2 + 2~1, Z) can be defined on the third cohomology group

analogous to the symplectic modular group for Riemann surfaces. This symplectic group contains the possible discrete symmetries of M 21· The mirror symmetry implies a corresponding action of Sp(2 + 2bu, Z) on the space

describing possible discrete symmetries of M 11· Thus altogether the target space duality of the (quantum corrected) Calabi-Yau space should be given by some subgroup Grv ~ Sp(2

+ 2b.n, Z) x Sp(2 + 2bu, Z).

Examples have been worked out in special cases. A potentially important application of the duality symmetries has been proposed in connection with the construction of low-energy effective actions. The idea is that these should be exact symmetries of the complete quantum theory and should still be present even after nonperturbative effects (such as those that break supersymmetry) are taken into account and after heavy fields are integrated out. This means that, in terms of a low-energy effective action in four dimensions with N=l supersymmetry,

11 the duality symmetries should be realized on the superpotential, which is therefore restricted to be a suitable automorphic function. This is a very significant restriction on the characterization of the low-energy theory. Therefore there is some hope for saying quite a bit about nonperturbative effects without solving the difficult problem of computing them from first principles. There is some evidence that the combination of gluino condensation and duality symmetry are sufficient to remove all flat directions from the potential. 19 This means that the size of the compact space, which is one of the moduli, is dynamically determined and all the other parameters that determine the vacuum configuration are also determined. Supersymmetry is broken and the cosmological constant typically comes out negative (corresponding to anti de Sitter space). However, there are some examples for which the cosmological constant vanishes. 20 Recently, there have been studies of automorphic prepotentials for general (2,2) compactifications. 21

4. Conceptual Challenges in Reconciling Gravity and Quantum Mechanics There are a variety of technical and conceptual obstacles that need to be overcome if a satisfactory understanding of the reconciliation of general relativity and quantum mechanics is to be achieved. These can be divided into two categories-amazing requirements and distasteful allegations. If string theory is the correct approach to constructing a fully consistent unification of all fundamental forces, then it should hold the keys to the right answers. In this case, our job is to discern the clever tricks that string theory employs. This may sound like a strange way to approach the problem, but string theory has proved to be a fruitful source of inspiration in the past. Examples range from the discovery of supersymmetry to unexpected anomaly and divergence cancellation mechanisms and much more. It could hold many more surprises in store for us. One "amazing requirement" is perturbative finiteness (or renormalizability). This is not yet fully established, but there is considerable evidence that this is achieved in string theory, even though it is apparently impossible for any point-particle field theory that incorporates general relativity (in four dimensions). A second requirement is that causality have a precise meaning when the space-time metric is a dynamical quantum field. This undoubtedly happens in string theory, but it would be nice to understand in detail just what is involved. Third, the theory should be applicable to the entire universe, perhaps describing it by a single wave function. An obvious question in this connection is whether string theory suggests some special choice of boundary condition, such as that proposed by Hartle and Hawking, and whether this could provide a rationale for selecting a particular vacuum configuration. The second category of conceptual issues consists of certain "distasteful allegations", which string theory might cleverly evade. The first of these is the claim that effects associated with virtual black holes cause pure quantum states to evolve into mixed states.22 If this were true, it would mean that the entire mathematical framework of quantum mechanics is inadequate. It seems reasonable to explore whether

12 string theory could avoid allowing pure states to evolve into mixed states. To the extent that string theory can be consistently formulated as an S matrix theory, it seems almost inevitable that this should work out. A second distasteful allegation is that wormhole contributions to the Euclidean path integral 23 render the parameters of particle physics stochastic. 24 In a previous paper, 2 I referred to this phenomenon as 'the curse of the wormhole,' since it would imply that even when the correct microscopic theory is known, it will still not be possible to compute experimental parameters such as coupling constants, mass ratios, and mixing angles from first principles. A third issue concerns the classification of black holes. According to the "no hair" theorems, in classical general relativity black holes are fully characterized by mass, electric charge, and angular momentum. On the other hand, they have a (large) entropy that is proportional to the area of the event horizon. 25 This amount of entropy corresponds to a number of degrees of freedom that is roughly what one would get from a vibrating membrane just above the horizon. 26 In fact, 't Hooft has tried to make sense of such a physical picture, interpreting the membrane as a string world sheet. 27 Alternatively, the black hole degrees of freedom might be accounted for in string theory in more subtle ways that utilize possibilities for evading the classical no-hair theorems by quantum effects. Recent studies have shown that black holes can have 'quantum hair,' which is observable (in principle) by generalized Bohm-Aharonov interference measurements. 28•29 Charges that can characterize quantum hair for black holes are precisely the same ones whose conservation cannot be destroyed by wormhole effects. Thus the nicest outcome might be for the correct fundamental theory to provide so many different types of quantum hair as to produce precisely the number of degrees of freedom that is required to account for the entropy of black holes. In a theory with enough distinct degrees of freedom to account for black hole entropy and to protect quantum coherence, there should be no deleterious effects due to wormholes. A mechanism that has been proposed as an origin for quantum hair is for a continuous gauge symmetry to break spontaneously leaving a discrete subgroup unbroken. As we have discussed, string theory has a. large group of discrete symmetries that can be understood as remnants of spontaneously broken gauge symmetries, namely the target space dualities. This fact led me to propose that these are the relevant symmetries for understanding quantum hair in string theory. 2 * However, following further studies and discussions with others, it has become clear that this suggestion has serious problems, mostly stemming from the fact that these symmetries are almost all broken for any particular choice of vacuum configuration. The first proposal for "quantum hair" of black holes, detectable only by BohmAharonov-type interference effects, was put forward a few years ago by Bowick et al. 28 As initially formulated, the analysis only applied to theories containing a massless

* This idea has been proposed independently in 'duality of the S field.'

Ref. [30], though the emphasis there is on

13 'axion.' However, a subsequent paper demonstrated that this restriction was not essential and that a suitable massive axion could do the same job. 32 Stripping away all interactions, the basic idea can be explained quite simply. Assume four-dimensional space-time and let Ap be a U(l) vector field and Bp 11 an antisymmetric tensor gauge field (called the axion). In the language of forms, the associated field strengths are given by H = dB and F = dA. The action consists of the usual kinetic terms, schematically given by S~;;,. = I d'x(H 2 + F 2 ), and a topological mass term of the form Smau = m I B /1. F. The equations of motion, d•H = mF and d•F = -mH, imply that both fields have mass m. What happens is that the antisymmetric tensor eats the vector to become massive. In four dimensions it is equivalent to say that the vector eats the scalar (which is dual to Bp 11 ) to become massive.

!

The axion charge in a region of three-dimensional space V with boundary oV is defined by Qaxion

=

I I H =

v

B.

av

For a space-time with nontrivial second homology, such as Schwarzschild space-time (whose topology is S 2 x R 2 ), it is possible to obtain nonzero axion charge while having the H field vanish outside some central region. In this case the B field on the enclosing two-surface is proportional to a two-form that is closed but not exact (i.e., belongs to the second cohomology group). In a theory with axions there are strings (cosmic or fundamental) that contribute a term to the action proportional to Bp 11 dxP /1. dx". As a result, a world sheet enclosing a black hole with axionic charge gives a Bohm-Aharonov phase exp[21riQu;0 ,.]. This makes the charge observable through interference effects (modulo unity). If there were nontrivial third homology, the axion charge itself would be quantized and nothing would be observable. 33 However, this is not the case for a Schwarzschild black hole.

IE

In the string theory context, the formula for the field strength H is embellished by various Chern-Simons terms that were omitted in the discussion above. Also, one-loop effects in ten dimensions give contributions to the effective action of the form I B II. tr(F4 ), which play a crucial role in anomaly cancellation. 34 Upon compactification it can happen that the ten-dimensional gauge fields acquire expectation values that result in a nonvanishing effective term of the form I B /1. F, where F is a U(l) gauge field in four dimensions, as required to give mass to the axion. This happens when the associated U(l) gauge symmetry in four dimensions appears to be anomalous by the usual criteria based on triangle diagrams. However, as in ten dimensions, B,_.11 has nontrivial gauge transformation properties that give compensating contributions and render the quantum theory consistent. Axion charge appears to be a good candidate for quantum hair in string theory. Of course, if this particular charge were the only type of quantum hair in string

14

theory, we would still be very far from achieving the goal of finding enough quantum degrees of freedom to account for all the entropy of black holes and overcoming the other problems in quantum gravity that we have discussed. Fortunately, string theory seems to allow various categories of generalizations of axion charge that could provide many more kinds of quantum hair. For example, the field Bp11 (z,y) is defined in ten dimensions. (Here x refers to four-dimensional space-time and y to six compactified dimensions.) In the usual Kaluza-Klein fashion, this represents an infinite family of four-dimensional fields BL~>(z) corresponding to an expansion in harmonics of the compact space of the form 'ECn(y)BL~(z). The analysis above only utilized the axion corresponding to the leading term in this series for which C(y) is a constant. The other terms describe fields that naturally have masses of the order of the compactification scale. It seems plausible that they could provide additional types of quantum hair. (When the analysis is done carefully, target-space duality may yet prove to be important!) Even this infinite collection of charges may not be the end of the story. The massive string spectrum contains an infinite number of gauge fields of every possible tensor structure. The particular gauge field Bp 11 is special by virtue of its coupling to the string world sheet, which played a crucial role in the reasoning above. Other gauge fields enter the world sheet action with couplings given by their associated vertex operators. For fields that are not massless in ten dimensions these give nonrenormalizable couplings in the sigma model, and are therefore difficult to analyze. Still, from a more general string field theory point of view, they are not really very different, and so there may be many more possibilities for quantum hair associated with the massive string spectrum. In addition to string symmetries altering some consequences of general relativity at the quantum level, it is also possible that special features of string theory play an important role at the classical level. One indication of this appears in a recent study of chargea black holes, 35 where effects of the dilaton field make qualitative changes already at zeroth order in a'. Specifically, whereas a Reissner-Nordstrom black hole of mass M and charge Q has its horizon at the radius Ru = M + M2 - Q2, the corresponding string solution has Ru = 2M y'1 - Q2 /2M2. Also, as the charge of the black hole approaches its maximum allowed value, the entropy S -+ 411' M 2 and the temperature T-+ 0 in the Reissner-Nordstrom case. In the string case one finds S-+ 0 and T-+ 81rM. (All these results are to leading order in a' and li.) As one might expect, the thermodynamic description breaks down in either of these extreme limits. 36

J

5. The Importance of Supersymmetry In order to gain the attention and respect of our experimental colleagues it is important to make predictions that bear on near-term experimental possibilities. It is pretty clear that the best prospect in this regard is supersymmetry. It would be nice if we could honestly assert that supersymmetry (broken at the weak scale) is an inevitable feature of any quasi-realistic string model and thus a necessity if string

15 theory is the correct basis of unification. Experimentalists could then be in a position to demonstrate that "string theory is false" or to discover important evidence in its support. But can we honestly make such an assertion? Certainly no string models that are remotely realistic have been constructed without low-energy supersymmetry. Also, in the context of string theory any alternative mechanism for dealing with the hierarchy problem seems very unlikely. Still, if as is generally assumed, string theory has a clever way of preventing a cosmological constant from arising once supersymmetry is broken, then maybe it could also have a clever stringy alternative for preventing Higgs particles from acquiring unification scale masses through radiative corrections. No plausible alternative to supersymmetry is known, but how sure can we be that one doesn't exist? There is always the possibility that some mechanism, not yet considered, could be important, but that shouldn't completely prevent from us ever sticking our necks out a bit. I don't think it would be dishonest for string theorists to assert that according to our present understanding, supersymmetry broken at the weak scale is required by string theory. In fact, the experimental prospects are beginning to look up. The requirement that the three couplings of the standard model should become equal at a unification scale fails badly without supersymmetry. On the other hand, for a susy scale ranging from 100 GeV to 10 TeV they merge very nicely at about 1016 GeV. 37 While this is far from conclusive, it is a very impressive bit of evidence. Given the present experimental situation, together with various theoretical and astrophysical considerations, it seems quite plausible that the lightest supersymmetry particles are at the low end of this range. Others, such as squarks and gluinos, maybe be around a TeV or so. If this is correct, it is unlikely that any of these particles will be produced and detected before the LHC or SSC comes on line. However, supersymmetry has important implications for the Higgs sector that could be confirmed sooner. Low-energy supersymmetry has two Higgs doublets, which after symmetry breaking result in a charged particle H±, and three neutral particles h, H, and A. The minimal supersymmetric standard model (MSSM) requires, at tree level, that h is the lightest of these and that its mass not exceed Mz. When radiative corrections are taken into account, it can be somewhat heavier, depending on the mass of the top quark. For example, if the top quark mass does not exceed 160 GeV then the h mass should not exceed 120 GeV. For a top quark mass below 130 GeV the bound is lowered to 100 GeV. These bounds need not be saturated, so there is a reasonable chance for a mass in the range 50-100 GeV, making it open to discovery at LEP 2. Another interesting possibility is that the top quark could decay into H+ plus a bottom quark. If it is kinematically allowed, this could be a significant branching fraction. (The precise prediction depends on the parameter tan(J = v 2 jv 1 .) I am optimistic that some of these particles will turn up during this decade and that this will open up an exciting era.for string theorists (as well as all particle physicists).

16

REFERENCES 1. J. H. Schwarz, Int. J. Mod. Phys. A2 (1987) 593-643; Int. J. Mod. Phys. A4 {1989) 2653-2713. 2. J. H. Schwarz, "Target Space Duality and the Curse of the Wormhole," CALT68-1688, to be published in Beyond the Standard Model II, the proceedings of a conference held at the University of Oklahoma (World Scientific 1991). 3. S. Ferrara and A. Strominger, in Strings 89, Proc. Superstring Workshop, Texas A&M Univ., ed. R. Amowitt et al., (World Scientific 1990); P. Candelas and X. C. de Ia Ossa, Nucl. Phys. B355 {1991) 455. 4. W. Lerche, C. Vafa, and N. Warner, Nucl. Phys. B324 (1989) 427. 5. P. Candelas, M. Lynker, and R. Schimmrigk, Nucl. Phys. B341 {1990) 383. 6. B. R. Greene and M. R. Plesser, Nucl. Phys. B338 (1990) 15. 7. P. S. Aspinwall, C. A. Lutken, and G. G. Ross, Phys. Lett. 241B (1990) 373; P. S. Aspinwall and C. A. Lutken, Nucl. Phys. 355 (1991) 482. 8. S. Ferrara, M. Bodner, and A. C. Cadavid, Phys. Lett. 247B (1990) 25; CIGSs. Quant. Grav. 8 (1991) 789. 9. P. Candelas, X. C. de Ia Ossa, P. S. Green, and L. Parkes, Nucl. Phys. B359 (1991) 21; Phys. Lett. 258B {1991) 118. 10. M. B. Green, J. H. Schwarz, and L. Brink, Nucl. Phys. Bl98 (1982) 474. 11. K. Kikkawa and M. Yamasaki, Phys. Lett. 149B (1984) 357; N. Sakai and I. Senda, Prog. Theor. Phys. 75 (1984) 692. 12. E. Alvarez and M. A. R. Osorio, Phys. Rev. D40 (1989) 1150; D. Gross and I. Klebanov, Nucl. Phys. B344 (1990) 475; E. Smith and J. Polchinski, "Duality Survives Time Dependence," Univ. of Texas preprint UTTG-07-91, Jan. 1991; A. A. Tseytlin, Mod. Phys. Lett. A16 (1991) 1721. 13. K. S. Narain, Phys. Lett. 169B (1986) 41; K. S. Narain, M. H. Sarmadi, and E. Witten, Nucl. Phys. B279 (1987) 369. 14. V. P. Nair, A. Shapere, A. Strominger, and F. Wilczek, Nucl. Phys. B287 (1987) 402; A. Shapere and F. Wilczek, Nucl. Phys. B320 (1989) 669; A. Giveon, E. Rabinovici, and G. Veneziano, Nucl. Phys. B322 (1989) 167. 15. S. Ferrara, D. Lust, A. Shapere, and S. Theisen, Phys. Lett. 225B (1989) 363; A. Giveon and M. Porrati, Phys. Lett. 246B (1990) 54; Nucl. Phys. B355 (1991) 422. 16. M. Dine, P. Huet, and N. Seiberg, Nucl. Phys. B322 (1989) 301. 17. N. Seiberg, Nucl. Phys. B303 (1988) 286.

17

18. M. B. Green, J. H. Schwarz, and P. C. West, Nucl. Phys. B254 (1985) 327. 19. A. Font, L. E. Ibanez, D. Liist, and F. Quevedo, Phys. Lett. 245B (1990) 401; S. Ferrara, N. Magnoli, T. R. Taylor, and G. Veneziano, Phys. Lett. 245B (1990) 409; H. P. Nilles and M. Olechowski, Phys. Lett. 248B (1990) 268; P. Binetruy and M. K. Gaillard, Phys. Lett. 253B (1991) 119. 20. M. Cvetic, A. Font, L. E. Ibanez, D. Liist, and F. Quevedo, Nucl. Phys. B361 (1991) 194. 21. S. Ferrara, C. Kounnas, D. Liist, aud F. Zwirner, "Duality-Invariant Partition Functions and Automorphic Superpotentials for (2,2) String Compactifications," preprint CERN-TH.6090/91; D. Liist, "Duality Invariant Effective String Actions and Automorphic Functions for (2,2) String Compactifications," preprint CERN-TH.6143/91. 22. S. W. Hawking, Phys. Rev. D14 (1976) 2460. 23. S. Coleman, Nucl. Phys. B307 (1988) 867; S.B. Giddings and A. Strominger, Nucl. Phys. B307 (1988) 854. 24. S. Coleman, Nucl. Phys. B310 (1988) 643; B. Grinstein and M. Wise, Phys. Lett. 212B (1988} 407; J. Preskill, Nucl. Phys. B323 (1989} 141; S. W. Hawking, Nucl. Phys. B335 (1990) 155. 25. S. W. Hawking, Commun. Math, Phys. 43 (1975) 199; Phys. Rev. 13 (1976) 191. 26. W. H. Zurek and K. S. Thorne, Phys. Rev. Lett. 54 (1985) 2171; K. S. Thorne, R. H. Price, and D. A. Macdonald, Black Holes: The Membrane Paradigm (Yale University Press, 1986). 27. G. 't Hooft, Nucl. Phys. B335 (1990) 138; "The Black Hole Horizon as a Quantum Surface," Nobel Symposium 79, June 1990, and references therein. 28. M. J. Bowick, S. B. Giddings, J. A. Harvey, G. T. Horowitz, and A. Strominger, Phys. Rev. Lett. 61 (1988) 2823. 29. J. Preskill and L. M. Krauss, Nucl. Phys. B341 (1990) 50; J. Preskill, "Quantum Hair," Caltech preprint CALT-68-1671, Nobel Symposium 79, June 1990; S. Coleman, J. Preskill, and F. Wilczek, Mod. Phys. Lett. A6 (1991) 1631. 30. S. Kalara and D. V. Nanopoulos, "String Duality and Black Holes," preprint CTP-TAMU-14/91, March 1991. 31. J. H. Schwarz, "Can String Theory Overcome Deep Problems in Quantum Gravity?" preprint CALT-68-1728, May 1991. 32. T. J. Allen, M. J. Bowick, and A. Lahiri, Phys. Lett. 237B (1990) 47. 33. R. Rohm and E. Witten, Ann. Phys. (N.Y.) 170 (1986) 454. C. Teitelboim, Phys. Lett. 167B (1986) 69.

18

34. M. B. Green and J. H. Schwarz, Phys. Lett. 1498 (1984) 117. 35. D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D43 (1991) 3140; see also G. Gibbons, Nucl. Phys. 8207 {1982) 337 and G. Gibbons and A. Maeda, Nucl. Phys. 8298 (1988) 741. 36. J. Preskill, P. Schwarz, A. Shapere, S. Trivedi, and F. Wilczek, "Limitation on the Statistical Description of Black Holes," preprint IASSNS-HEP-91-34. 37. J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. 2498 (1990) 441 and 2608 (1991) 131; U. Amaldi, W. de Boer, and H. Fiirstenau, Phys. Lett. 2608 (1991) 447; P. Langacker and M.-X. Luo, preprint UPR-0466T (1991).

19

SIGMA MODEL ANOMALIES, NON-HARMONIC COUPLINGS AND STRING THEORY Gabriel Lopes Cardoso and Burt A. Ovrut 1 Department of Physics University of Pennsylvania Philadelphia, PA 19104-6396 ABSTRACT We discuss the one-loop sigma model anomalies, both gauge and gravitational, in the effective field theories of (2,2) symmetric ZN orbifolds. By comparison with explicit superstring calculations, we show that the universal part of these anomalies must be removed by a sigma model generalization of the Green-Schwarz mechanism. The non-universal part, when it occurs, is not cancelled by such a mechanism and leads to non-harmonic gauge and gravitational couplings. In several recent papers, the existence of non-vanishing sigma model anomalies, which violate both Kahler and sigma model coordinate invariance, has been demonstrated 1 •2•3 in a large class of four-dimensional field theories associated with the low energy limit of superstring theory. Furthermore, it was shownl.2 how to modify such theories so as to cancel the universal part of these anomalies. This is accomplished via the sigma model, four-dimensional analog of the Green-Schwarz mechanism using the dilaton, dilatino, axion linear supermultiplet. The non-universal part of the sigma model anomalies, however, can not be cancelled by this mechanism unless one introduces several linear multiplets; an unappealing extension of these theories. As first noticed in reference 2, these sigma model anomalies lead to non-harmonic contributions to the gauge couplings 4 • It is important, therefore, to know which, if any, of these anomalies is cancelled by the Green-Schwarz mechanism. This issue can be resolved, however, by comparing the field theory results with explicit string theory calculations of amplitudes related to both the non-universal4 and universal 6 parts of these anomalies. Such calculations have recently been carried out for (2,2) symmetric ZN orbifolds. In this paper, we resolve this controversy for the (2,2) symmetric ZN orbifolds. The Z3 case is discussed in detail. For Z3 orbifolds, we show that the anomaly is entirely universal and must, to be consistent with explicit string theory calculations', be lWork supported in part by the Department of Energy under Conbad No. DOE-AC02-76-ER03071, and NATO Grant No. 860684.

20 completely cancelled using the Green-Schwarz mechanism associated with the single supermultiplet containing the dilaton, dilatino and axion. The same conclusion is reached for the Z1 orbifold. Finally, we discuss the cases of ZN with N :/= 3, 7. We emphasize that only in reference 1 and in this paper are the supergravity contributions to the sigma model anomalies, in addition to the gauge field contributions, included in the derivation. The tree level Kahler potential associated with the massless chiral superfields of the z3 orbifold has been computed8 •7 , and is given by

K =

-ln det ( b;:;

+ b;; - 24>~1: ~E) + (det (b + ;;)

+ (det (b +

b)i,r

213

i,r911 Yi~ Yi1

c; c;

{1)

where qn = ~ and q12 = q13 = ~· The first term on the right hand side is valid to quadratic order only in q,;E . It is well known8that, in component fields, there are two composite connections. The first is the composite connection, a,., which transforms in such a way as to insure invariance under Kahler transformations. The second composite connection, r ,.., is the Christoffel connection associated with the Kahler metric. It transforms in such a way as to insure invariance under general coordinate transformations on the complex Kahler manifold. It is of interest to ask whether there are sigma model anomalies, Kahler and sigma model coordinate anomalies respectively, associated with these connections. One-loop supergraphs yield two non-local contributions to the effective action. The first is associated with two external gauge superfields and is a function of the gauge superfield strengths W,?". The second non-local term is associated with two external supergravity fields and is a function of supergravity superfield Wap.,.. The complete non-local one-loop effective action, r anom, is given by fanom

=

4(4l1r) 2

Jcl'xd"O [30g~1 (w:•")

2

+ 30g;u(3) (w;u( 3)"f

+ 30gi;, (w.;:•"r + 470(Wap.,.) 2 ] ~D 2 KM +he where KM

= -In det ( b +b);)'

(2)

Note that all terms are proportional to the universal

factor KM. Under a Kahler transformation 6KKM = F+F, where F is a holomorphic function of chiral superfields. It is easy to check that 6K r anom :/= o. Therefore, r anom is a non-local term that violates Kahler invariance and, hence, there is a Kahler anomaly at the one-loop level. Let us now Taylor expand KM around the vacuum expectation values of the moduli. That is, b;; = (b;;} + b:;. where b:; are chiral superfields. r anom can then be written, in terms of component fields, as

21

!

+30gi, (F:•)'- 2 5(C,_p)2] A(b,b) _ _ i_ [aOg2 (FEe• jEe•"") 2( 411' )2 Be IW

+ 309SU(3) 2 (FSU(a)o jSU(a)o,..,) ,..,

!

+30gi. (F:•P 8 •.,..,)- 2 5 (R,_pRaP"")] (9•1jb~;+9,,j6h) + ...}

(3)

where (4) Note that A is not harmonic. Related to this one notices, using (4), that (5) and, hence, the 9 equations in (4) are not integrable. It is of interest to ask how this result compares with direct calculations in string theory. In a recent paper6 , it has been shown that for the (2,2) symmetric Z3 orbifold an explicit string calculation yields (6) It follows that KM can be, at most, a constant. This result is incompatible with having K M = -In det ( b + b) C. How can one reconcile the effective field theory and the direct string theory result? . As discussed in references 1 and 2, the answer is to add a real linear supermultiplet, i, to the theory whose field strength is modified to contain E 8 , SU(3), Es and Lorentz group Chem-Simons terms. To the lowest non-trivial order in background supergravity fields, i satisfies9

-z-l = -8 """ D LJ Tg ( waH•)2 -

8TL

(Walh) 2

(7)

H

where Tg and TL are arbitrary real constants whose values will, essentially, be determined below. It is also necessary to include in the theory the four-dimensional analog of the ten-dimensional Green-Schwarz term. This term, to lowest order in the supergravity fields, is given by

(8) where (3 is an arbitrary real constant. As was first discussed in reference 1, the reducible tree level graph , which is constructed from one vertex associated with (7) and the second vertex with (8), will exactly cancel r anom given in (2) as long as we choose

22

/3TE1

=

/3TsU(3)

=

30

/3TEe

= 4( 41r) 2 gi_,

(9)

Under these circumstances, the sigma model anomalies of the one-loop theory are exactly cancelled, and the field theory predicts ~(b,li) =

o, e.= e, = o

(10)

which is consistent with the exact string theory result. We conclude that for the (2,2) symmetric Z3 orbifold, the effective field theory must have a Green-Schwarz mechanism associated with gauge and gravitational sigma model anomalies. It is essential to realize that the Green-Schwarz mechanism in Z 3 is possible precisely because the sigma model anomalies are "universal". We have shown that the situation is identical in the orbifold. For ZN with N =/: 3, 7, the anomalies contain both a universal and a non-universal part and, hence, cannot be completely cancelled by a Green-Schwarz mechanism involving a single linear multiplet. In fact, the explicit string calculations in reference 4 show that for ZN orbifolds where N =/: 3, 7 there are, in fact, non-vanishing, nonharmonic moduli dependent gauge couplings that are associated with non-vanishing, non-universal sigma model anomalies in the effective field theory. Hence, the universal part of these anomalies must be cancelled using the Green-Schwarz mechanism, as it was in the Z3 and Z 7 case. However, the non-universal part of the anomalies is not cancelled. These non-universal anomalies break Kahler and sigma model coordinate invariance and, as a. consequence, target space duality symmetry. However, it has been shown that superstring amplitudes and, hence, the associated effective field theory must indeed be duality invariant. How can one reconcile the two results? The answer was given in reference 3 where it was shown, within the context of orbifolds with gauge group E 8 x U(1) 2 x E 8 , that duality invariance can be restored by adding the appropriate counterterms constructed from Dedekind functions of the moduli superfields. Such counterterms exist4 for all (2,2) symmetric ZN orbifolds when N =/: 3, 7.

z7

Acknowledgements: We would like to thankS. Ferrara, T. Taylor and V. Ka.plunovsky for informative discussions. We are indebted to Jan Louis for correcting a mistake in our Kahler potential and other helpful remarks. References 1. G.L. Cardoso and B.A. Ovrut, University of Pennsylvania preprint UPR-0464T.

2. J.P. Derendinger, S. Ferrara., C. Kounnas, and F. Zwirner, CERN preprint TH.6004/91.

23

3. J. Louis, SLAC preprint PUB-5527. 4. L.J. Dixon, V.S. Kaplunovsky, and J. Louis SLAC preprint PUB-5138. 5. I. Antoniadis, K.S. Narain, and T.R. Taylor, preprint NUB-3025, IC/91/124, CPTH-A050.0491. 6. M. Cvetic, J. Louis, and B.A. Ovrut, Phys. Lett. B206 (1988) 227; M. Cvetic, J. Molera, and B.A. Ovrut, Phys. Rev. D40 (1989), 1140; L.J. Dixon, V.S. Kaplunovsky, and J. Louis, Nucl. Phys. B329 (1990) 27. 7. J. Louis, private communication 8. B. Zumino, Phys. Lett. B87 (1979) 203; E. Witten and J. Bagger, Phys. Lett Bll5 (1982) 202; J. Bagger, Nucl. Phys. B211 (1983) 302. 9. P. Binetruy, G. Girardi, R. Grimm, and M. Miiller, Phys. Lett. Bl95 (1987) 389; B.A. Ovrut and S.K. Rama, Nucl. Phys. B333 (1990) 380; University of Pennsylvania preprint UPR-0433T.

24

THE COMPLETE ACTION OF CHIRAL D=lO, N=2 SUPERGRAVITY

Leonardo Castellani Istituto Nazionale di Fisica Nucleare, Sezione di Torino Via P. Giuria 1, I-10125 Torino, Italy and

Igor Pesando Dipartimento di Fisica Teozica, UniversitB. di Torino Via P. Giuria 1, I-10125 Torino, Italy

Abstract We construct the complete superspace action l of chiral D = 10, N = 2 supergravity. The selfduality condition on the field strength F 41 - 41 is derived as an equation of motion in superspace. The spacetime restriction of l is not supersymmetric, since the selfduality condition does not foUow from the spacetime variational equations. However, supersymmetry can be restored via the introduction of a lagrangn multiplier term.

Chiral D=10, N=2 supergravity (2b supergravity for short) is an interesting theory in many respects. Like D=ll supergravity, it has maximal supersymmetry, it cannot be derived from a higher dimensional theory, and cannot be coupled to supermatter. Unlike D=ll supergravity, it is a limiting caore of a superstring theory (type liB). The liB superstring has intriguing features that make it simultaneously discouraging and attractive: the absence of gauge fields in the ten dimensional theory implies a low rank gauge group G in the compactified D=4 theory, in contradistinction to the high rank gauge groups typical of D=4 heterotic models. In some compactifications the fermions fall into chiral representations of G. On the other hand, some general arguments seem to prevent G to be (or to contain) the whole SU(3) ® SU(2) ® U(1) phenomenological group [1) (see however ref. (2)). Even accepting these no-go theorems, type II superstrings still deserve their share of attention: indeed a canonical map, called the l1-map in ref.s (3), relates two different modular invariant heterotic models to every consistent modular invariant type II superstring theory. As emphasized in (3b), the h-map is the analogue, at the level of two-dimensional conformal field theories, of the spin connection embedding into the gauge connection coupled to the heterotic fermions. The massless theory limit, i.e. 2b super1~avity, is the object of the present paper, which summarizes the results of ref. (4). This theory has been shown to be anomaly-free in ref. !5), and to be (probably) related to interesting gauged D = 4 supergravities with vanishing cosmological constant discussed in ref.s (6).

25 The field equations of 2b supergravity have been derived in ref.s (7), essentially by requiring closure of the supersymmetry algebra. In ref. [8) we have rederived them in a geometric framework (9) based on a particular free differential algebra (for reviews on free differential algebras, and their use in supergravity theories containing antisymmetric tensors, see ref.s (9,10)). This geometric framework provides an algorithmic way to construct lagrangians. We have applied this algorithm and constructed the complete lagrangian for 2b supergravity, so far missing in the literature. Partial results, up to quadratic order in the fermi fields, had been previously obtained in ref.s (11). An improved version of SUPERGRAV (12), a symbolic manipulation package for Bose-Fermi exterior calculus (and tailored to implement the above algorithm), has been used in most of the calculations. What we obtain is actually a superspace lagrangian, whose variational equations, restricted to spacetime, correctly reproduce all the field equations derived in ref.s (7], and include the self-duality condition on the field strength of the 4-rank antisymmetric tensor. This situation is similar to what happens in the D = 6 supergravity of ref. (13). The self-duality condition, necessary to matcl1 on-shell bosonic and fermionic degrees of freedom, can be obtained by varying the superspace lagrangian C, but not its spacetime restriction. In other words, the spacetime restriction of our C is not supersymmetric. However, it can be made supersymmetric by introducing an additional term containing a lagrange multiplier (this has been discussed in (14) in the context of theories with selfdual field strengths). The theory contains a complex anti-Weyl gravitino t/J,. and a complex Weyl spinor .A. The bosonic fields are: the graviton g1111 , a complex antisymmetric tensor A,.,, a real antisymmetric tensor B 1111 p, (restricted by a self-duality condition) and a complex scalar t/J. There is a global U(1) symmetry that rotates the two supersymmetry charges into each other. Scalar fields in supergravity theories are interpreted as coordinates of noncompact coset spaces. Here the complex scalar t/J can be seen as parametrizing the coset SU(1, 1)/U(1) (7). As in ref.s (7), a real auxiliary field a is added to the complex scalar t/J, and both are combined into a SU(l, 1) group matrix:

( v~ v~

vlV

1)

= exp (

ia

t/J*

t/J )

-ia

,

EofJ

V~V:

= 1.

(1)

The U(1) becomes then a local symmetry, compensating for the extra field a. Indices a, /3 refer to SU(1, 1), subscripts ± to U(1) charges ±1. It is moreover convenient to define the SU(l, 1) invariant quantities:

= -ieapV~a,.v_e, P,. = -eaplt'.fo,.v{

Q,.

(2a) (2b)

Q,. transforms as a U(1) gauge field under U(1), and is used as U(1) connection in covariant derivatives. As discussed in ref.[8), the algebraic structure of the theory is encoded in a particular free differential algebra (FDA), which extends the Cartan-Maurer equations of the chiral

=

=

N 2 super-Poincare Lie algebra in ·n 10. The algorithm described in ref.s (9] for the gauging of this FDA yields the following Lagrangian (the fields are written as forms and we have omitted wedge symbols):

with

C.., =R"•"•V"•-"••e .. ,-.. ,.

"v -336 i -

41

1 R"·T.r ~R";\.r 1111 •"•.XV"•-•••~~;~oot-«to '¥' ot-oa •'¥'·V"•-"•- 4

1 CA =(-84i-T.rll·-"·· 'I' 'I'··v··-"·vfle + a {J- 2..t.r"·-"·.XV"·-·••e llt -1110 vfle - a {J+ 12 'I'

+ ea{J + c.c. ) R(A)"' + F fJ, yfl• e(J1 (J2 v••-••• ea -a 111 yfJ 1 Cs =( -560~r"•-"•t/JV"•-"•- : F"•-••v••-"••e .. ,-.. ,.+ 111 _ 118

_

1

+ :o>.r"•-"•.XV"•-"•) X (R(B) + ~ ea{JA"'R(A)fl)- 210ieafJdA"'dAfJ B

c Fa --120e{J,{J, 1 yflt - p,fl• b,-b.efJafJ• yfJa + p,fl• bt-ba v••-"•• e .. ,-.. ,. + 1 (e{J,{J, vfl•p,fl• + 12 - ...... .T.r•·•·' 'I' " + c.c. )V"•-"•• eu,-.... "

~wF,

" lwP+P-

(4)

1 Flb, -•• Flb, -bo V"•-llto e .. , -a,o =108 =902 P"''V"' a+ + ea1a P"'"V"'• 4__ ea a V"•-"•• e111 - 410 2

2

8

4

Cv: =~r .. ,,. • .X*(VV.f'- V!'~*.X)V.f'ep,fJ2 V"•-" 10 ea,- .. ,.2(VVfJ' -9 +

Cw

- yfl•.T.• - 'I' "')VfJ• + e{J 1 {J 2 pfla 41 _ yfl• efJ8 fJ4 V"•-"•• e41 _410

+ c.c.

= ~ ~r"'-""Vt/JV"•-"••ea,- 410 + c.c

" i 'r"'V'V"•-"•• e111 ~w.x=g"

"

=-

1110 +c.c

+ c.c " _ ~-t.r•·'· ~-t.r•"'"'·'· lw(1,1) -32 'I' 'I' >.r•"•"• .XV"•-.... e llt-1110 + 32 'I' 'I' >.r• .XV""-"'" e lit-litO

£(2,2)

42~*r 6 t{J~r"• -"•t!J* V 6V"•-••

- 1~r"' -"•t{J>.r"•-•• .XV"•-•• 2

21 ~r·•-"•t{J>.r••-•• .XV"'-... 2

L is invariant under the supersymmetry variations:

c5V" =- 2Im(ET"T/I)

(5) c.J. "" O'f'=vE+

5 ·r..,-... E V"•("" rca 1 -ca 1 96 z

+ SlE 1

41 _ 410

F"·-···

+ 9r••"•e•v••)va + .l(-r••-"•E*V"' 32 + FfJ ••-"• ea{J + 2.(r••>..I[Jr"•e•- - 1-r••-••>..I[Jrh-••e*) 32

1680

+ - 1-i( -3r" 142 EXr"• >. V" 1

+ 21EXr"• >. V"') 128 - _!_i(-r••-••eXr 256 ... - ... >.V"' + 5r"•••eXr"•-"• >. v••)

+ - 1-ir•·-··EXr·•-..•>..v.. •)

3072 Full details on the derivation of the lagrangian (4) can be found in re£.[4]. References

[1] L. Dixon, V. Kaplunovski and C. Vafa, Nucl. Phys. B294 (1987) 43. [2) L. Castellani, Phys. Lett. B245 (1990) 417. [3a] W. Lerche, D. Liist and A.N. Schellekens, Nucl. Phys. B287 (1987) 477 and Phys. Lett. B187 (1987) 45. F. Englert, H. Nicolai and A.N. Schellekcms, Nucl. Phys. B274 (1986) 315. [3b) D. Gepner, Nucl. Phys. B296 (1988) 757; Phys. Lett. B199 (1987) 380; Trieste lectures at the Superstring school 1989. [4] L. Castellani and I. Pesando, "Chiral D=lO, N=2 Supergravity on the group manifold (II). The complete action", Torino preprint DFTT-13/91.

28 [5) L. Alvarez-Gaume and E. Witten, Nucl. Phys. B234 {1983) 269. [6) M. Giinaydin, L.J. Romans and N.P. Warner, Phys. Lett. 164B {1985) 309. (7a) M.B. Green and J.H Schwarz, Phys. Lett. 109B {1982) 444; 122B {1983) 43; J.H. Schwarz and P.C. West, Phys. Lett. 126B {1983) 301; J.H. Schwarz, Nucl. Phys. B226 {1983) 269. [7b) P. Howe and P.C. West, Nucl. Phys. B238 {1984) 181. (8) L. Castellani, Nucl. Phys. B294 {1987) 877. [9) L. Castellani, R. D'Auria and P. Fre, "Seven Lectures on the group manifold approach to supergravity and the spontaneous compactification of extra dimensions", Proc. XIX Winter School Karpacz 1983, ed. B. Milewski (World Scientific, Singapore); L. Castellani, R. D'Auria and P. Fre, "Supergravity and Superstrings: a geometric perspective", World Scientific {1991), Singapore. [10) D. Sullivan, Infinitesimal computations in topology, Bull. deL' Institut des Hautes Etudes Scientifiques, Pub!. Math. 47 {1977). R. D' Auria and P. Fre, Nucl. Phys. B201 (1982) 101. L. Castellani, P. Fre, F. Giani, K. Pilch and P. van Nieuwenhuizen, Ann. Phys. 146 {1983) 35. P. van Nieuwenhuizen, Free graded differential algebras in: Group theoretical methods in physics, Lect. Notes in Phys. 180 (Springer, Berlin, 1983). [11) A.R. Kavalov and R.L. Mkrtchyan, Ere''8.1l preprints EPHI-938{89)-86 and EPHI963 (13)-87. [12) L. Castellani, Int. Jou. Mod. Phys. A6 (1988) 1435. (13) R. D' Auria, P. Fre and T. Regge, Phys. Lett. B128 (1983) 44. (14) W. Siegel, Nucl. Phys. B238 (1984) 307.

29

Topological Defects in the Moduli Sector of String Theory*

MII.JAK CVETIC Dep~~rtment of Plar•ie•

Unioer•itr of PeramrlNraio

Plailadelplaio, PA

1110-1-1~11

ABSTRACT We point out that the moduli aector of the (2, 2) striq compadification with its nonperturbatively preserved non-compact s:ymmetries is a fertile framework to study global topological defects, thus providing a natural source for the large seale structure formation. Baaed on the target space modular invariance of the nonperturbative superpotential of the four-dimensional N 1 supersymmetric striq vacua, topologically stable stringy domain walla are found. They are supenymmetric solutions, thus saturating the Bogomolnyi bound. It is also shoWD that there are moduli aecton that allow for the global monopoletype and texture-type configurations whose radial stability is ensured by higher derivative terms.

=

Topological defects occur during the spontaneous break-dOWD of ga\lle symmetries, u a consequence of the nontrivial homotopy group n. of the vacuum manifolds. Their existence has important cosmological consequences. In particular global topological defects, • 12 u u like textures ' and more recently, global monopoles ' u well u global n2 textures ' were proposed u a source of large scale structure formation. On the other hand, in the framework of grand-unified theoria (or theories beyond the standard model) it is often unnatural to impose global non-Abelian symmetries that would ensure the existence of such global topological defects. Here we would like to point out that in the string theory, the moduli sector of (2, 2) string compactification provides a natural framework for such global defects, with its potentially important physical implications.

30 In (2, 2) string compactifications, where (2, 2) stands for N = 2 left-moving as well as N 2 right-moving world-sheet supersymmetry, there are massless fields - moduli M which have no potential, i.e. V(M) 0, to all orders in string loops? Thus perturbatively there is a large degeneracy of string vacua, since any vacuum expectation value of moduli corresponds to the vacuum solution. On the other hand it is known that nonperturbative stringy effects like gaugino condensation 8 and axionic string instantons' give size to the nonperturbative superpotential. In the case of the modulus T associated with the internal size of the compactified space for the so-called flat background compactifications (e.g., orbifolds, self-dual lattice constructions, fermionic constructions) the generalized target space duality is characterized by noncompact discrete group PSL(2,Z) = SL(2,Z)/Z2 specified by T-+ :,).~~with a, b, c, d E Z and ad - be = 1. H one assumes that the generalized target space duality is preserved even nonperturbatively:o,n the form of the nonperturbative superpotential is very restrictive! 1 The fact that this is an exact symmeiry of string theory even at the level of nonperturbative effects is supported by genus-one threshold calculations: 2•u which in turn specify the form of the gaugino condensate! 4 ' 15 This phenomenon has intriguing physical implications leading to the stable supersymmetric domain walls!' This phyaica of modulus Tis actually a generalization of the well known axion phyaica 17 introduced to solve the strong CP problem in QCD. Spontaneously broken global U(1) Peccei-Quinn symmetry is non-linearly realized through a paeudo-Goldstone boson, the invisible axion 9. Nonperturbative QCD effects through the axial anomaly break explicitly U(1) symmetry down to ZN1 , by generating an effective potential proportional to 1- cosN/9. This potential leads to domain wall solutions 18 with N1 walls meeting at the axionic strings! 7 As an instructive example let's first consider a global supersymmetric theory by with

=

=

(1)

=

Here, Grr [)z.IJ.t.K(T,T) is the positive definite metric on the complex modulus space and the superpotential, W, is a rational polynomial P(j(T)) ofthe modular-invariant functionj(T)1' i.e. amodularinvariantformofPSL(2,Z). The potential V aTrl[)z.W(T)I 2 = aTri8;P(j)f)zoj(T)I 2 has at least two isolated zeros at T = 1 and T = p eiff/f in the fundamental domain 'D for T, i.e. when l{)zoj(T)1 2 = O.tt Other isolated degenerate minima might as well arise when I8;P(j)l 2 = 0.

=

=

Then, the mass per unit area of the domain wall can be written as: 20 00

j

p.=

dzGni8..,T-eiiGTriJ.t.W(T)I 2 +2Re(e- 11 AW)

(2)

-oo

where AW:: W(T(z

= oo))- W(T(z = -oo)).

The arbitrary phase 9 has to be chosen

31

e'

such that 9 = .1.W/I.1.WI, thus maximizing the cross term in Eq. (2). Then, we find K = 2I.1.WI, where K denotes the kink number. Since is analytic in T, the line integral over T is path independent as for a conservative force. The minimum is obtained only if the Bogomolnyi bound 8,.T(z) = aTf e198j.W(T(z)) is saturated. In this case 8,.W(T(z)) =oTt ei91cn-W(T(z))l 2, which implies that the phase of 8,.W does not change with z. Thus, the supersymmetric domain wall is a mapping from the z-a.xis (-oo,oo] to a straight line connecting between two degenerate vacua in the W-plane. We would like to emphasize that this result is general; it applies to any globally supersymmetric theory with disconnected degenerate minima that preserve supersymmetry. For the superpotential, e.g. W(T) = (a')- 312j(T) the potential has two isolated degenerate minima at T = 1 and T = p e'"/ 6 • At these fixed points, j(T = p) = 0 and j(T = 1) = 1728. Therefore, the mass per unit area is I"= 2 x 1728(a')- 312. Other cases 21 can be worked out analogously.

cn-w

I'~

=

=

The case with gravity restored has a Kihler potential K -3log(T + T) and the superpotential should transform as a weight -3 modular function under modular transformations. 22 ' 11 The most general choice, nonsingular everywhere in the fundamental domain 'D, is

W,.,,.(T)

= H~;~;),

H,.,,.:: (i(T) -1728)"'12 • j"13(T)P(i(T)), m,n

= R+

(3)

Here, 77(T} is the Dedekind eta function, a modular form of weight 1/2 and P(i(T)) is an arbitrary polynomial of j(T). The potential is of the following form:

V,.,.(T,T)

3IHI 2

(T + T)

en-a

3 •

= (T+ T)3177112(1-3-(H + 2'11"02)1 2 -1)

(4)

where G2 = -4'11"cn-77/77- 2'11"/(T + T). In general the scalar potential (4) has an anti-de Sitter minimum with broken supersymmetry?1 However, one can see that form ~ 2, n ~ 2 and P(i) = 1, the potential is semi-positive definite with the two isolated minima at T = 1 and T = p with it unbroken local supersymmetry just like in the global supersymmetric case. We now minimize the domain wall mass density. By the planar symmetry, the most general ltatic Ansatz for the metric 24 is 4. 2 = A(lzl)(-dt2 + dz 2 ) + B(lzl)(dz2 + dy2 ) in which the domain wall is oriented parallel to (z,y) plane. Using the supersymmetry transformation laws

i 1 = (Vp(w)- 2Im(GTVpT)]ta + 2(o-pl)ae 0 , 1 Tf 5xa = 2(11~'l)aVpT-eTG Gtta Q

5'1/Jpa

G

(5)

with commuting, covariantly constant, chiralspinors t:t 1 the ADM mass density I" can be

32 expressed as

25

(6) The i,j indices are for spatial directions. The minimum of the Bogomolnyi bound is achieved if Eq.(5) vanish. Again, the stringy domain wall is stabilized by the topological kink number. Unfortunately, the nice holomorphie structure of the scalar potential is lost. In other words, there is now a bolomorpbie anomaly in the scalar potential due to the supergravity coupling. This implies that the path connecting two degenerate vacua in superpotential space is not a straight line. In fact, one can understand the motion as a geode1ie path in a nontrivial Kibler metric, thus in G(T,T). One can show (numerically) that in our example the path along the circleT= expi8(z) with 8 = (0,'11'/6), i.e., the self-dual line ofT--+ 1/T modular transformation, is the geodesic path connecting between T = 1 and T p in the scalar potential spa.Ce. Thus, we have again established an existence of stable domain walls. The superpotential is quite complicated, however, the numerical solution in can be obtained. 21

=

It is interesting to note that stringy cosmic strings 11 can be viewed as boundaries of our domain walls. Because the domain wall number is two, the intersection of two such domain walls is precisely the line of stringy cosmic strings. On the other hand such stable domain walls are disastrous from the cosmological point of view. One possible solution to this problem is that after supersymmetry breaking, the degeneracy of the two minima is lifted. In that ease, the domain wall becomes unstable via the false vacuum decay. 28 We would now like to point out 27 the existence of other global topological defects, like global monopole-type and texture-type defects in the moduli sector of string theory. Such defects could exist in the study of the symmmetry structure of the effective theory when there are more than one modulus (which of course is a generic situation). We shall illustrate the idea using examples based on the so called flat backgrounds, i.e. generalization of SL(2,Z). For that purpose we shall study the simplest example of Zt manifold with continuous symmetry SU(2,2) on the four moduli

T= [Tu

T2l

(7)

of compactified space. Note that the moduli T live on the coset SU(2, 2)/SU(2) x SU(2) x U(1). The continuous non-compact symmetry SU(2,2) is an ezact symmetry 28 at least at 11 It is intrigaiug that the pre~e~~t kink solito1111 also appear in integrable, supersymme&ric two-dime1111ional N 2 Landau-Gillllburs models~

=

33 the string tree-level. Note that this continuous symmetry in the modulus could be broken down to the discrete subgroup 8U(2,2, Z) due to nonperturbative effects, e.g. gaugino condensation and/or axionic instanton effects. However, at this point we shall stick to the continuous symmetry. For the time being we shall keep in mind that SU(2,2,Z) is the vacuum symmetry and thus the T fields should live in the fundamental domain of 8U(2,2,Z). The maximal comapct symmetry of SU(2, 2) is 8U(2)A x SU(2)B x U(1) C SU(2)A+B· Note also that in projective coordinates?' Z = (1- T)/(1 + T). Z transforms as 1 + 3 under 8U(2)A+B· The ansatz Z E!=l u .. v.. with v.. = l(r)z01 /r ensures the map of Z on the S 2 •

=

Let us concentrate now on the Lagrangian for the Z field and thus a specific solution for l(r). Note that the Z fields have no potential to all orders in string loops. Thus the kinetic energy term 28 shrinks I ~ 0 due to Derrick's theorem and thus should be stabilized by higher derivative terms. Such higher derivative terms arise even at the tree level of the string theory. They should respect the noncompact SU(2,2) symmetry. Also, if one sticks to terms with at most two time derivatives, one has a unique form for the terms that involve four derivatives, which is very similar in nature to the Skyrme term 30 in the Skyrmion model and can serve the same role as the stabilizing term. In this case I = Or as r ~ 0 and I = D/r2 as r ~ oo The energy stored in such a configuration is finite. This is different from the standard global monopole configuration~ which has I ~ lo as r ~ oo which has linearly divergent energy and thus long range interaction relevant for large scale formation. Another interesting observation would be to study the texture-type configurations, which have a chance to occur within this sector. Namely, the Z fields transform as 4 under the compact symmetry SU(2)A X SU(2)s,..., 80(4) and thus the ansatz: Z = a(r)+ b( r) E!=l u01 z 01 /r is mapped onto 8 3 • The potential problem in this case is an impossibility to ensure a 2 (r)+b2 (r) = j2 where I is a constant. Interestingly, {a(r),b(r)} ~ 0 as r ~ oo and thus the knot configuration disappears at large distances. The above studied configurations are much milder defects than strings and domain walls and they finite range and thus finite energy. Further study of cosmological implications of such global defects is under consideration. I would like to thank my collaborators S. Griffies, F. Quevedo, and S.-J. Rey, for many fruitful discussions and enjoyable collaborations. I would also like to thank the Aspen Center for Physics, the International Centre for Theoretical Physics, Trieste, and CERN for their hospitality. The work is supported in part by the U.S. DOE Grant DE-22418-281 and by a grant from University of Pennsylvania Research Foundation and by the NATO Research Grant #90o-700.

34

REFERENCES 1. R. Davis, Gen. Rei. Grav. 19, 331 (1987). 2. N. Turok, Phys. Rev. Lett. 63, 2625 {1989). 3. M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341 {1989). 4. D. Bennett and S. Rhie, Phys. Rev. Lett. 65, 1709 (1990). 5. L. Perivolaropoulos, BROWN-HET-775 (November 1990). 6. D. Bennett and S. Rhie, UCRL-IC-104061 (June 1990). 7. M. Dine and N. Seiberg, Nucl. Phys. B301, 357 {1988). 8. J. P. Derendinger, L. E. Ibanez, and H. P. Nilles, Phys. Lett. USB, 65 {1985); M. Dine, R. Rohm, N. Seiberg, and E. Witten, Phys. Lett. 156B, 55 (1985). 9. S.-J. Rey, Azionic String In.tantom and Their Low-Energ11 lmplicatiom, Invited Talk at Tuscaloosa Workshop on Particle Theory and Superstrings, ed. L. Clavelli and B. Harm, World Scientific Pub., (November, 1989); Phys. Rev. D 43, 526 (1991). 10. S. Ferrara, D. Liist, A. Shapere, and S. Theisen, Phys. Lett 225B, 363 {1989). 11. M. Cvetic, A. Font, L. E. Ibanez, D. Liist, and F. Quevedo, Nucl. Phys. B361 (1991) 194. 12. V. Kaplunovsky, Nucl. Phys. B30'1, 145 (1988). 13. L. Dixon, V. Kaplunovsky, and J. Louis, Nucl. Phys. B355, 649 (1991); J. Louis, SLAC-PUB-5527 (April1991). 14. A. Font, L. E. Ibanez, D. Liist, and F. Quevedo, Phys. Lett. 245B, 401 (1990); S. Ferrara, N. Magnoli, T. R. Taylor, and G. Veneziano, Phys. Lett. 24GB, 409 (1990); P. Binetruy and M. K. Gaillard, Phys. Lett. 253B, 119 (1991). 15. H. P. Nilles and M. Olechowski, Phys. Lett. 248B, 268 (1990). 16. M. Cvetic, S. J. Rey and F. Quevedo,"Stringy Domain Walls and Target Space Duality", UPR-Q445-T (April1991) to appear in Phys. Rev. Lett. 17. J. E. Kim, Phys. Rev. Lett. 43, 103 (1979); M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. 104B, 199 (1981); and Nucl. Phys. B189, 575 (1981); M. B. Wise, H. Georgi, and S. L. Glashow, Phys. Rev. Lett. 4'1, 402 (1981); A good review is by J. E. Kim, Phys. Rep. 150, 1 (1987). 18. P. Sikivie, Phys. Rev. Lett. 48, 1156 (1982); G. Lazarides and Q. Shaft, Phys. Lett.USB (1982) 21; For a review, see A. Vilenkin, Phys. Rep. 121, 263 (1985). 19. B. Schoeneberg, Elliptic Modular .Functions, Springer, Berlin-Heidelberg (1970); J. Lehner, Discontinuous Groups and Automorphic Functions, ed. by the American Mathematical Society, (1964). 20. P. Fendley, S. Mathur, C. Vafa and N.P. Warner, Phys. Lett. B243(1990) 257.

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36 HOMOGENEOUS KAHLER AND QUATERNIONIC MANIFOLDS IN N=2 SUPERGRAVITY

B. de Wit 1 Theory Division, CERN CH-1211, Geneva 23, Switzerland ABSTRACT Motivated by the study of string compactifications we discuss the implications of special geometry for the symmetry structure of Kahler and quatemionic manifolds. We exhibit the relation between real, complex and quaternionic spaces that couple to N = 2 supergravity in d = 5, 4 and 3 space-time dimensions, respectively. Special attention is devoted to the homogeneous spaces. New homogeneous quaternionic and special Kihler spaces are identified.

1. Introduction Non-linear sigma models that appear in supersymmetric theories are subject to intriguing geometrical restrictions whose precise nature depends on the dimension of space-time and the degree of extension of the supersymmetry. For instance, in d = 2 space-time dimensions, Kahler, hyper-Kahler and quaternionic sigma. models appear naturally, possibly accompanied by torsion originating from a. Wess-ZuminoWitten term. These models have been extensively studied, in particular because they can be interpreted as models for string propagation in non-trivial space-times (for an (incomplete) list of references, see [1]). An interesting feature of two-dimensional non-linear sigma. models, which plays an essential role in their application to string theory, is the existence of a. connection between the geometry of target space and the short-distance behaviour of the theory. This aspect was already emphasized in the first paper of [1]. In four space-time dimensions the situation is similar. Kibler manifolds arise naturally for N = 1 chiral [2] and N = 2 vector (3] multiplets, whereas hyper-Kahler and qua.ternionic sigma. models appear for N =2 scalar multiplets (4-7]. The presence of sca.la.r and vector fields in some of these models gives rise to the phenomenon of duality transformations, invaria.nce transformations of the equations of motion but not of the action, which cannot be defined for the vector fields but act on the corresponding (abelian) field strengths [8, 9, 3]. These duality transformations are intertwined with the isometries of the non-linear sigma. model. In particular, in some of the extended supergra.vity theories the duality transformations are entirely fixed by supersymmetry. The most renowned example is perhaps N = 8 supergra.vity, where the duality transformations constitute the exceptional group E7 , with the scalar fields parametrizing the E7 /SU(8) coset space (9]. 10n

leave of absence from the Institute for Theoretical Physics, Utrecht, The Netherlands

37 In other space-time dimensions the situation is similar. Of particular importance in the context of this talk is the work on five-dimensional vector multiplets coupled to N = 2 supergravity [10]. In these theories an important class of non-linear sigma models corresponds to four (irreducible) symmetric spaces that are associated with Jordan algebras and the "magic square". Much work has been devoted to supersymmetry and supergravity in other space-time dimensions as well (see, for instance, the contributions reprinted in [11 ]). Needless to say, many of these theories are related by dimensional reduction or by dimensional compactification. Often it turned out to be productive to exploit such a relationship in studying the symmetry structure of the various theories. In this talk, where I describe work done in collaboration with A. Van Proeyen and F. Vanderseypen, the relation between supergravity theories in different dimensions and the symmetry structure of the corresponding non-linear sigma models plays an important role [12]. The reason for the renewed interest in these topics stems from the work on superstring compactifications, in particular on (2, 2) superconformal backgrounds with c = 9. These compactifications lead to effective "low-energy" theories in four space-time dimensions. For the heterotic string one thus ends up with a N = 1 and for type-II superstrings with a N = 2 supergravity theory. In the absence of a potential the Poincare-invariant ground states of the effective four-dimensional theory are characterized by the vacuum-expectation values of the scalar fields. Therefore, under certain circumstances the target manifold of the sigma model corresponds to the moduli space of the superconformal theories used for the superstring compactification [13]. It was found that the target space of the non-linear sigma models originating from compactifications of the heterotic or type-11 superstring, must factorize in a number of sub!lpaces that are of a restricted type. For the heterotic string one obtains a product of so-called special Kahler spaces [13-17]. All properties of a special Kahler geometry are encoded in a holomorphic function, which is homogeneous of second degree. Special Kahler geometry is intimately related to N = 2 supersymmetry and was discovered quite some time ago in the study of general couplings of vector multiplets to N = 2 supergravity [3]. The intrinsic geometric characterization of the special Kahler spaces was discussed more recently in [18]. A new feature arises when considering type-II theories, as N = 2 supersymmetry allows both Kahler and quaternionic sigma models in four dimensions, associated with the coupling of vector and scalar multiplets, respectively. The fact that both IIA and IIB theories can be compactified on the same superconformal system proved a powerful tool in analyzing the structure of the resulting non-linear si~ma models. Their target space is again a product space, but now consisting of a special Kahler and a quaternionic space. When comparing the compactification of the type- IIA to the compactification of the type-liB superstring (on the same superconformal system), one discovers that the vector and the scalar multiplets are interchanged (apart from the so-called universal multiplet, which is related to the operator 1 of the (2,2) superconformal system) [13]. This means that, roughly speaking, the Kahler and the

38

quatemionic spaces are interchanged when going from compactified IIA to compactHied liB, and it implies that also the quatemionic manifold has special geometry 2 in the sense that its properties are encoded in a homogeneous holomorphic function [14). For quaternionic manifolds, special geometry is not implied by N = 2 supersymmetry, as N =2 supergravity allows the coupling to more general quaternionic spaces [5). Hence, there exists a mapping between the special Kahler and quaternionic manifolds, which in (14) was called the c map. More precisely, the c map relates a special Kahler manifold of complex dimension n to a special quaternionic manifold of quaternionic dimension n + 1. As was shown in [14) this map is induced by ordinary dimensional reduction from a four-dimensional supergravity theory coupled to vector multiplets to three dimensions. As it turns out, the map is also related to Alekseevskii's classification of homogeneous quaternionic spaces [19), because in this work one identifies a so-called principal Kahlerian subalgebra for the normal quaternionic spaces other than the projective ones, which leads to a normal Kahler space. Therefore this procedure is somehow related to the inverse c map, at least for homogeneous spaces. In this talk I will concentrate on the symmetry structure of the various special manifolds. As was shown in [20) the symmetry of the special quaternionic manifolds can be understood systematically in terms of the symmetry of the corresponding Kahler manifold. One always encounters a number of extra symmetries that find their origin in the symmetries of the higher-dimensional supergravity theory, and possibly a number of hidden symmetries. A condition for the existence of these hidden symmetries was derived. Here we further extend this analysis. One class of special Kahler manifolds can also be obtained by dimensional reduction from five-dimensional supergravity coupled to vector multiplets, which contains a non-linear sigma model associated with a real manifold that is again "special" in that its properties are encoded in a single function. Thus, there exists a map from special real (n - 1)dimensional manifolds to special Kahler manifolds of complex dimension n. This map will·be called the r map. Hence one encounters the same situation as for the c map. The resulting Kahler manifold has a number of symmetries in common with those of• the special real manifold; then there are a number of extra symmetries that have·a well-defined origin in the five-dimensional supergravity, and, in addition, there may be extra hidden symmetries, which have no obvious origin. The condition for the existence of the hidden symmetries was already analyzed in (21). The r and c maps are a convenient tool for studying the special homogeneous spaces. One can show that a homogeneous space in the image of one of these maps must originate from a theory in which the combined transformations of scalar and vector fields act transitively on the scalars and vice versa. Motivated by this relation Cecotti [22) constructed all the N = 2 supergravity theories that are related by the c map to the normal quatemionic spaces classified in [19) (with the exception of the projective quaternionie spaces, which are not in the image of the c map). Here we 2 Theae

special quaternionic spaces were also called "dual-quaternionic".

39 consider the special Kahler spaces that are in the image of the r map (which include those classified in [22]) and construct all the corresponding holomorphic functions for which the duality transformations act transitively on the Kahler manifold (so that every two points on the manifold are related by a duality transformation). These functions thus define a class of homogeneous special manifolds that are real, Kahler or quaternionic, which may be compared to the spaces classified in [19, 22]. This comparison then leads to the identification of new homogeneous quaternionic manifolds, which were not considered in (19) (and neither were the corresponding Kahler manifolds in [22]). As real spaces a subclass of them was constructed in [23].

2. Symmetries and dimensional reduction Let me start by introducing the bosonic part of the Lagrangian for N = 2 supergravity coupled to n vector supermultiplets in d = 4 space-time dimensions. The corresponding non-linear sigma models lead to the so-called special Kahler mani5 folds. Then I discuss how a subclass of these d 4 theories originate from d supergravity, which allows sigma models based on special real manifolds. The relation between the isometries of corresponding real and Kahler manifolds is explained. Subsequently the reduction to d = 3 supergravity theories is presented, which leads to special quaternionic manifolds. The relation between the symmetry structure of corresponding Kahler and quaternionic manifolds is then discussed.

=

=

!U. Special Kahler manifolds The coupling of n vector multiplets to N 2 supergravity is characterized in terms of a single holomorphic function F(X), which is homogeneous of second degree 0, 1, ... , n. Therefore in terms of the n + 1 variables X 1 labelled by indices I it satisfies identities such as F = lFrX 1, Fr = FuXJ, X 1FuK 0, where the subscripts I,J, . .. denote differentiation with respect to X 1 , XJ, etc. The bosonic Lagrangian takes the following form (3)

=

=

e-•c =

=

-lR+(XNX)- 1 Mu8,..X 1 8"'XJ

+~ {Nu F:,} F+,..vJ + Jlu F;,} F-,..vJ} ,

(1)

where R is the Ricci scalar and F!,1 are the (anti)selfdual components of then+ 1 field strengths. The tensors Nu, Ma and Nu are defined by

Nu

=

i

MrJ

=

Nu-

Nu

=

4

(Fu

+ Fu),

(NX)r(NX)J XNX ' 1(NX)r(NX)J -Fu-

XNX

where we used an obvious notation where (NX)r

'

(2)

= NuXJ, XNX = X 1NuXJ, etc..

40

The Lagrangian (I) contains n+ I vector fields, with one of them belonging to the field representation of pure N =2 supergravity (the so-called graviphoton). However, it depends on only n scalar fields, because MIJ has a null vector proportional to X 1 and, J(J, as one easily verifies, while the tensors MIJ, llu and Nu depend only on the ratios of the fields by virtue of the homogeneity of the function F(X). Therefore it is convenient to introduce n independent fields through the ratios zA = XA f X 0 (A= I, ... ' n). Including z 0 =I we can replace all fields X 1 in the previous equations by z 1 without loss of generality. The Lagrangian of these n scalar fields zA takes the form of a non-linear sigma model corresponding to a Kabler manifold. Its Kabler potential is equal to K(z, z) = lnY(z,z) = lnNuz 1 zJ, so that the sigma model metric is given by

DAB

=

lPK(z,z)

{)zA {)zB

=

MAs zNz .

(3)

The curvature tensor corresponding to this metric equals

A D A 6D I -EAD R sc = -26(B c)- (zNz) 2 QscEQ ,

(4)

where

(5) The Lagrangian (I) leads to field equations, which can exhibit an invariance under generalized duality transformations acting on both the scalar fields and the field strengths. When present these transformations constitute an invariance of the scalar Lagrangian separately and thus define certain isometries of the corresponding nonlinear sigma model. However, we cannot exclude the possibility that this sigma model possesses more isometries, which are broken by the interactions with the vector fields. The generalized duality transformations constitute a subgroup of Sp(2n + 2, ffi.). On generic 2(n +I)-dimensional vectors (u 1 ,vJ) they act by means of constant real matrices 0 satisfying

o-l = fl OT flT,

where fl = (

~l ~ )

.

(6)

Infinitesimal transformations can thus be decomposed as 6u1 6v1

= =

B 1J uJ - DIJ VJ, Cu uJ- BJ1VJ,

(7)

where B 1J, Cu and DIJ are real constant (n +I) x (n +I) matrices (note that C and Dare symmetric). The 2(n +I)-component vectors (X 1 , -~iFJ), (F;};}, 2iiiJK F;j.,K) and their complex conjugates transform under the duality transformations according to (7). The fact that the transformation of F1 is already determined by the transformation of X 1, (8)

41 places restrictions on the duality transformations. This is expressed by the consistency equation [3] (9) Note that the function F(X) is usually not invariant under duality transformations. We should also point out that different functions F(X) can give rise to the same theory in the sense that their field equations are equivalent [14, 22, 24]. A special class of functions is given by .

F(X) = tdABC

xAxsxc xo ,

(10)

with dAsc a symmetric real tensor. Supergravity theories based on these functions (or on functions leading to equivalent field equations) can lead to flat potentials, as was shown in [21]. Furthermore they appear in the low-energy sector of superstring compactifications on (2,2) superconformal theories [14] and exhibit Peccei-Quinn-like symmetries, as is appropriate for many (classical) superstring compactifications. The coefficients dAsc are related to certain three-point functions of the superconformal theory, (11) According to [22] this class of functions also comprises all the normal Kahler spaces that can be coupled to N = 2 supergravity (with the exception of the complex projective spaces, which are described by minimal coupling). The functions (10) lead to the following Kibler metric, which only depends on the imaginary parts of the coordinates zA, gAB= MAB zNz

= 6 (dx)As

(dxxx)

_ 9 (dxx)A (dxx)s (dxxx)2 '

(12)

where zA : i(zA- zA), (dx)AB = dABC z 0 , (dzz)A = dABC x 8 x 0 and (dxzx) = dAscxAx 8 x 0 • Furthermore we have QAsc = ~idABC· The theory based on (10) is always invariant under certain duality invariances, which were analyzed in [21]. They are characterized by parameters /3, bA Bland aA. Under the corresponding isometries the coordinates zA transform as follows:

(13) While the isometries associated with the parameters /3 and bA do always exist, the presence of isometries associated with the parameters Bl and aA depends on whether the following conditions are satisfied, -D

=

0,

(14)

aaE~BCD = 0.

(15)

B (A dsc)D

•bl

iJ3 a2

•b3

iJ

p •b2

Figure 1: Root lattice corresponding to then = 3 theory characterized by F(X) = 3i (X 2I X 0 ) (X2X 1 - (X 3)2). The subgroup associated with the parameters iJ1 has two generators denoted by B2 and B3 • Their roots correspond to the solvable algebra of SU(1, 1), which is extended to a six-dimensional solvable algebra by the roots associated with the parameters /J, bt, b2 and b3 • In this case there is one hidden symmetry associated with the parameter a 2 and indicated in the diagram by . Here the tensor EXscv is defined by E~BCD

= cEFG dE(AB dcv)F -

6e. dscD)·

(16)

with

(17)

C" 8 cac

The condition (15) ensures that, in the coordinates employed here, is constant and so is R"scD av. When the maximal number of aA symmetries is realized, it thus follows that E~scv vanishes and R"scD is constant. This situation corresponds to a symmetric Kahler space (21, 25]. Clearly the transformations characterized by B1 and P form two subgroups of the full group of duality transformations. The root lattice corresponding to these duality transformations consists of the root lattice for the subgroup corresponding to B1 extended with one dimension associated with the eigenvalues of the roots under the /J-symmetry. This leads to a characteristic lattice such as shown in Fig. 1 for a particular example with n = 3. For higher rank one obtains a similar lattice by projecting on a suitably chosen plane. 1!.1!. Special real manifolds The theories corresponding to the functions (10) can be obtained by dimensional reduction from Maxwell-Einstein supergravity theories in five space-time dimensions

d =5 metric n vectors n -1 scalars total

n metric 1 vectors 1 scalars 1 0 0

II

II

1 1 n n n-1 0 ln+11 2n

Table 1: Decomposition of the d = 5 bosonic fields into d = 4 fields. [10]. This theory contains n- 1 real scalar fields and n vector fields (one of them corresponding to the graviphoton). The relevant Lagrangian is e- 1 £

=

-~R- ~dABchAo,.h 8 lJ"hC

+~ (6(d h)AB- 9(d hh)A (d hh)B) F:.,(A) F 8 ""(A)

+e- 1it."""'>.dABC F:.,(A) F,!(A) Af,

(18)

where A~ and F,1.,(A) denote the gauge fields and their corresponding abelian field strengths, and the scalar fields hA are subject to the condition

{19) After reduction to four dimensions, the Lagrangian (18) becomes equal to (1); the imaginary part of the four-dimensional scalar fields zA originate from the fifth component of the gauge fields A~, while their real part corresponds to the n - 1 independent fields hA and the component g55 of the metric. The n + 1 gauge fields in four dimensions are related to the n gauge fields and the off-diagonal components g,.5 of the metric in five dimensions. The conversion of the bosonic fields is summarized in Table 1. Clearly in four dimensions we have n + 1 extra scalar fields. The Lagrangian (18) is invariant under linear transformations of the fields (20)

provided that the matrices iJ satisfy the condition (14). These transformations still constitute a symmetry in four dimensions where they leave the function F(X) invariant. After reduction to four space-time dimensions a number of extra symmetries emerge, which have their origin in the five-dimensional theory. First of all, the extra vector field emerging from the five-dimensional metric has a corresponding gauge invariance related to reparametrizations of the extra fifth coordinate by functions that depend only on the four space-time coordinates. Then there are special gauge transformations of the n vector fields with gauge functions that depend exclusively and linearly on the fifth coordinate. Under these transformations the fifth component of each gauge field transforms with a constant translation, whereas the remaining

44

four-dimensional gauge fields transform linearly into the gauge field originating from the five-dimensional metric. These transformations are the ones associated with the parameters 6A. Finally there are the scale transformations of the fifth coordinate which, after dimensional reduction, corresponds to the transformations associated with the parameter {J. The five-dimensional origin of the duality invariances in four dimensions is thus concisely summarized by ~A

Bs gauge transformation oc x 5 scale transformation of x 5

•A

==> Bs ==> bA ==> {J

Hence altogether we have n + 1 extra symmetries associated with the parameters bA and {J, and n + 1 extra scalar fields. What remains are the possible extra duality invariances of the four-dimensional theory that are proportional to the parameters aA. These transformations have no five-dimensional origin and their presence depends entirely on the non-trivial restriction (15). We shall denote such symmetries as hidden symmetries, to distinguish them from those whose existence can be deduced on more general grounds, as explained above. Decomposing the full symmetry algebra W into eigenspaces of the generator associated with the {J symmetry (in the adjoint representation), we have (21)

where the subscript denotes the eigenvalue with respect to the {J symmetry. As it turns out, W 0 corresponds to the subalgebra associated with the parameters B'}, and {J, w2/3 contains the generators corresponding to the parameters 6A, and all possible generators corresponding to the hidden symmetries belong to W_ 2t 3 (see, for instance, the example shown in Fig. 1). The dimension of W_ 2t 3 is thus at most equal to n, whereas the dimension of W2t3 is always equal to n. Unless we have maximal symmetry (i.e. unless there are n independent symmetries associated with the parameters aA, in which case the space is symmetric) the isometry group of the corresponding Kahler space is not semi-simple. !.9. Special quaternionic manifolds We now return to the general four-dimensional Lagrangian (1) based on a general function F(X) and consider its reduction to three space-time dimensions. In this case an extra feature is present, because the standard (abelian) gauge field Lagrangian in three dimensions can be converted to a scalar field Lagrangian by means of a duality transformation. Only derivatives of the new scalar field appear, so that it has a corresponding invariance under constant shifts. Each four-dimensional gauge field thus gives rise to two scalar fields with two corresponding isometries. One is its component in the fourth dimension and the other is the scalar field that results from the conversion of the three-dimensional gauge field. The n + 1 four-dimensional vector fields A~ thus lead to 2( n +1) scalar fields, which will be denoted by A I and B I. The

45

d =4 metric n + 1 vectors 2n scalars total

I metric I scalars I 1 0 0 11

1

2 2n+2 2n 1

4n + 4

11

Table 2: Decomposition of the d = 4 bosonic fields into d = 3 fields. corresponding invariances have parameters oJ and fh. The same conversion can be used for the vector field that emerges from the three-dimensional metric, so that the four-dimensional metric gives rise to a three-dimensional metric and two scalar fields. These scalar fields are denoted by q, and u, and they also lead to two invariances, one related to the scale transformation of the extra coordinate with parameter f 0 and another one corresponding to the converted three-dimensional vector field with parameter f+. Altogether, the Lagrangian (1) thus gives rise to 4(n + 1) scalar fields, as shown in Table 2, coupled to gravity with 2n + 4 additional invariances. The four-dimensional origin of these invariances can be summarized as follows: duality invariance gauge transformation oc x 4 Lagrange multipliers scale transformation of x 4

==> duality invariance ==> ar ==> f3r ' (.+ ==> fo

As the Lagrangian is still supersymmetric the scalar fields define a quaternionic non-linear sigma model of quaternionic dimension n + 1. The corresponding quaternionic spaces are called special quaternionic spaces and obviously depend on a homogeneous holomorphic function of second degree. Like all quaternionic spaces, they are irreducible Einstein spaces [26]. The explicit Lagrangian was determined in [27], where the quaternionic structure was explicitly verified, and in [20] (we use the notation of [20]). Under the extra 2n + 4 transformations the fields transform as follows:

h'Al h'Br

= -~fo AI+ 0 r,

c5q, = -(.0 q,,

= -~f0 Br + f3r,

h'u = -f0 u+ Ha 1 Br- f3rA 1 )

+ f+,

(22)

while zA remains inert. Under the duality transformations the scalar fields A 1 and B 1 transform as indicated in (7), while the transformations of zA follow from (8); the fields q, and u are invariant under the duality transformations. Consider now the root lattice corresponding to all these symmetries. It consists of the root lattice of the duality invariance extended with one dimension associated with the eigenvalue of the roots under the scale symmetry with parameter f.0 • This leads to a root lattice such as shown in Fig. 2, where we have exhibited the case

46

c,o

oo Ot

01

•

(ll

/3

f-

fo

f+

•

• f3t

f3t

bl .

f3o

•

f3o

Figure 2: Root lattice corresponding to the SU(1, 1) duality invariance of then = 1 Kahler manifold based on F(X) = i(X 1 ) 3 /X 0 , extended with the roots belonging to the extra transformations that emerge from the reduction to three dimensions. Observe that there are no duality invariances associated with the matrices B~ in this case. Additional hidden symmetries are located on the left half-plane. In the case at hand, there are five such symmetries indicated by , which extend this diagram to the root lattice of G2 c+ 2 ). Its solvable subalgebra consists of the solvable subalgebra of SU(l, 1), associated with the parameters /3 and b\ extended by the generators of the six extra symmetries corresponding to f 0 , f+, o1 and /31 • where the duality invariances constitute a group of rank 1 (namely SU(l, 1)) and n = 1. In the general case we obtain a similar diagram after projecting all the roots on a suitably chosen plane. The algebra V corresponding to these roots, which is obviously non-semisimple, can generally be decomposed into eigenspaces of the generator e0 associated with the parameter fo in the adjoint representation. Hence we have

V

= Vo + Vt/2 + Vt,

(23)

where the V.. contain the generators that satisfy [eo, V.. ] = a V... The generators in V0 correspond to the generators of the duality invariance, previously denoted by W, supplemented by the generator e0 , V 112 contains the 2n + 2 generators corresponding to the parameters o 1 , /3r, and vl contains the generator corresponding to f+. Again one may wonder whether additional hidden symmetries are possible, in analogy with the symmetries associated with the parameters a A in the Kahler case. This question was analyzed in [20] and it was found that the hidden symmetries are always The root lattice corresponding to associated to roots with eo eigenvalues -1 or all symmetries of the Lagrangian (1) thus decomposes according to

-!·

(24)

47 As shown in [20], for symmetric spaces, the dimension of V_ 1 and V- 112 is equal to 1 and 2n + 2, respectively. Otherwise V_ 1 is empty and the dimension of V_ 1t 2 is less than or equal to n + 1 (the latter follows from the requirement that [V_ 1, 2 , V _1, 2 ] must vanish in this case). The fact that there are no new generators with positive eo-eigenvalues shows that V 1 is always of dimension 1. The parameter associated with the new symmetry in V- 1 is denoted by t-; those corresponding to the symmetries in v-1/2 are denoted by & 1 and The full symmetry structure of the quaternionic manifold is encoded in a single function h,

Pr-

(25)

=

where 81 B1 + ~iFuAJ = NuBJ. The function h is a real quartic polynomial in the fields A 1 and Br. Furthermore it is homogeneous of zeroth degree in X and X separately, as well as invariant under duality transformations. The maximal number of symmetries exist if and only if the function h is independent of X 1 . In that case both the Kahler manifold and the associated quaternionic manifold are symmetric. Hidden symmetries belonging to V- 112 exist for each independent linear combination of first-order derivatives with respect to A 1 and B1 that is independent of X 1• In [20] it was also shown that the quaternionic spaces based on (10) always have at least one hidden symmetry associated with the parameter /Jo. When the Kahler manifold corresponding to (10) has hidden symmetries corresponding to parameters aA, then the corresponding quaternionic manifolds must have hidden symmetries as well with parameters =a We refer to [12] for further details.

PA

A.

3. Homogeneous spaces The three types of special manifolds are related by dimensional reduction of the corresponding supergravity models as outlined above. A special real (n - 1)dimensional manifold JR,._ 1 is thus related to a special Kahler manifold ~ of complex dimension n. Likewise, a special Kahler manifold of complex dimension n is related to a special quaternionic manifold IH,.+l of quaternionic dimension n + 1. These relations define a mapping from special real to special Kahler manifolds, and from special Kahler manifolds to special quaternionic manifolds. As explained in the introduction, these mappings will be called the r and the c maps, which act according to3

(26) 3 When the coupling of a certain space to supergravity is not unique, the result of these maps will depend on the type of coupling as, for instance, characterized by the embedding of the duality transformations in Sp(2n + 2,1R).

48 It should be dear that the inverse r and c maps do not always exist. For instance, only the Kahler manifolds based on the functions (10) (and the functions that lead to equivalent field equations) can be in the image of the r map. Under these maps the dimensionality of the manifold increases and so does the rank of the symmetry algebra and of its solvable subalgebra. The latter is obvious from the discussion in the previous section. When performing a dimensional reduction, the scale transformations associated with the extra space-time coordinate, parametrized by /3 (or f 0 ), extend the dimension of the Cartan subalgebra of the symmetry group by one unit. If the rank of the maximal solvable subgroup of the symmetry group in higher dimensions were equal to r, then its extension after dimensional reduction with the roots parametrized by /3 and bA (or f 0 , a1 , /3r and f+), which have non-negative eigenvalues with respect to the extra generator of the Cartan subalgebra, defines again a solvable algebra. This solvable algebra is thus of rank r + 1. Although it is not a priori excluded that there are extra symmetries which can be included in this solvable subalgebra, this will leave the rank unchanged. For our purposes the following considerations are important. If the result of the c map is a homogeneous quaternionic space, then the duality invariance (the symmetry of the scalar-vector sector of the theory) of the original theory must act transitively on the corresponding manifold parametrized by the scalar fields. The proof of this result, which applies also to the r map, is given in [12]. Also the converse is true: if the vector-scalar symmetries act transitively on the manifold parametrized by the scalars, then one can show that the symmetry group after dimensional reduction gives rise to additional symmetries, which leave the original scalar fields invariant but act transitively on the new scalar fields. In this respect it is important that, after dimensional reduction, the number of new symmetries is always larger than or equal to the number of new coordinates. The above results show that homogenous quaternionic spaces that are in the image of the c map correspond to special homogeneous Kahler spaces. On the other hand, special homogeneous Kahler spaces give rise to special homogeneous quaternionic spaces, provided that the scalar-vector symmetry transformations act transitively on the Kibler manifold. Likewise, such Kibler spaces that are themselves in the image of the r map correspond to special homogeneous real spaces. Again, special homogeneous real spaces give rise to homogeneous Kahler spaces, provided that the vector-scalar symmetries act transitively on the real manifold. As was emphasized in [14] the inverse c map is related in a certain way to the classification of normal quaternionic spaces as given by Alekseevskii [19]. Normal quaternionic spaces are quaternionic spaces that admit a transitive completely solvable group of motions. It was conjectured in [19] that the homogeneous quaternionic spaces consist of compact symmetric quaternionic and normal quaternionic spaces. The algebra corresponding to the group of solvable motions in the latter case exhibits the same decomposition as in (23) (in that analysis e0 is defined as a one-dimensional invariant subalgebra). According to Alekseevskii there are two different types of nor-

49

mal quatemionic spaces characterized by their so-called canonical quatemionic subalgebra. The first type with subalgebra Cl turns out to correspond to the quatemionic projective spaces Sp(m, 1)/(Sp(m) ® Sp(1)). Their solvable algebra v• decomposes as in (23), where ~ contains only the generator eo, while and have dimension 4n and 3, respectively. Obviously the quaternionic projective spaces are not in the image of the c map. The second type has a canonical subalgebra A~ and the structure of the solvable algebra V" is as follows: V~ is a direct sum of e0 and a normal Kahler algebra W of dimension 2n, 1, has dimension 2n + 2 and has dimension 1. In order to be quaternionic, the representation of W induced by the adjoint representation of on 12 must generate a solvable subgroup of Sp(2n + 2, IR). Therefore each normal quatemionic space of this type defines the basic ingredients of a special normal Kahler space, encoded in its solvable transitive group of duality transformations. Alekseevskii's analysis thus strongly indicates that the corresponding N = 2 supergravity theory should exist, so that under the c map one will recover the original normal quaternionic space. To establish the existence of the supergravity theory, one must prove that a corresponding holomorphic function F(X) exists that allows for these duality transformations (i.e., that satisfies (9)). This program was carried out by Cecotti (22], who explicitly constructed the function F(X) corresponding to each of the normal quaternionic spaces with canonical subalgebra A}. With the exception of the so-called minimal coupling, where F(X) is a quadratic polynomial, all functions F(X) can be brought into the form (10). The corresponding special Kahler manifolds were denoted by H(p,q) and K(p,q). Under the c map, they lead to the normal quaternionic manifolds V(p,q) and W(p,q) defined in (19]. If Alekseevskii's classification is complete, there can be no other special Kahler spaces with solvable transitive duality transformations. It is possible to start from the other end and classify directly all functions of type (10) that have a group of duality transformations that acts transitively on the corresponding Kahler manifold. This approach was followed in (29]. One starts by choosing a reference point on the manifold where the kinetic terms for the various fields of the corresponding supergravity Lagrangian have the proper sign. Subsequently one performs certain reparametrizations, such that this reference point equals zA = ( 0, ... , 0) (corresponding to xA = (1, 0, ... , 0)) and

vr,,

v:

v•

v:

v:

v:

-li,

dn"

= 0,

(a,b = 2, ... ,n)

(27)

with dm > 0. This parametrization is called the canonical parametrization [21] (see also (10]). Now consider the transformation rule (13). Obviously then independent transformations associated with the parameters ~ then induce n independent real shifts of the coordinates zA, while the transformations proportional to possible parameters aA can always be combined with the ones proportional to the parameters ~, such that they leave the reference ~int invariant. Therefore, if the transformations associated with the parameters BAs and P induce n independent translations

50

of the imaginary parts of zA, then the duality transformations act transitively and the space is homogeneous. One such transformation corresponds to the parameter {J, so that the remaining condition for homogeneity is that the matrices iJA8 contain at least n- 1 independent parameters iJA 1 • On the other hand, the matrix iJA8 is restricted by the fact that it should leave the tensor dAsc invariant. In the canonical parametrization, this implies that iJA8 takes the following form [21]

• • d,.bc B"& = B"t -d +A,.& ,

(28)

111

where A,.& is an antisymmetric matrix subject to the condition (29) where

(30) Now we observe that transformations associated with the matrices A,.&, that are independent of the parameters B"t. leave the canonical reference point invariant and thus correspond to the isotropy group. Hence we are left with the requirement that the symmetry group should contain n-1 independent parameters B" 1 • According to the previous equation this requires4

r,.&cd = dut de(Gb Ac)e;d,

(31)

where A,.&= iJ•1A,.&;c. The possible coefficients d,.bc that satisfy this condition were classified in [29] and the results for the possible functions F(X) can be summarized as follows. First the complex coordinates are split into Xt, X 2 , X" and xm, where the range of the indices I' and m is equal to q + 1 and r, respectively. Hence we have n

= 3+q+r,

(32)

so that n ~ 2. Using the results of the classification for the coefficients d,.&. it turns out that the possible functions F(X) can be written as

F(X)

=;

0{

X 1 (X 2 ) 2 - X 1 (X") 2 - X 2 (Xm) 2 + -y,.m,. X" xm X"} ,

(33)

where the coefficients -y,.mn are r x r gamma matrices that generate a real (q+ I)dimensional Clifford algebra of positive signature. This property constrains the possible values for q and r. Observe that (33) is no longer in the canonical parametrization, due to a linear redefinition that we have performed on X 1 and X 2 • Before discussing the various possibilities for the functions (33), we first note that the tensor (16), when evaluated at the reference point in the canonical parametrization, is just proportional to r Gbcd• Therefore it follows that the Kahler spaces with 4 A special case of this equation, namely r otc 0) the real spaces corresponding to L(-1,r) are symmetric, but not the Kahler and quaternionic spaces. The reason for this extra symmetry is that the real sigma models have extra invariances that are not preserved by the interactions with the vector fields in d = 5 supergravity. Therefore these isometries are not preserved under the r map. Observe that the root lattice for the real and the Kahler spaces corresponding to L( -1, 1) was discussed in Fig. 1. Subsequently consider q = 0, in which case we have one gamma matrix with eigenvalues ±1, each corresponding to a representation of the one-dimensional Clifford algebra. These spaces are denoted by L(O,P, F), which is equivalent to L(O, F, P), where P and F specify the multiplicity of the two representations, so that n = 3 + P+F. In [19, 22] these spaces were denoted by W(P,F) and K(P,F), corresponding to the quaternionic and the Kahler cases, respectively. We have a symmetric space whenever P or F vanishes (see Table 3). Finally we have the case of positive q, denoted by L(q, P), where P defines the number of irreducible representations of the (q+ 1)-dimensional Clifford algebra contained in the gamma matrices 'Y~ 0 are also listed in 'l&ble 3 and exist only for P = 1 and n = 6, 9, 15 or 27. The corresponding real manifolds were discussed in (10) in connection with the magic square. Their Kibler counterparts were derived in (21). The work reported in this talk was done in collaboration with A. Van Proeyen and F. Vanderseypen. I thankS. Cecotti, S. Ferrara and P. Fre for valuable discussions and the organizers of this conference for their hospitality and providing a most stimulating atmosphere. 4. References 1. L. Alvarez·Gaume and D.Z. Freedman, Commun. Math. Phys. 80 (1981) 443,

2. 3. 4.

5. 6.

7. 8.

S.J. Gates, C.M. Hull and M. Roeek, Nucl. Phys. 8248 (1984) 157, T. Curtright and C.K. Zachos, Phys. Rev. Lett. 53 (1984) 1799, V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. 8247 (1984) 83, P. DiVecchia, V.G. Knizhnik, J.L. Petersen and P. Rossi, Nucl. Phys. 8253 (1985) 701, C.M. Hull and E. Witten, Phys. Lett. 1808 (1985) 398, C.M. Hull, Phys. Lett. 1788 (1986) 357; in Super Field ThetJries, proc. of the NATO Workshop, Vancouver, 1986, eds. H.C. Lee, V. Elias, G. Kunststatter, R.B. Mann and K.S. Viswanathan, Plenum, 1987, E. Bergshoeff, E. Sezgin and H. Nishino, Phys. Lett. 1878 (1987) 167, A. Sevrin, Ph. Spindel, W. Troost and A. Van Proeyen, Nucl. Phys. 8308 (1988) 662; 8311 (1988/89) 465, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. 8312 (1989) 58. B. Zumino, Phys. Lett. 878 (1979) 203. B. de Wit and A. Van Proeyen, Nucl. Phys. 8245 (1984) 89. U. Lindstrom and M. Roeek, Nucl. Phys. 8222 (1983) 285, N.J. Hitchin, A. Karlhede, U. Lindstrom and M. Roeek, Commun. Math. Phys. 108 (1987) 535. J. Bagger and E. Witten, Nucl. Phys. 8222 (1983) 1. C.K. Zachos, Phys. Lett. 788 (1976) 329, B. de Wit, J.-W. van Holten and A. Van Proeyen, Phys. Lett. 958 (1980) 51, P. Breitenlohner and M. Sohnius, Nucl. Phys. 8187 (1981) 409, · K. Galicki, Nucl. Phys. 8289 (1987) 573, J.A. Bagger, A.S. Galperin, E.A. Ivanov and V.I. Ogievetsky, Nucl. Phys. 8303 (1988) 522. B. de Wit, P. Lauwers and A. Van Proeyen, Nucl. Phys. 8255 (1985) 569. S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. 8121 (1977) 393, E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 748 (1978) 61, B. de Wit, Nucl. Phys. 8158 (1979) 189, M.K. Gaillard and B. Zumino, Nucl. Phys. 8193 (1981) 221.

54

9. B. Julia and E. Cremmer, Nucl. Phys. B159 (1979) 141, B. de Wit and H. Nicolai, Nucl. Phys. B208 (1982) 232. 10. M. Gunaydin, G. Sierra and P.K. Townsend, Phys. Lett. 133B (1983) 72; Nucl. Phys. B242 (1984) 244, B253 (1985) 573. 11. A. Salam and E. Sezgin, Supergravities in Diverse Dimensions, NorthHolland/World Scientific (1989). 12. B. de Wit, F. Vanderseypen and A. Van Proeyen, in preparation. 13. N. Seiberg, Nucl. Phys. B303 (1988) 286. 14. S. Cecotti, S. Ferrara and L. Girardello, Int. J. Mod. Phys. A4 (1989) 2457. 15. L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B329 (1990) 27. 16. S. Ferrara and A. Strominger, in Strings '89, eds. R. Arnowitt, R. Bryan, M.J. Duff, D.V. Nanopoulos and C.N. Pope (World Scientific, 1989), p. 245. 17. P. Candelas, P. Green and.T. Hiibsch, Nucl. Phys. B330 (1990) 49, P. Candelas, X. C. de Ia Ossa, P. Green and L. Parkes, Phys. Lett. 258B (1991) 118, M. Bodner and A.C. Cadavid, Class. Quantum Grav. 7 (1990) 829, M. Bodner, A.C. Cadavid and S. Ferrara, Class. Quantum Grav. 8 (1991) 789. 18. A. Strominger, Commun. Math. Phys. 133 (1990) 163, L. Castellani, R. D'Auria and S. Ferrara, Phys. Lett. B241 (1990) 57; Class. Quantum Grav. 7 (1990) 1767, R. D'Auria, S. Ferrara and P. Fre, Nucl. Phys. B359 (1991) 705. 19. D.V. Alekseevskii, Math. USSR lzvestija 9 (1975) 297. 20. B. de Wit and A. Van Proeyen, Phys. Lett. B252 (1990) 221. 21. E. Cremmer, C. Kounnas, A. Van Proeyen, J.P. Derendinger, S. Ferrara, B. de Wit and L. Girardello, Nucl. Phys. B250 (1985) 385. 22. S. Cecotti, Commun. Math. Phys. 124 (1989) 23. 23. M. Giinaydin, G. Sierra and P.K. Townsend, Class. Quantum Grav. 3 (1986) 763. 24. P. Fre and P. Soriani, preprint SISSA 90/91/EP, June 1991. 25. E. Cremmer and A. Van Proeyen, Class. Quantum Grav. 2 (1985) 445. 26. S. Ishihara, J. Diff. Geom. 9 (1974) 483. 27. S. Ferrara apd S. Sabharwal, Nucl. Phys. B332 (1990) 317. 28. F. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Nucl. Phys. B304 (1988) 348, 371, C.M. Hull, Phys. Lett. 240B (1990) 110; Nucl. Phys. B353 (1991) 707, L.J. Romans, Nucl. Phys. B352 (1991) 829. 29. B. de Wit and A. Van Proeyen, preprint CERN TH.6302/91.

56

A DUALITY BETWEEN STRINGS AND FIVEBRANES

M. J. Duft"f and J. X. Lut

Center for Theoretical Physics Physics Department Texas A&M University College Station, Texas 77843

ABSTRACT Not only does the heterotic string admit a heterotic fivebrane as a soliton but the heterotic fivebrane admits as a soliton a heterotic string. This provides further evidence that the two theories may be dual descriptions of the same physics, with the strong coupling regime of one theory described by the weak coupling regime of the other. To illustrate this, we show how the energy-momentum tensor of the quadratic Yang-Mills· action associated with the string reduces to that of an elementary fivebrane, and how the energy-momentum tensor of the quartic Yang-Mills action associated with the fivebrane reduces to that of an elementary string.

t Work supported in part by NSF grant PHY~9045132. t Supported by a World Laboratory Scholarship.

56 1. Introduction

At the start of the superstring revolution of 1984, many physicists believed that the heterotic string [1] might provide that Holy Grail of theoretical physics: a unified theory of

all the forces and particles of nature, including gravity. Six years later this initial euphoria has been dampened by the realization that many of the really basic questions of string theory (e.g. How do strings break supersymmetry? How do strings choose a vacuum state?) cannot be answered within the framework of a weak-coupling perturbation expansion.

In the meantime, a small but dedicated group of theorists were posing a seemingly very different question: If we are going to replace 0-dimensional point particles by 1dimensional strings, why not 2-dimensional membranes or, in general, p-dimensional objects or "p-branes"? (2] This work has revealed that many of the original objections to membranes ("membranes must have ghosts", "membranes have no massless particles in their spectrum", "supermembranes cannot exhibit the

~~:-symmetry

crucial for superstring

consistency", "membranes cannot describe chiral fermions", etc) were without foundation.

In this paper, we wish to describe some recent advances in supermembrane theory concerning the "heterotic fivebrane", an object whose existence was conjectured in 1987 (3] but which has recently been rescued from obscurity by Strominger (4], who showed that the heterotic string admits the heterotic fivebrane as a soliton solution. Our new result is that the converse is also true (5]. This lends support to the idea that the two theories are dual descriptions of the same physics, with the strong coupling regime of one theory described by the weak coupling regime of the other. Specifically, we shall see that the dimensionless loop expansion parameters g(string) and g(fivebrane), are related by 1 g (fivebrane) = -::-;:=;==;===? (string)

Vg

(1.1)

and that the string tension T2 and fivebrane tension T6 obey a Dirac quantization rule

[6,7] n =integer

where ~~: 2 is the D=10 gravitational constant.

(1.2)

67 2. Extended obje,cts Consider some extended object of dimension p moving through spacetime. It sweeps out a worldvolume of dimension d a d

= 1 worldline,

sweeps out a d

a p

= p + 1.

Thus a p

= 1 string sweeps out

= 3 worldvolume and so on.

a d

= 0 point

= 2 worldsheet,

particle sweeps out a p

= 2 membrane

Clearly the dimension of the object cannot

exceed the dimension of the spacetime so we must have d :5 D where Dis the dimension of spacetime, sometimes called the "target space". Its trajectory is described by the

xM is the spacetime coordinate (M = 0, 1, ... ,D- 1) and ei are the worldvolume coordinates (i = 0, 1, ... , d - 1). In what follows, we shall frequently make the so-called "static gauge choice" by making the D = d + (D - d) split

functions XM({) where

(2.1)

where p

= 0, 1, ... d- 1 and m = d, ... D- 1, and then setting X"({)={"

(2.2)

Thus the only physical degrees of freedom are given by the (D- d) ym({). To describe a super p-brane, we augment the D bosonic coordinates XM({) with anticommuting fermionic coordianates 9°(e). Depending on D, this spinor could be a Dirac spinor, a Weyl spinor, a Majorana spinor or a Majorana-Weyl spinor. The action for the super p-brane also has fermionic It-symmetry which reflects the fact that half of the spinor degrees of freedom are redundant. We may therefore adopt a physical gauge which eliminates the unphysical half. The net result is that the theory exhibits a d-dimensional supersymmetry with equal number of bose and fermi worldvolume degrees of freedom, where the number of supersymmetries is exactly one half of the original spacetime supersymmetries. However, the range of d and D which permit bose-fermi equality (assuming only scalar and spinor fields) is severely limited to the twelve points on the "brane-scan" of Fig. 1. In the next sections we shall focus on the meaning of the horizontal line that connects the (d

= 2,D = 10) superstrings and the (d = 6,D = 10) superfivebranes.

58 3. Antisymmetric tensor ftelds and duality It is well known· that a charged particle couples to an abelian vector potential AM which displays a gauge invariance

(3.1)

and has a gauge invariant field strength

(3.2)

Similarly a string couples to a rank-2 antisymmetric tensor potential AM N

= -AN M with

gauge invariance

(3.3)

and field strength

(3.4) In general, a (d- 1)-brane couples toad-form AM,M•···M4 with

(3.5)

and

FM, M····MH,

= (d + 1)B[M, AM····M4+1]

(3.6)

In the language a£ differential forms we may write

(3.7)

and

(3.8)

59 from which the Bianchi identity (3.9)

dF::O

follows immediately. In the absence of other interactions, the equation of motion for the d-form potential is

(3.10) where the source J is ad-form. Here we have introduced the Hodge dual operation • which converts ad-form into a (D- d) form e.g.

and

eM'·"MD

is the D-dimensional alternating symbol with e012 "·D-t

= +1.

A familiar example of the field strength (3.8), Bianchi identity (3.9) and the field equation (3.10) is provided by Maxwell's equations in D = 4:

FMN =(}MAN- (}NAM otMFNP] 0rM *FNP)

(3.11)

=0

(3.12)

= *JMNP

(3.13)

The asymmetry between the equation for F and that !or • F corresponds physically to the statement that there are no magnetic: monopoles. If we want to restore the symmetry by introducing magnetic monopoles then we rmmt give up equation (3.11) in favour of

(3.14) so that

(3.15)

with

eo (3.16) and hence (3.17)

If X is singular,

xl23 = gc53(y)

(3.18)

we speak of an "elementary monopole" in analogy with the elementary electric charge (3.19) Whereas if it is smeared out so as to be regular at the origin we speak of a "solitonic monopole". In both cases, we have

(3.20) (3.21) In this language, the Dirac monopole is elementary whereas the 't Hooft-Polyakov monopole is solitonic. The solitonic monopole is obtained by embedding Maxwell theory in a nonabelian gauge theory with Higgs scalars. It is the presence of the Higgs field term

XMNP

that smears out the monopole and gives it a finite "size" p. From (3.21), however, as we shrink the size to zero we must find lim xl23 = g.S 3 (y) ,_o

(3.22)

The electric charge is conserved by virtue of the field equations and hence corresponds to a Noether charge, whereas the magnetic charge is identically conserved and corresponds to a topological charge. According to the Dirac quantization rule

81 eg

n

- - 411' - 2

(3.23)

n = integer.

The argument generalizes to arbitrary d and D (8). The usual equations

Fo~+l =dA.,

dF4+1

(3.24)

=0

(3.25) (3.26)

imply the presence of an "electric" charge i.e. a (d- 1) brane but no "magnetic" charge i.e. no (D- d- 3)-brane. To restore the symmetry by introducing a (D- d- 3)-brane we must modify (3.24) to

(3.27) so that the Bianchi identity (3.25) becomes

(3.28) with

(3.29)

If X is a singular,

(3.30) we speak of an "elementary (D- d- 3}-brane" whereas if it is smeared out so as to be regular at the origin we speak of a "solitonic (D- d- 3)-brane". We then have

j = j • 9D-, ~21, ~27)

=

Kt(rp,ij)) + K3(!f>, '¢)

+ e-(K,-K2)/3 G1~ 27"J27 + e-(K2-Kt)/3 02~21

(1.5)

The special geometry of M 1, M 2 implies a quasi-holomorphic structure for the metric

G and the three-index tensor W. In fact, G and W can both be expressed in terms of a holomorphic prepotential F [28, 27). The entire effective action will then only depend on F 1(rp) and F2(!f>), which will then encode the dynamical information of the internal manifold [7, 8]. For C-Y manifolds defined by some polynomial in CP"', the Picard-Fuchs equations [2G-22) satisfied by the periods of the holomorphic three-form 0( If>) allow one to determine G, Wand the duality group r completely. These equations are equivalent to those which determined the perturbed (by marginal deformations) fusion rule coefficients for the O.P.E. of chiral primary fields of the corresponding N = 2 superconformal field theory (17-19]. The identification of points in the moduli space connected by r actually means that the moduli space of string theory is not a manifold M, but an orbifold M/f, with fixed points. In the effective action r appears as an exact symmetry, and there are arguments that this symmetry survives (higher genus) string perturbation theory [29, 30]. In Section 2 we will show how target space modular anomalies [31-34] are cancelled in string theory, and how this mechanism determines the running of gauge couplings induced by dynamical (moduli) scales [35-37]. To prepare the ground for discussion on target-space modular anomalies, it is important to realize that a duality transformation r on the moduli space If> --+ f!f> acts with a Kahler phase on the Yukawa couplings, and with rescaling and U(1) phases on all the fermions of the theory (31-33). Since fermions are coupled to gauge and gravitational fields, these (field-dependent) phases can induce mixed u-model-gauge (or gravitational) anomalies through ordinary (one loop) triangular graphs, in which a u-model U(1) connection couples to gauge or gravitational gauge fields.

84 Here, we will give the transformation properties, under a generic discrete isometry r, of all relevant quantities appearing in the effective Lagrangian for a N = 1 heterotic string compactified on a C-Y manifold, with gauge group Ea x Ea. These transformations, since they act on sections of line bundles, will generally act accompanied by Kahler phases (38]. Let 1/J,

wt(rr.p)-+ e2At-3-ni: wt(r.p) and accordingly, the scalar fields of chiral

W2(ri{J)-+ e2A.-:ty,I: W2(1/J) ;

(1.7)

Ea families

1)27-+ e"'li:t+(A,-A,)/31)27 and lliif-+ e'12l:o+(At-A2)/31)21'

(1.8)

ensure that the total superpotential W defined by

w = W1 t~7 + W2~1

(1.9)

just cancels the variation of K in the 'norm' over the line-bundle

II W II= eo= WW*eK.

(1.10)

The change of the moduli and family metrics is given by Gt

-+

at e-..,.(I:t+Et)

(1.11)

0?- -+ 02,.... e-'l>(l:o+E•)+(A,+A•)/3-(At+At)/3

27 27 Equations (1.11) are easily derived from Eqs. (1.3) and (1.4). Therefore the extra (holomorphic) phase -yE(~) (7 is a real normalization constant) is due to the discrete isometry r of the modular group, which acts on the fermions as a U(1) gauge transformation with gauge connection {31-33]:

B,.

= i(B,;fJ,.~;- B,;EJ,."f;)

and B;

= fJ;(log det G)

(1.12)

( ~ is a generic modulus coordinate). The holomorphic phases E, A are actually related to one another since E is given by log det fJf[ /8~; (~;-+ ff(~)) and the W 1 •2 tensors and charged matter fields are required to be tensor densities under modular transformations (the Kahler phase could also be obtained from the special geometry formulae). The tranformation laws of the fermions are now determined by the bilinear fermionic terms in the effective action, which are not chiral invariant (bilinear with an even number of 7 metrics), namely 2) Here we 888ume for simplicity only one chiral weight r for all families.

85

+ W 1 Xif Xi7 +n-) G~ Xn ~ t;T, ~ Xi7 ~ ti7

eKf2(W1 X2T X27 +21

eK/ 2 t/J,.t~""t/1,,

x'G;j 8,. tfl7"7"t/J,

~t~""F,.,x•.

(1.13}

(1.14) (1.15) (1.16)

Equations (1.13) and (1.14), i.e. the Yubwa coupliqs, fix the phues of the charged Ee fermions and of the gauginos. Equations {1.15) and (1.16) fix the phues of the gravitino, moduli fermions, and dilatino. The result is X27 -+ Xn e-n/2(Es-Es)-5/12(As-As)-1/12(At-Ao} e-n/2(Es-E1 )-1/8(As-As)+1/8(A2 -A2 ) (1.17)

(the same for Xif with 1 +-+ 2) ~

-+

~ e1/4(As-A1 )+1/4(At-it)

{1.18)

(1.19) (the same for the other type of moduli fermions with 1 +-+ 2). Note that ~.t/J,.,x. undergo a pure U(1) Kahler phase, while the matter and moduli fermions acquire extra. terms owing to the coordinate change. Furthermore, the pure Kahler phase of charged fermions is not standard because the charged chiral scalars transform [12) u holomorphic sections under a Kahler transformation on M 1 x W. Target-space duality transformations can be discussed in great detail for Z,. orbifold models which are limiting cases of C-Y manifolds [39). In this case, the moduli space of orbifolds (40, 7, 12) corresponds to the so-called untwisted moduli, and the duality groups are explicitly known (4, 5, 41-43). We will not report here this analysis, which hu been discussed in great detail in recent papers, but rather try to extract from it some general conclusions.

2

Moduli dependence of gauge couplings and target-space duality anomaly cancellation Recently, loop corrections to gauge couplings have been computed in heterotic string compactification [32-37, 44, 47) (in four dimensions) with an unbroken gauge group G. The analysis hu mainly been confined to (2,2) orbifold vacua with G = Ea x G', by a number of authors, with string and field theory calculations, exploiting the relation between renormalization of gauge couplinp and anomalies [45, 46) in N = 1 supersymmetric gauge theories. The peculiar outcome of these investigations is that moduli corrections to gauge couplings are not holomorphic (in the moduli chiral fields) so the question whether these results, which are manifestly target-space duality invariants, can be compatible with

86 standard supergravity arguments [38) was raisE!) is completely absent [37, 31, 33). When a Green-Schwarz term is present, the S-Kii.hler potential becomes modified with a non-trivial mixing to the moduli [31, 33, 34): -log(S + S)-+ -log(S +

S +fl.( I/>, 4)))

(2.8)

where, in orEier to preserve modular invariance,

(2.9)

fl. is the universal term, in Eq. (2.5), which is cancelled by the GS mechanism. The non-holomorphic term E .. C~ H.(t/>, 4)) comes from the computation of a mixed cr-model gauge anomaly and depends only on the effective couplings of the massless states and their U(1) charges under modular transformations, as defined by Eqs. (2.26)-(2.28). For smooth C-Y manifolds, A= Es or Et;, and a refers to (1,1) or (2,1) moduli, so that we have only four terms to determine. For A = Es C~Ht(r.p,T{)) =

(T(G = Es)- 5/3T(R = 27)h(t,t) -1/3T(R = 27)h(t,2))Kt(= g- 2

~ 2.

(6)

This in turn determines the string tension T which is related to the reduced Planck mass: mpz = (81rGN)-! ~ 1.8 x 1018GeV by:

(7) giving, up to additional renormalization effects between the unification scale and the string scale, the result stated in the introduction. However, if there are complete SU(5) matter supermultiplets in addition to those present in the MSSM, this will raise the value of aa, with no other effect in one-loop order. For example, for three generations of chiral supermultiplets filling E6 27 representations (note that !::.r = 0 in this case), the one-loop ,8-function vanishes for

118

SU(3)c, so that the QCD coupling ceases to evolve once the SUSY threshold ms is reached, giving

What is the interpretation of the unification scale AauT? Several different scenarios are possible, with different interpretations. For "four-dimensional" string vacua, 14 where all the extra dimensions of the heterotic string are interpreted as internal dimensions, the "radius of compactification" R T-! is fixed by the string scale and there are no breathing modes, or moduli. One obtains15 an enlarged gauge group of rank 22 or less, that breaks via the conventional Higgs mechanism to the SM gauge group, possibly via intermediate symmetry breaking steps. In "ten-dimensional" string vacua, the "observed" Es group might break via compactification to some group G that in turn breaks via the Higgs mechanism to the Standard Model. For example, the flipped SU(5) model could emerge from the symmetry breaking scenario16

=

Es SU(4~onomw 80(10) H~ni SU(5) ® U(l)' ~SM. Then the value of AauT < R- 1 , as determined by the extrapolation of the LEP data, is identified with the scale where Higgs breaking occurs. Since the U(l) of the SM is not embedded in SU(5) in this model, one generally expects a2 = a3 -=/: a 1 at this scale. 5 In all the scenarios just described, AauT is interpreted as the scale where the gauge symmetry of an effective four dimensional field theory is broken by the Higgs mechanism to a smaller gauge group. In such cases the threshold effects are well understood, provided AauT < ../T, and for "10-dimensional" strings AauT < R- 1 • The LEP data provides no information on the radius R of compactification in such scenarios. A different interpretation applies if Es is broken at the compactification scale n-l AauT to the SM gauge group times some factor group, or to a larger group that subsequently breaks at an intermediate scale to the SM group. In the latter case the fit to an RGEanalysis of the low energy data would include both A1nt and AGUT = R- 1 , as for the 80(10) model2 mentioned in Sect. 2. In these scenarios one directly extracts the value of R, for example, R ~ 102 T-! for the fit to the MSSM quoted in the Introduction. However, at the two-loop level, this

=

119

result is sensitive to the total number of complete SU(5) matter representations that contribute to the RGE. For example, each additional generation of SM fermions was found 17 to increase the value of Aaur by a factor 2; one would expect an even larger effect in the MSSM, as well as additional contributions from extra generations of (5 + 5) as in (4). Since in this second class of scenarios, the GUT scale now characterizes the transition from a 10-dimensional field or string theory to a four dimensional field theory, threshold effects are much less well understood. Calculations 18•19 of the finite corrections20 from integration over heavy string and Kaluza-Klein modes suggest that the a; do not precisely unify at a single scale. The deviation from unification is in principle calculable for a given string vacuum, but is different for different vacua. 18•21 Another problem is in the precise interpretation of the compactification scale, since the metric g;J of the complex compact manifold (or orbifold) is not in general specified by a single scale. When one extracts22 - 24 an effective 4-dimensional supergravity theory from a ten-dimensional one, the Kiihler potential K(Z, Z) that defines the low energy theory for the chiral multiplets Z, is related to the metric g;J, i,j = 1,2,3, by e

K

=

1 (Res)2det -49iJ· 16

(8)

A manifestly supersymmetric loop expansion of the effective (nonrenormalizable) 4-dimensional supergravity theory requires a field-dependent cut-off; consistent results have been found 25 •26 with the cut-off

Aaur mpl

J 16 K) = \ (Res)2e

.1. 6

=< detg;J >

_a 3,

(9)

which for compactification on three tori of equal radius, 22 g;J = 8;j(Rmp1 )~, reduces to the intuitively expected value, Aaur = R- 1 • However even if (9) is the correct scale to be used in the RGE, there could be additional threshold effects due to a nonuniformity of the background metric. Assuming that these ambiguities can be resolved in such a way that the parameters of the string vacuum can be reliably inferred from the data, one has then to produce predictions of the theory with which to compare them.

120

5. Can the Parameters of the String Vacuum be Calculated? In this final section I will describe some attempts to answer the above question in the affirmative. There is a class of models based, for example, on orbifold compactification that have a "no-scale" structure,27 with the Kahler potential (8) of the form 22·24

K = -ln(S + S) -In det(7i1 + T.;-

E ~i'~j) +twisted sector terms,

..

(10)

where S is the chiral superfield that contains the dilaton Res, and 7i1 = T1; are the moduli superfields; the vev 's of their real scalar components < Ret,1 > determine the structure of the background metric and hence the compactification scale (9). The ~f are gauge nonsinglet matter superfields with scalar components cpf. Neglecting possible massless modes in the twisted sector, the classical vacuum is supersymmetric and the vev 's < s > and < t > are undetermined classically in these models. To make contact with observed physics one must appeal to nonperturbative sources of SUSY breaking. A popular candidate mechanism is gaugino condensation28 in the "hidden" E~ SUSY Yang-Mills sector of the effective supergravity theory in four dimensions. This induces a superpotential29 for the dilaton superfield S: W = W(~) + 'he-3S/2bo 1 (11) where W(4>) is the classical superpotential for matter and bo determines the hidden sector ,8-function. However, the vacuum energy density for the theory defined by (10) and (11) is positive semi-definite. The potential vanishes at its minimum, which occurs for< W >= 0, that is, for < cp >= 0 and for either < s >-+ oo (g-+ 0) or h = 0, so condensation does not occur and supersymmetry remains unbroken. Note that the parameter h can be interpreted30•31 as the vev of the scalar component h of the lightest chiral supermultiplet H that occurs as a bound state in the confined hidden Yang-Mills sector:

had= 4 < e-&/2boh3 eKI2 >= - -g2- A3c , where - -1/2bo!J3AGUT Ac-e

is the scale of gaugino condensation.

(12)

121

Another possibility is that E~ is broken at compactification to a product gauge group32 n Gi with different .8-functions bi. This scenario has recently been reconsidered33 by Dixon et al. who took into account finite threshold corrections 18- 21 ~i that separate the values of the different coupling constants at the GUT scale. Then the effective superpotential takes the form 33

W = W(~)

+ I:hie(-35/:lb;+A;).

(13)

i

Now a vanishing energy density can occur for hi =f 0; for example a solution with 1 ~ 25 was estimated33 in the context of "four dimensional" >~ 2, strings. However the superpotential vanishes at this minimum, and therefore so does the gravitino mass

< Res

aa

(14) so supersymmetry remains unbroken. Provided one does not consider an effective theory with an unbounded potential,34 - 36 this result that gaugino condensation alone cannot break supersymmetry appears to be quite general. 3 :1 The same result was found for vanishing vacuum energy in models37•38 with several gaugino condensates using the composite bound state formalism and including threshold corrections. An additional source of SUSY breaking could be29 the (quantized) vev of the field strength Htmn of ten-dimensional supergravity. Then the effective superpotential takes the form

c ex:

J

dVImn


= 2?rn =f 0, l,m,n = 4, ... ,9,

= Y'LBMN.

L, M, N

= 0, ... '9.

(16)

For c =f 0 this effective theory has the following properties both at the classicallevel29 and at the one-loop39 level: the cosmological constant vanishes, the gravitino mass m 0 can be nonvanishing, so that local supersymmetry is broken, in which case the vacuum is degenerate, and there is no manifestation of SUSY breaking in the observable sector. Nonrenormalization theorems for supergravity, together with classical nonlinear, noncompact symmetries suggest 26 that these results will persist to all orders of the effective theory defined by (10)

122 and (15). The gravitino mass (14) is determined as (assuming AauT = R- 1 ) m ......

cA3

., GUT ~

0 ...... 4!f'm~1

3C X 1011 GeV

(17)

where the number on the right is the value suggested by the analysis3 of Amaldi et al. if AauT = R- 1 • At one loop the degeneracy in the moduli space is lifted; for the case of a single modulus, 22 9iJ = Ci;(ResRet)i, one obtains39 m0 ~ 0.1AauT ~ 0.1mp,fboc

[which is not necessarily compatible with the last expression in (17)). The effective superpotential (15) may be obtained31 •26 by first constructing an effective theory for the composite superfield H, solving for the vev of its scalar component hand setting H =< H >=< h >;for a suitably defined theory, this is equivalent to integrating out the superfield H at tree level. If instead the effective low energy theory is defined by integrating out the composite superfield at the one loop level, the chiral and conformal anomalies induce terms that break the nonlinear, noncompact symmetries in such a way that SUSY breaking is transmitted to the observable sector. Specifically a gaugino mass is generated that is of order1 m 0 mkA~ 1 _ m9 ...... ( 4 ) "'c ?rmpt 4

x MeV,

(18)

where mH is the mass of the H-supermultiplet, and the last figure is again the one suggested by (17) and the analysis3 of DELPHI data for AauT = R- 1 • The numbers in both (17) and (18)-which may be overkill-should probably be regarded as lower bounds in the general context of string models. They are very sensitive to the value of AGUT; for AauT = 1011GeV, they become, for example, me;~ 3c x 1014 GeV,

m 9 ...... c x 10TeV.

The effective theory defined by (10) and (15) does not respect modular invariance, or target space duality, under which R --+ R- 1 • Restoration of modular invariance can be achieved34•35 by replacing the constants

h --+ h(T),

c--+ c(T)

by appropriate functions of the moduli. At the level of the effective theory for the composite field H, the required T-dependence35 of h can be interpreted19

123 as threshold corrections18 from integration over heavy Ka!uza-Klein and string modes. This modification lifts the degeneracy in t at the classical level of the effective low energy theory, but the resulting potential34- 36 is unbounded in the direction s -+ 0, g -+ oo. However, a reinterpretation38 of the effective superpotential for H in terms of a wavefunction renormalization results in an effective theory that is once again positive semi-definite and of the no-scale form, 27 with a degenerate vacuum and no SUSY breaking in the observable sector at tree level. Local SUSY breaking, m 0 'I 0 can occur provided that modular invariance is broken by the classical nonperturbative vev (16): c = constant 'I 0. For this theory the vacuum degeneracy is fully tifted at one loop. The results depend to some extent on uncertain threshold effects (that can be adjusted within reasonable limits to assure a vanishing cosmological constant at the one-loop level). A preliminacy investigation40 of the one-loop effective potential in the case of three moduli and a hidden E~ gauge group shows a minimum of the potential with, for example, the parameters:

a(/~ 25,

R ~ 2.8T-~ ~ 2.2

X

w- 18Gev-l'

ma ~ 2

X

1016GeV,

where the gauge nonsinglet matter and twisted sector fields were set to zero. To determine whether or not observable SUSY breaking is sufficiently strongly suppressed in this model requires an understanding of how the threshold corrections depend on the gauge nonsinglet superfields ~i-note that for this model (see (8) and (10)] the radii of the three tori are actually given by with a= 1,2,3.

Ra =< t"'-

~jcp 01 j 2

>

The moral of this discussion is that, while the gap between superstring theory and phenomenology remains a wide one, attempts to bridge it may ultimately bear fruit. Acknowledgements. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DEAC03-76SF00098 and in part by the National Science Foundation under grant PHY-90-21139. References 1. S. Weinberg, Phys. Lett. 91B: 51 (1980); D.R.T. Jones, Phys. Rev. D25: 581 (1982); W.J. Marciano and B. Selijanovic, Phys. Rev. D25: 3092

124 (1982); M.B. Einhorn and D.R.T. Jones, Nucl. Phys. B196: 475 (1982); M.E. Machacek and M.T. Vaughn, Nucl. Phys. B222: 83 (1983}; U. Amaldi et al., Phys. Rev. D36: 1385 (1987); J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B249: 441 (1990). 2. P. Langa.cker and M. Luo, Univ. of Penn. preprint UPR-0466T (1991). 3. U. Amaldi, W. de Boer and H. Fiirstenau, CERN-PPE/91-44 (1991). 4. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32: 438 (1974). 5. J. Ellis, S. Kelley and D.V. Nanopoulos, CERN-Texas A&M preprint CTPTAMU-97/90, ACT-19, CERN-TH.5943/90. 6. B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Nucl. Phys. B278: 667 (1986) and B292: 606 (1987). 7. M. Green and J. Schwarz, Phys. Lett. B149: 117 (1984); 8. D. Gross, J. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54: 502 (1985). 9. P. Candelas, G. Horowitz, A. Stromihger and E. Witten, Nucl. B258: 46 (1985).

Phys.

10. Y. Hosotani, Phys. Lett. 129B: 75 (1985). 11. I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett B194: 231 (1987). 12. E. Witten, Nucl. Phys. B258: 75 (1985); J.D. Breit, B.A. Ovrut and G. Segre, Phys. Lett. 158B: 33 (1985). 13. F. del Aguila, G.D. Coughlan and M. Quiros, Nucl. Phys. B307: 633 (1988), erratum: ibid. B312: 751 (1989). 14. K.S. Narain, Phys. Lett. 169B: 41 (1986); I. Antoniadis, C. Ba.chas, C. Kounnas, and P. Windy, Phys. Lett. 171B: 51 (1986); H. Kawai, D.C. Lewellen and S.-H. Tye, Nucl. Phys. B288: 1 (1987).

125

15. I. Antoniadis, C. Ba.chas, C. Kowmas, Nucl. Phys. B289: 87 (1987); 1886); S. Ferrara, L. Girardello, C. Kounnas and M. Porrati, Phys. Lett. 192B: 368 (1987). 16. B.A. Campbell, J. Ellis, J.S. Hagelin, D.V. Nanopoulos and R. Ticciati, Phys. Lett. 198B: 200 (1987). 17. J. Ellis, M.K. Gaillard, D.V. Nanopoulos and S. Ruda.z, Nucl. B176: 61 (1980).

Phys.

18. L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B355: 649 (1991). 19. V. Kaplunovsky and J. Louis, SLAC-PUB to appear, and J. Louis, SLACPUB-5527 (1991), to appear in the proceedings of the 2nd International Symposium on Particles, Strings and Cosmology. 20. V.S. Kaplunovsky, Nucl. Phys. B307: 145, (1988). 21. S. Ferrara, C. Kounnas, D. Liist, and F. Zwirner, CERN-TH.6090/91, ENS-LPTENS-91/14 (1991). 22. E. Witten, Phys. Lett. 155B: 151 (1985). 23. S.P. Li, R. Peshanski and C.A. Savoy, Nucl. Phys. B289: 206 (1987). 24. S. Ferrara, C. Kounnas and M. Porrati, Phys. Lett. 181B: 263 (1986). 25. P. Binetruy and M.K. Gaillard, Phys. Lett. 220B: 68 (1989) and work in progress. 26. P. Binetruy and M.K. Gaillard, Annecy-CERN preprint LAPP-TH-273/90, CERN-TH.5727/90 (1990), to be published in Nuclear Physics. 27. N. Chang, S. Ouvry and X. Wu, Phys. Rev. Lett. 51: 327 (1981); E. Cremmer, S. Ferrara, C. Kounnas and D.V. Nanopoulos, Phys. Lett. 133B: 61 (1983); J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B247: 373 (1984); U. Ellwanger and M.G. Schmidt, Phys. Lett. 145B: 192 (1984). 28. H.P. Nilles, Phys. Lett. 115B: 193 (1982).

126 29. M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. 156B: 55 (1985). 30. G. Veneziano and S. Ya.nkielowicz, Phys. Lett. 113B: 231 (1982); T.R. Taylor, Phys. Lett. 16413: 43 (1985). 31. P. Bint!truy and M.K. Gaillard, Phys. Lett. 232B: 83 (1989). 32. N.V. Krasnikov, Phys. Lett 193B: 37 (1987). 33. L.J. Dixon, V.S. Kaplunovsky, J. Louis and M. Peskin, to appear; J. Louis, private communication; V .S. Kaplunovsky, these precedings. 34. A. Font, L.E. Ibanez, D. Liist and F. Quevedo, Phys. Lett. B245: 401 (1990). 35. S. Ferrara, N. Magnoli, T. Taylor and G. Veneziano, Phys. Lett. B245: 409 (1990). 36. M. Cvetic, A. Font, L.E. Ibanez, D. Liist and F. Quevedo, CERN-TH.5999/91, UPR-0444-T (1991). 37. T. Taylor, Phys.Lett. B252: 59 (1990). 38. P. Binetruy and M.K. Gaillard, Phys. Lett. 253B: 119 (1991). 39. P .. Binetruy, S.Dawson, M.K. Gaillard and I. Hinchliffe, Phys. Lett. 192B: 377 (1987). and Phys. Rev. D37: 2633 (1988). 40. M.K. Gaillard, A. Papadopoulos and D. Pierce, in progress.

127

Lagrangian Chiral Coset Construction of Heterotic String Theories in (1,0) Superspace S. James Gates Jr. 1 •2 and S. V. Ketov3

Department of Phy1ic1, Univer1ity of Maryland at College Park, College Park, MD !014!-4111, USA and S.M.Kuzenko and O.A.Soloviev Tomd: Univer1ity, 36 Lenin ave., Tom1k 634050, USSR

We present invariant actions describing (1, 0) supersymmetric non-Abelian leftons and rightons on coset manifolds. Our discussion covers both chiral and nonchiral cases of the (1, 0) supersymmetric WZNW actions. We analyze the supergravitational, chiral and Siegel anomalies of D < 10 heterotic strings within the Lagrangian realization of GKO coset construction. General conditions for anomaly cancellation are found and various anomaly-free solutions are discussed. Among these are the "realistic" ones, such as the Gepner's models and the four-dimensional heterotic strings compactified on orbifolds. Coupling of the heterotic string coset models to bosonic massless modes and the Sie~el anomalies of the associated nonlinear u-models are briefly considered. 1. S.J.Gates Jr., S.V.Ketov, S.M.Kuzenko and O.A.Soloviev, Lagrangian Chiral Coset Construction of Heterotic String Theories in (1,0) Superspace, Mary-

land preprint

# UMDEPP 91-193 (February 1991),

2. S.J .Gates and S. V.Ketov, More about (2, 0) Supersymmetric Two-Dimensional WZNW Model in (2,0) Superspace, Maryland preprint # UMDEPP 91-231 (April1991).

1 Research

supported in part by NSF grant# PHY88-16001-A02 of Physics and Astronomy, Howard Univenity, Washington, D.C. 20059, USA 8 0n leave of absence from: HCEI SB AS USSR, 4 Abdemicheskii, Tomsk 634066, USSR 2 Department

128 DUALITY INVARIANT EFFECTIVE STRING ACTIONS

Dieter Liist CERN, CH-1211 Geneva 23, Switzerland

Abstract We present some results concerning duality invariant· effective string actions and automorphic functions for (2,2) string compactifications.

Target space duality symmetries (like R

-+

1/R) provide very strong con-

straints (for a recent review and a more extensive list of references see [1]) when constructing the four-dimensional effective supergravity action of (2,2) string compactifications. As first shown in [2], the target space modular invariance of the effective action establishes a connection between the N = 1 superpotential and the theory of modular functions. Let us first consider heterotic string compactifications on a six-dimensional orbifold. In this case the target space duality group acting on the overall complex modulus field Tis given by the modular group PSL(2,Z). Modular functions appear, for example, in higher genus corrections to the effective string supergravity action. Specifically, the threshold corrections !::.. to the gauge coupling constants from the massive fields are of the form [3), [4]1/g 2 ex log[(T + f')I!7(T)I 4], where 77(T) is the Dedekind function. Moreover, one can also investigate the possible form of a purely T-dependent, non-perturbative superpotential. A possible choice is [2] W(T) = 17(T)-6 • The relation of this type of non-perturbative superpotential with the threshold effects can be understood [3) in the context of gaugino condensation. Now let us discuss general (2,2) Calabi-Yau compactifications with moduli fields Ti ( i = 1, ... , h). Consider the following norm llt::..(Ti)W = t::..(Ti)eK .&(11) where t::..(Tt) is a holomorphic section of a line bundle over the moduli space. Since duality transformations, being discrete isometrics of the moduli space, act as K&hler transformations on the K&hler potential K, i.e K-+ K +log IA(Tt)1 2 ,

129 it follows that for IIAII 2 to be duality invariant A has to transform as A -. (A(7i))- 1A. Thus A is an automorphic function of the discrete duality group. In order to give an, at least formal, construction of A, one uses the fact the moduli space of Calabi-Yau compactifications is a special Kabler manifold (see the contributions of S. Ferrara and J. Schwarz to this conference). Then one IM Xl+iNI:F 12 obtains the following expression [5]log IIA(7i)ll 2 = - 'EM1 ,ntlog 1 +XI:F: · Here the XI (I = 0, ... , h) are holomorphic functions (sections) of the moduli: XI = xi(Ji). F(X) is a homogeneous holomorphic function of degree 2 in the h + 1 variables X I and :FI = {}FI ax I. The sum in the above expression goes over a restricted set of integers MI and NI which is determined by the action of the duality group on X 1 and :FJ. Considering for example the previous orbifold case, one derives, using F = i( X 1) 3I X 0 , T = -iX 1I X 0 and the appropriate restriction on the summation integers (see (5] for details), A(T) = 'l(T)- 6 •

liJ:

In summary, we have described the relation between automorphic functions of

discrete duality groups and the effective string action. This is relevant for studying perturbative as well as non-perturbative effects in string theory.

REFERENCES [1] D. Liist, preprint CERN-TH.6143I91 (1991), to appear in the proceedings of the 'Workshop on String Theory', Trieste, April1991. (2] S. Ferrara, D. Liist, A. Shapere and S. Theisen, Phys. Lett. B225 (1989) 363. [3] A. Font, L.E. Ibanez, D. Liist and F. Quevedo, Phys. Lett. B245 (1990) 401; S. Ferrara, N. Magnoli, T.R. Taylor and G. Veneziano, Phys. Lett. B245 (1990) 409. (4] L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B355 (1991) 649; I. Antoniadis, K.S. Narain and T.R. Taylor, preprint NUB-3025 (1991); J. Louis, preprint SLAC-PUB-5527 (1991); J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwimer, preprint CERN-TH.6004I91 (1991).

[5] S. Ferrara, C. Kounnas, D. Liist and F. Zwimer, preprint CERN-TH.6090I91 (1991), to appear in Nucl. Phys. B.

130

CHERN- SIMONS THEORIES IN THREE DIMENSIONS FROM HETEROTIC STRING IN TEN DIMENSIONS Hitoshi NISHINO Department of Ph!11ic1 & A1tronom11 Univer1it11 of Mar11land College Park, MD l07~l, USA.

The possibility that the ten-dimensional (D = 10) heterotic superstring [1] generates a class of Chern-Simons theories (2] in three-dimensions, via compactifications on the internal seven-dimensional manifolds is presented (3]. As an explicit example of such compactifications, we give (Calabi-Yau)e X S1 X (Minkowski)3, using what is called dual formulation (4] of the heterotic string, which has the seventh-rank tensor NM,···Mr dual to the ordinary field strength GMNP· Requiring the usual quantization condition on the Yang-Mills coupling (5], as well as the consistency of the Wess-Zumino-Witten vertex operator (6] for the sixth-rank potential M 11 , ••. 110 , we get the string tension 1/o:' and the Yang-Mills coupling constant g2 in D = 10 double-quantized in terms of two integers m and n through a condensate of the antisymmetric field strength:

1 Ell'"P e 26 there are negative (as well as positive) values of k that give c = 26. In this case SO(d+l,l) can provide 1 time and d- 1 space coordinates. For the supersymmetric generalization the negative k solution appears for d> 10. 1

138 need is GI fi and the strings described by these cosets propagate in a curved space-times of d = 1 + dim(GIH) dimensions. Note that dim(GIH) is even since GIH is a Kahler manifold. The complete list of these GIH spaces is known: SU(n,m)ISU(n) x SU(n), SO(n,2}ISO(n), so•(2n}ISU(n), Sp•(2n}ISU(n), EiiS0(10), E;IE8 • The st;u.; on so• ,Sp• ,E• imply that the real forms for these groups are taken such that the GIH generators are purely non-compact. The Kihlerian cosets GI H are precisely the ones used in [6] to construct N = 2 superconformal theories. The supersymmetric generalization of these models is done in a Kazama-Suzuki style for non~ompact groups as in [6], except that the U(1) and its supersymmetric partner have to be put back in. This model has only N = 1 world sheet supersymmetry because the extra super U(1) admits only N = 1. As an example consider the super generalization of the SU(2, 1)1SU(2) model. The central charge is now c :;:: 6kl(k- 3} + 312 which gives c :;:: 15 for k = 2715. The central charge of the general supersymmetric model follows from [6] by adding 312. Thus c = l[k dim(GI H)l(k- g) + 1] where g is the Coxeter number. Note that the simplest case of putting back the U(1) occurs for the 2d Euclidean black hole SU(1,1}IU(1). The new model then is a 1+2 dimensional string described by the SU(1,1) group manifold. This is the only case of a group manifold with single time coordinate. This particular model has already been studied in [1,3].

3-Gauged WZWN Models for anti-de-Sitter Strings and Black Holes We consider the equations of motion [4] that follows from the gauged WZWN model [8-12].

(3.1a) where D:t. g = a:t. g - [A:t., g] and F+ _ = a+ A_ - a_ A+ - [A+ , A_). We concentrate on the anti-de-Sitter strings. Then A:t. is the gauge field for the Lorentz subgroup H=SO(d1,1) . The subscripts H, GI H means that the matrices that represent the quantities in parantheses have to be restricted to the H-subspace or GIH-subspace. To further develop these equations we parametrize the group element as g :;:: ht where h is an element in the left Lorentz subgroup and t is an element in the left coset. Furthermore we explicitly write t as a matrix parametrized by the Lorentz vector X" (o+ , o-) which represents a string coordinate

139 t

=

( b bX,.

(3.2)

-bX" ) ('lZ +®X,. X")

where G = (1- V1+)(2)/X2, b = l/Vl+X2 and,,.., = diGg(l,-1, ... -1) is the Minkowaki metric and Lorentz indices are lowered or raised by it. It is easy to verify that t is an SO(d- 1, 2) group element. The equations of motion now take the form

(3.16) where (D:t X)" = a:t X"- (A:t ):X", and the brackets indicate anti-symmetrization of the indices. For light-like strings ](2 = 0 in two dimensions we have X" = (t/1, ±t/1). t/1 is gauge invariant. In this case the equations of motion simplify tremendously, yielding a+ a_ t/1 = 0, which is completely solved by t/1 = t/I£(C7+) + flR(C7-), with flL,flR arbitrary functions except for periodicity conditions. The currents take the form = (a+ ¢JL, ±a+ ¢lL) and J; = (a+¢lR,±a+¢lR)· The conformal constraints Jf = J~ = 0 are also solved without any conditions on ¢lL.R.

Jr

In 2 dimensions, for follows

spac~like

JCl = tGnh(2r) .9inh(A), JCl = tGn(2t') c0.9h(A),

or

tim~like

regions we parametrize the coordinates as

X 1 = tGnh(2r) co.9h(A), X 1 = tGn(2t') .9inh(A),

11: = e:r:p (

h: = e:r:p (

~t ~)

2~ 2~') .

(3.3)

where the first line corresponds to spac~like -1 ~ ](2 ~ 0 with -oo < r,t < oo, while the second line correspond.9 to tim~ like ](2 ;::: 0 with -oo < r' < oo and -1r /4 ~ t' ~ 1r /4. Note that under Lorentz gauge transformations A' = A+~ while r,t,r,t' are gauge invariants. (We may choose A = t in the spac~like region or A = r' in the tim~ like region, if we like.) Using the equations of motion we determine D:t A in the spac~like and tim~like regions

(3.4)

140 which is valid for any gauge (since A± is not fixed). Replacing these results in equations of motion for the gauge invariant variables r, t, r, t' for any gauge

a+ a_ r -

coth(r)

. h ( ) a+ ta_ t s1n 2 r

= 0,

a a

+ - t

- a+ ra_ t +a_ ra_ t . h( r) - 0. cos h( r) s1n

(3.5)

The same form of equation holds with r replacing t and t' repacing r respectively and with trigonometric functions replacing the hyperbolic ones. They are derived from the following gauge invariant actions

The black hole occurs at r = 0 or t' = 0 corresponding to the origin X~' = 0. We have thus arrived at a black hole background of the type described by Witten, except for the cotangent functions replacing Witten's tangent function. The reason for this difference is that we have gauged the vector U(1) subgroup while Witten has gauged the axial U(1) subgroup. Thus, in the Euclidean version of our black hole we do not get Witten's cigar for the shape of the space, but rather the shape of a trumpet that diverges at r = 0. As pointed out in a number of recent papers [4,13,14,15,16] there is a duality similar to the R- 1/R duality in this case by noting tanh(r)- coth(r). It was argued in [4] that this duality became evident in a particular gauge in which the physics of both gaugings reduced to the same equation for X". It is interesting to clarify this point further by relating our X" to the Kruskal coordinates used by Witten. For the vector gauging, as above, we identify the Kruskal coordinates and through (3.3) relate them to X± = XO ± X 1 as follows (we take the gauge A = t, r' as indicated above) v

= e'coshr

u

= e-•coshr;

x+

= 2vy'uV=l

X-

2uv -1

= _ 2uv'uv 2uv -1

1

V,VI

II III v = e•' cost' u = e-•' cost'; x+=2v~ x- = 2 u~ B B' 2uv -1 2uv -1

(3.7)

The first line is for space-like and the second line for time-like X2. The roman numerals correspond to regions of the manifold as explained below and as in Figs.l-3. Similarly, for the axial gauging these equations take the form

v = e'sinhr u

= e-• sinhr; x+ =

x-

uv = 2uy'l1- 2uv

I,IV

v = e•' sint'

= e-•' sint';

x-

uv = 2uv'11- 2uv

IIA,IIIA

u

2vy'l- uv 1-2uv 2vv'l- uv x+ = 1- 2uv

(3.8)

141

In both cases all apace and time like regions in the entire X• plane are covered by each one of these parametrization&. However, the vector and axial gauginga, when viewed from the point of view of the X• variables cover different regions of the Kruskal u- v plane. When the u- v plane is covered once, the X• plane is covered twice (see figs. 1,2,3). We are led to think of the X• plane as double sheeted. One sheet is described by (3.7) the other by (3.8). The black hole singularity appears on the sheet described by (3.7) at the origin of the X" universe. We can identify the various regions of the u- v plane and Xp. plane as follows. Let us use Witten's identifications of the regions I, II, III, IV, V, VI in the u- v plane, however we will divide his regions II and II I into two parts IIA, I Is, I IIA , II Is by drawing the curves that correspond to uv = 1/2. The B-regions are sandwiched between uv = 1/2 and uv = 1. Then we see that the black hole singularity that occurs at uv = 1 must appear in the middle of the regions IIs - V or II Is - VI which correspond to (3.7). The duality transformation relates the regions (!, IIA, IV, IliA) to the regions (V, lis, Ills, VI). Thus duality transforms the first sheet of the X" plane to the second sheet. The universe looks quite interesting from the point of view of the X" variables. Spacelike regions V, VI (or I, IV) are finite and expanding with time in the future and contracting in the past, since X: < 1 + X:. Signals from the space-like edge of the universe (r -+ oo or )(2 = -1) can cross to the time-like regions. So observers in regions 11,111 can see "background radiation" coming from the edge of the universe in V, VI or I, IV. This is an expanding universe that starts with a Big-Bang in the sheet described by (3.7). The Big-Bang at the origin X" = 0 is represented by the black hole singularity. The sheet (3.8) which has no singularity is dual to the sheet (3.7).

4-The quantum theory The study of the quantum theory for the curved space-time strings by using current algebra techniques has revealed a problem with negative norms [1,3]. By now we have realized that this is a general problem that seems to be shared by all curved space-time theories whether they are described by current algebras or not. Perturbative beta function techniques do not really have a control over the metric or other background fields, since they need to be adjusted at each order to keep the beta functions at zero for conformal invariance. By contrast in the current algebra approach it has been possible to keep exact conformal invariance and solve for physical states exactly. The negative norm problem revealed by this technique is missed in the perturbative approach. One message is that more constraints than conformal invariance are needed to define a physical theory. Another

142 possible message is that the negative norms are sipals of instability that need to be understood in physical terms (e.g. Hawking radiation). Following [4) we will suggest some solutions below. Let us first explain the general source of negative norms. For the string models discussed here there are two sources of negative norms. The first is the familiar negative norm time-like oscillators that are present even for the flat string. These are eliminated by the Virasoro constraints. The naive counting of the number of negative norm oscillators and number of constraints match and therefore it can be expected that this source of negative norms is not the problem. The no ghost theorem of flat strings can be extended to take care of these negative norms. The second source of negative norms is the curvature of space-time and will happen even without negative norm oscillators as follows. From the study of the space-like Hermitian symmetric space cosets G / H in [6] we know that unitarity puts a limit on the representations that can occur at the base. This result is analogous to the unitarity conditions for SU(2) current algebra j :$ k/2 or SU(1,1)/U(1) coset j + 1 < k/2 (6,7]. Note that in all these cases the coordinates are space-like. H the representations exceed the unitarity limit there are negative norm states. The source of these negative norms is the curvature of the space. One may be tempted to deal with these negative norm states by simply throwing them out, in the same way that we do for SU(2). In the context of a complete string theory (as our c = 26 models are) the higher excited states of the string directly lead to higher representations that exceed the unitarity bounds. Therefore limiting the representations is equivalent to limiting the string to Jive in only low excitations. This possibility exists but seems rather unlikely for the generic cases, since one must be able to show that scattering of low excitations never leads to the high excitations (i.e. operator products must close among the low excitations). In particular, we know that for SU(2) when k is not an integer, it is not possible to cut- off the Hilbert space to a finite dimensional unitary subspace. In the generic case of curved space-time strings we do not have the luxury of integer k, let alone the fact that such a theorem is not known to exist for the non- compact current algebras. We must emphasize one additional point. In comparison to the previous study in ref.[6,7) we need to further apply the Virasoro constraints and analyse the remaining physical subspace. As it turns out, from the examples studied so far [1,3,4] there does remain negative norms even after the Virasoro projection. Thus, these negative norms should be expected quite generally and an appropriate solution must be found. In the detailed analysis of the 2d black hole string in (4] we have noticed that if we restrict ourselves to states that correspond to wavefunctions or vertex operators that are single valued as a function of the black hole manifold, then negative norm states seem to

143 be projected out. A complete proof of this fact is still lacking, but the calculations so far are encouraging. Perhaps this is a sufficiently general approach toward eliminating the undesirable negative norm states. A comparison of the states listed in [4] after the above projection, and the so called special states of 2d gravity as described by a Liouville theory coupled to a c=1 matter system [17] can be made [18]. It is seen that the black hole states of [4] match half of the states of the Liouville theory as far as the quantum numbers are concerned, but the degeneracies for each quantum number are not yet known to match. In particular, the zero norm structure of the states identified in [4) remains to be understood. Another solution was suggested in [4]. This involved the construction of current algebra representations that automatically lead to modules in the continuous series of SU(1,1) that do not have any negative norms. In this case it turns out that the string cannot have any excitations. Thus the physics is quite different than the solution of the previous paragraph. This second case is also studied in [15]. There remains one other possibility to deal with the negative norms. In our discussion of the norms above we concentrated only on the left-moving sector. The full theory is constructed as a product of states from the left and right moving sectors in such a way that they are modular invariant. It is possible that the modular invariant combinations of left and right moving states project to overall positive norms even though the purely left or right theory may have negative norms. Unfortunately, modular invariants remain to be understood for non-compact current algebras, even for SU(1, 1).

5-Discuaaion The physics of the string in the black hole remains still open. Does the string get excited at all in the black hole? Is there a Hilbert space that is complete, unitary as well as modular invariant? We have offered four possiblities: (i) single cover of the manifold, (ii) continuous series, (iii) Modular invariant projection to positive norms, (iv) truncation to only low excitations if operator products close in some cases. They describe very different physics. The resolution of these issues in two dimensions is likely to generalize to the higher dimensional Anti-de-Sitter strings. H we would imagine that there is no consistent Hilbert space then what does it all mean? It is possible that such inconsistency is the signal that the string would rather radiate (like Hawkins radiation) than set exci~ed into nesative norm states. It may also

144 be a signal analogous to the Klein paradox, which means that we are using the wrong formalism to describe the physics. A formalism that can deal with multi-strings, like string field theory, may be the answer, just like field theory is the answer to the Klein paradox. It seems to us that, answering these questions is an important physics problem concerning string theory. Their resolution may be of relevance to understanding string singularities and conditions similar to the early stages of the Universe.

REFERENCES

[1J I. Bars and D. Nemeschansky, Nucl. Phys. B348 {1991) 89. [2) C.Callan, D.Friedan, E.Martinec, M.Perry, Nucl.Phys. B262 {1985) 593. [3J J. Balog, L. O'Raifeartaigh, P. Forgacs and A. Wipf, Nucl. Phys. B325 {1989) 225. P.M.S. Petropoulos, Phys. Lett. B236 {1990) 151. (4) I. Bars, String Propagation on Black Holes, USC-91/HEP-83, May 1991. [SJ E. Witten, On String Theory and Black Holes, lAS preprint, IASSNS- HEP-91/12 {March 1991). [6J I.Bars, Nucl.Phys. B334 {1990) 125. [7) L.Dixon, J.Lykken and M.Peskin, Nucl.Phys. B325 {1989) 329. [8J K. Bardakci, E.Rabinovici and B.Saering, Nucl. Phys. B299 {1988) 151. [9J Y. Frishman and A. Sonnenschein, Nucl. Phys. B301 {1988) 346. [10) P. Bowcock, Nucl. Phys B316 {1989) 80. [UJ K.Gawedski and A. Kupiainen, Nucl. Phys. B320 (1989) 625. [12) H.J. Schnitzer, Nucl. Phys. B324 (1989) 412. D. Karabali, Q-Han Park, H.J. Schnitzer and Yang, Phys. Lett. B216 {1989) 307. ; D. Karabali and H.J. Schnitzer, Nucl. Phys. B329 {1990) 649.

z.

145 (13) A. Giveon, Target space duality and Stringy black holes, Berkeley preprint LBL-30671, May 1991. (14) E. Kiritsis, Duality in Gauged WZW models, Berkeley preprint, LBL-30747, May 1991. (15) R.Dijkgraaf, E.Verlinde, H.Verlinde, String propagation in black hole geometry, lAS preprint IASSNS-HEP-91/22, May 1991. (16) A.A. Tseytlin, Duality and the Dilaton, Johns Hopkins preprint, JHU-TIPAC-91008, May 1991. (17) D. Gross, I. Klebanov, M. Newman, Princeton preprint, PUPT-1192. B.H.Lian and G.J.Zuckerrnan, Yale preprint YCTP-P18-91 , May 1991; A. Polyakov, Lecture in the conference. (18) J.Distler, private communication.

Figure Captions Fig.1 - Regions in the u - v plane. Fig.2- Regions in the X" plane. Vector. Fig.3 - Regions in the X" plane. Axial.

Fi~. 2.

146

Physical States of the String in a Black Hole Background* Jacques Distler Joseph Henry Laboratories Princeton University Princeton, NJ 08544 USA Philip Nelson Physics Department University of Pennsylvania Philadelphia, PA 19104 USA

I review the construction of the physical spectrum for the "black hole" solution recently proposed by E. Witten as an exact string background with a 2-dimensional target space. The spectrum contains some of the states found in studies of d = 1 noncritical string theory but, in addition, has new states not previously found in the d = 1 noncritical string. Along the way I will note a remarkable "stringy" symmetry of the spectrum relating massive states to massless ones and comment on the relation between this theory and the d = 1 noncritical string theory.

9/91 *This work was supported by NSF grants PHYSS-57200 and PHYS0-19754.

147 The subject of this talk is discussed in great detail in the recent preprint [1). For these Proceedings, I will try to give an overview of that work (3) and make some comments on future directions. The interpretation of certain noncompact coset models (4) as string black holes in a two-dimensional spacetime was first given by Witten [2). The SL(2,R)/S0(1, 1) coset model corresponds to a Minkowski signature black hole, while the SL(2,R)/U(l) coset corresponds to what might be called the Euclidean black hole. Since string theory is supposed to provide a theory of quantum gravity, it is interesting to probe how the exact string theory in these backgrounds compares with the geometrical picture one obtains from the classical a-model. More generally, one is interested in probing the consistency of string propagation in curved spacetime backgrounds. We are fairly used to considering background in which the spatial directions are curved. From the world sheet point of view, we replace some of the 26 free bosons which describe the string in flat space with a more general nonlinear a-model, or more abstractly, by some unitary conformal field theory with the same central charge. In flat spacetime, the time direction is represented by a wrong-signature free boson. This is a nonunitary (albeit free) CFT. When we look at curved spacetimes, we must therefore consider more general nonunitary CFT's. To obtain a consistent quantum mechanics out of such a theory, we need some generalization of the No

Gho~t

theorem (5-7) which, in flat

spacetime, guarantees the existence of a consistent truncation of the theory to a space of only positive norm states. The 2-d black hole is the perhaps the simplest string theory in which one can explore the generalization of this to non-flat spacetimes. The modern approach to the no-ghost theorem is to define the BRS operator Q associated to the gauge symmetry of the problem. One then passes from the full (indefinite) Hilbert space to the (hopefully positive definite) cohomology of Q. In the present case of SL(2, R)/U(1) coset conformal field theories, there are two alternative viewpoints one can take: 1) We first construct the coset CFT by imposing the the U(l) highest weight condition and then couple the resulting theory to gravity. The BRS operator that results is

Q

=

f

c Tco•et

+ bc{}c

which is associated to gauging the diffeomorphism symmetry. 2) Alternatively, we can take seriously the realization of the coset model as a gauged WZW model. Therefore we are gauging both diffeomorphisms and the U(l).

148 These two are, in fact equivalent. One can view the first as computing an iterated cohomology, HQ(HiJucl) ). The latter amounts to computing the cohomology of the "total" BRS charge Qr = Q+Qu(t)· It turns out that the spectral sequence of the double complex (where the bigra.ding is by diffeomorphism- and U( 1)-ghost numbers) degenerates at the

Eh term which shows that the cohomology of Qr is isomorphic to the iterated cohomology. Unfortunately, unlike the case of compact SU(2), we don't really know how to construct these SL(2,R) coset models as full-fledged string theories. Indeed, by identifying which current algebra representations give rise to nontrivial BRS cohomology, we are really taking the first step in that direction. Having identified the relevant coset modules, we need a prescription for putting together left- and right-moving degrees of freedom (more on this later) and finally for assembling the whole shebang into a modular invariant theory. In particular, it is crucial that the operator algebra close. In the case of compact SU(2), the fusion rule are a truncCJted version of the usual rules for the addition of angular momenta. The representations with spins j $ k /2 close upon themselves. This unfortunately depends rather delicately on the null-vector structure of tbe SU(2) current algebra representations, and seems unlikely to hold in the SL(2, R) case. We'll take the conservative approach and consider all j. Since this is a somewhat confusing point, I'd like to emphasize the difference between the BRS approach (or any other approach) to the no-ghost theorem and what is commonly done in constructing unitary conformal field theories

a l4 BPZ. When one starts with a

positive-semidefinite Hilbert space, one can construct by hand a positive-definite Hilbert space as the quotient by the subspace of null vectors. In an indefinite Hilbert space, this simple proceedure is not possible. Instead, we must

fir~t

restrict to the (positive-

semidefinite) subspace of BRS-closed states before we can take the quotient. Without loss of generality, we may consider representations of the current algebra built upon irreducible representations of global SL{2,R) at the base. These are classified by the spectrum of m eigenvalues at the base. If m is unbounded, as occurs for generic values of j, m then we label the resulting coset module C. H m is bounded from below at the base, we have a lowest weight representation, and we can distinguish two subcases: 1) the minimum value of m is j

+ 1, in which case we label the resulting coset module v+, or

2) the minimum value of m is - j, in which case, we label the coset module i5+. Similarly, for highest weight representatious, m is bounded from above at the base, and these give rise to coset module v- and 15-. Finally, starting with representations in which m at the base is bounded from both above and below, we obtain the coset modules U.

149 The BRS cohomology that one finds in these modules comes in two basic flavours. First are states which come from the base of the current algebra representation. These have the form (Cn are the ghost oscillators)

CJij,m) with

m = ±3(j + 1/2) . These states have zero oscillator number and are the analogues of the tachyon states in the d = 1 noncritical string. Of course, "tachyon" is something of a misnomer. In two spacetime dimensions, such states are massless. The states at higher oscillator number come paired. For each state in the cohomology at ghost number one, there is a state at

~host

number zero. If there are two states in the

cohomology at a given mass level, then there is another state in the cohomology at ghost number two. The situation is summarized in the following table.

Representation

c

1515vv15+ 15+ v+ v+ u

J

m

dimH~

dimH~

dimH~

~(s+r-1)

~(s- r)

1

1

0

l{2s +4r -5)

-~(2s- 4r- 1)

1

0

~(2s + 2r- 3)

-~(2s- 2r + 1)

0 1

Hs+2r-3)

~(s-2r+1)

0

0 1

0

l(2s + 4r- 5)

~(2s-4r-1)

1

0

0

i(2s + 4r- 5) ~(2s + 2r- 3)

~(2,-4r-1)

0 1

1 0

0 0

0

1

0

-~(2"- 4r- 1)

1

0

0

~(s- r)

1

0

1

Hs + 2r- 3) l(2s +4r -5) ~(s + r- 2)

~(2"- 2r + 1) -Hs- 2r + 1)

0

All of this was for the Euclidean SL{2,R)/U(1) coset. The case of the Minkowskian SL( 2, R) /SO( 1, 1) coset diJfers in that there we wish to diagonalize one of the noncompact generators. That would not be neoessary were we only concerned with constructing states at mass level zero (as was the focus in in (8] ). The massless states are annihilated by the positive frequency modes of a.ll of the currents. To get the whole parafermion module, we need to construct states annihilated by the positive frequency modes of the so{1, 1)

150 generator, but not necessarily by the other two currents. Hence it is essential that the so( 1' 1) generator be diagonalizeable.

In an indefinite Hilbert space, the familiar statement that a Hermitian operator has only real eigenvalues does not hold. Nevertheless, for an irreducible representation, the eigenvalues, p of the so(1, 1) generator cannot be arbitrary complex numbers. Rather, we must have Im(p) E !Z. In fact, the spectrum that we find has either J.l pure real or pure imaginary. The base states ("tachyons") have pER and j

= -! ± ip/3.

A priori, we also have the analogues

of a subset of the discrete states found above with j real and J.l = im

E iZ. Quit possibly,

however, it may be consistent to truncate the physical spectrum to real p, in which case, the only discrete states would have J.l

= 0.

As already alluded to, the tachyons one finds here are in precise correspondence with those that one finds in the d

= 1 noncritical string.

If we want a precise dictionary, we

need only compare the dispersion relations in the two theories. One finds that the correct identification is Px

= -2v'2 3 -(m or p),

The relation between the discrete states of the two models is a little more complicated. Using the above dictionary, the discrete states of Liouville [9] at ghost number 1 (the canonical ghost number) occur at j

= !{u + v- 2)

and m

= !(u- v) for

u,v positive

integers. From the above table, we see that these are in precise correspondence with the discrete states from the modules

v+, v- ,c, except that for u, v both even, there are two v+ and one from v- ). The states from the modules

states in the coset module (one from

£)± seem to have no correspondents in the Liouville theory. The discrepancy between the two theories may not be as large as it appears at first. As discussed in (1], there is a stringy isomorphism which relates

i>j

to Vj+l/&' so we are

perhaps double counting if we include both. This still does not explain the double degeneracy for u, v even. It also spereads a cloud over the guiding principle of our "dictionary". The isomorphism relates states of different j, m and different oscillator numbers, hence these are somewhat ambiguous quantum numbers for labeling states of the coset theory. The correspondence we have been trying to establish is perhaps best viewed as a semiclassical (large k) one. The existence of the discrete states in the coset model is an .. inherently stringy phenomenon. One can show that for k > 9/4, all but two of the discrete states disappear. The existence of an infinite number of discrete states is an artifact of

151 k

= 9J4 (that is, of having a two dimensional target space for the string).

Reasoning valid

at large k should not be expected to reproduce them accurately. The situation is somewhat reminiscent of the familiar R

1/ R duality symmetry.

-+

Semiclassical reasoning is certainly suspect at small R. The novel feature here is that the new symmetry does not commute with oscillator number (as R

-+

1/ R does), and thus

exchanges states of different spins! Probably, the existence of such a symmetry is, as is much else in this subject, a peculiarity of 2 dimensional target spaces. Nevertheless, it would be interesting to find other (higher dimensional) examples of this phenomenon. I should mention one final point on the relation between the "black hole" and the d = 1 noncritical string (with a compact free boson X). In both cases, one should properly

allow for the existence of "winding states"[8). That is, when we put together left- and right-moving degrees of freedom, the allowed values of m, m should be correlated

m = !(n + kn'),

m=

!(-n + kn')

for n, n' E Z. Note that this is in sharp constrast to the case of simply connected current algebras such as SU(2), where m- m E z. The proof of all of these statements [1) relies, in part, on the free-boson representation of the coset modules•. Since it is somewhat perilous to take the free-boson representation too seriously, we should clearly isolate what we actually used it for.

In fact, we used the free-boson representation to obtain two pieces of information: first, we used it to find the cohomology of the coset module Cj; second, we used it to establish the "stringy isomorphism" between the coset modules Vj and

i5j_ 118 • Either of

these could have been established, perhaps with greater difficulty, using other methods. Having computed the physical spectrum, we can return to the no-ghost theorem. Strictly speaking, the no-ghost theorem does not require the all physical states have positive norm. There is a limited scope for having discrete states with negative norm. Indeed, there are such states even in the critical bosonic string in 26 dimensional fiat space. However, in order not to screw up the unitarity of physical amplitudes, these states must be delicately paired with states of positive norm with the same quantum numbers. This holds true in the critical bosonic string, and it holds true in the Minkowski SL(2,R.)/SO(l, 1) coset. Propagating states, on the other hand, must be positive norm.

This also holds true,

• For some other recent applications of the free field representation in this context, see

[10].

152 although since the only propagating state is the "tachyon", it was never seriously in doubt. String propagation in this two dimensional background passes this most basic of consistency checks. Clearly, one would like to go beyond this and consider some higher dimensional examples, where the no-ghost theorem would be realized in a less "trivial" fashion. It is also clear that one would like to do better than we have, and actually construct the full conformal field theories associated to these noncompact coset models. As we have seen, some of the most novel features of the black hole, like those of the noncritical string, are unlikely to persist in higher dimensions. It will be interesting to see what new feature arise instead.

153 References

(1] J. Distler and P. Nelson, "New Discrete States of Strings Near a Black Hole", Princeton preprint PUPT-1262 (1991). (2] E. Witten, "On string theory and black holes," IASSNS-HEP-91/12. [3) See also: I. Bars, "String propagation on black holes", USC preprint USC-9-HE-B3 (1991) for a somewhat different view on this subject. [4) For earlier work on this class of theories, see e.g.: J. Balog, L. O'Raifeartaigh, P. Forgacs and A. Wipf, "Consistency of string propagation on curved spacetimes: an SU(1, 1) example", Nucl. Phys. B325 (1989) 225; P. Petropoulos, "Comments on SU(1, 1) string theory", Phys. Lett. 236B (1990) 151 and especially: I. Bars and D. Nemeschansky, "String propagation in backgrounds with curved spacetime," Nucl. Phys. B348 (1991) 89. [5] P. Goddard and C. Thorn, "Compatibility of the dual pomeron with unitarity and the absence of ghosts in the dual resonance model," Phys. Lett. 40B (1972) 235; R. Brower, "Spectrum-generating algebra and no-ghost theorem for the dual model," Phys. Rev. D6 (1972) 1655. [6) M. Kato and K. Ogawa, "Covariant quantization of string based on BRS invariance," Nucl. Phys. B212 (1983) 443. [7) l. Frenkel, H. Garland and G. Zuckerman, "Semi-infinite cohomology and string theory," Proc. Nat. Acad. Sci. USA 83 (1986) 8442. [8) R. Dijkgraaf, H. Verlinde and E. Verlinde, "String propagation in a black hole background", Princeton preprint PUPT-1252 (1991). [9) A. Polyakov, Mod. Phys. Lett. A6 (1991) 635; D. Gross and I. Klebanov, "S=1 at c=1", Nucl. Phys. B359 (1991) 3. [10] E. Martinec and S. Shatashvili, "Black hole physics and Liouville theory", Chicago preprint EFI-91-22 (1991); M. Bershadsky and D. Kutasov, "Comment on gauged WZW theory", Princeton preprint PUPT-1261 (1991).

154

ASHTEKAR'S APPROACH TO QUANTUM GRAVITY

GARY T. HOROWITZ Department of Phy1ic1, Univer1ity of Co.lifornio. So.nto. Barbara, CA 93106-9530, USA

ABSTRACT A review is given of work by Abhay Ashtekar and his colleagues Carlo Rovelli, Lee Smolin, and others, which is directed at constructing a nonperturbative quantum theory of general relativity.

1. Introduction

I have been asked to review the current status of an approach to quantum gravity which is being developed by Abhay Ashtekar and his colleagues Carlo Rovelli, Lee Smolin, and othersl. I should emphasize that I have not actively worked on this approach and as a result, my knowledge of it is somewhat incomplete. However I have followed the progress in this area and would like to describe for you the main ideas involved, the current status, and open problems. Ashtekar and colleagues are trying to quantize standard four dimensional general relativity without supersymmetry, higher derivatives, extra dimensions, extended objects, etc. The first question that probably comes to mind is why are they wasting their time on a program that is doomed to failure? Isn't it well known that general relativity cannot be quantized? Perhaps surprisingly, the answer is no. It is, of course, known that general relativity is not perturbatively renormalizable2 . But unlike the case for most quantum field theories, this may not be as bad as it sounds. General relativity is qualitatively different from other field theories in that the dynamical field is the spacetime metric. One might even argue that standard field theory perturbation techniques 1hould break down since they are based on the assumption that spacetime looks Minkowskian at arbitrarily short distances, which is not very plausible in quantum gravity. As I will describe, there are indications that quantum general relativity provides a natural cut-off at the Planck scale. Ashtekar works in the framework of canonical quantization. Thus it is analogous to the functional Schroedinger representation for ordinary field theories 3 • However, as we will see, the reparameterization invariance of general relativity leads to certain simplifications. Canonical quantization of general relativity has, of course, been tried before. But previous investigations have almost always used the spatial three metric and its conjugate momentum as the basic canonical variables. This leads to constraints which are difficult to solve (or even make sense of) in the quantum theory. Ashtekar instead chooses canonical variables which are analogous to those in ordinary gauge theories. The resulting constraints are simpler and more progress can be made toward constructing the quantum theory. To motivate Ashtekar's choice of canonical variables, we begin by considering general relativity in three dimensions. This theory can be described in terms of an

155

w'/!

and a triad of (dual) vectors e:. (The spacetime metric is S0(2,1) connection defined in terms of the triad by 9p.v = e:et'lob·) The action is

S

=

J

(1)

ea II Rbc fabc

where R = di...J + w II w is the curvature two form or field strength of w. This action in {act describes a slight extension of general relativity. When e: consists of three linearly independent vectors, one can show that (1) is equivalent to the usual Einstein action J R.;=g. But the action (1) and the resulting field equations remain well defined even in the limit that the triad becomes linearly dependent. Thus this theory includes degenerate metrics. We will return to this point in Section 3. Witten has shown4 that if one chooses the dynamical variables to be the spatial components of the connection Wi and its conjugate momentum Ei (which is just the dual of the spatial components of the triad), then the canonical quantization of this theory can be carried out exactly. The theory has two constraints:

(2) (3)

~i =0

The first is the familiar Gauss' law. The second says that the spatial connection Wi is flat. As a result of reparameterization invariance, the Hamiltonian is proportional to the constraints. Thus, to construct the quantum theory, one does not need to solve the time dependent Schroedinger equation. It suffices to find states which are annihilated by the quantum version of the constraints. One can represent states in terms of functional& of the connection. Imposing (2) requires that ,P(w) be gauge invariant and imposing (3) requires that ,P have support on just the flat connections. So physical states are functional& of gauge inequivalent flat connections.

2. Toward Quantum General Relativity Since the above approach works so well in three dimensions, it is natural to try it in four. This leads directly to Ashtekar's variables. (Actually, Ashtekar began his four dimensional work several years before the three dimensional case was considered5 !) In four dimensions, general relativity can be described in terms of an The action is S0(3,1) connection wf}' and a tetrad of (dual) vectors

e:.

S=

j ea

II e 11 II g:d fabcd

(4)

Using the three dimensional case as a guide, one is tempted to consider the spatial components of the connection and its conjugate momentum as the basic dynamical variables. Unfortunately, if one casts the theory into canonical form, one finds that some of the constraints are now second class. One can explicitly solve the second class constraints, but the remaining constraints become nonpolynomial6 • Ashtekar's key insight 5•7 was to replace Wp. with its self dual part A,. Wp.- i • w,.. (One can show that the action (4) with Wp. replaced by A,. is still equivalent to

=

156

general relativity.) At first sight this appears to be a rather minor change. One is essentially replacing a connection having 24 real components with one having 12 complex components. However closer examination reveals that the consequences are much deeper than that. This is most easily seen in the Euclidean context. The Euclidean Einstein equations can be obtained from the action (4) with either an S0(4) connection or ita self dual part A11 which is a real SU(2) connection. Thus one can eliminate half of the components of the connection without losing any information! (The reason is basically that the two actions differ by a term proportional to J R(#wpa) which does not contribute. Since one retains the full tetrad, one has the spacetime metric and in any solution, one can always reconstruct the complete connection and its curvature.) In addition to the obvious economy of fields, there are further advantages to working with A11 • For example one can show that Einstein's field equation (with arbitrary cosmological constant) is equivalent to the 1elf dual YangMill' equation for the connection A118 . (More precisely, it is equivalent to the self dual Yang-Mills equation in a curved background where A11 is equal to the self dual part of the spin connection.) Using this correspondence, one can find gravitational analogs of SU(2) Yang-Mills instantons: The one instanton solutions turn out to correspond to the four sphere, with the size of the instanton related to the radius of the sphere9 . Returning to the Lorentzian context, one finds further advantages of using the self dual connection when one constructs the canonical formulation of the theory. The dynamical variables are the spatial components of the connection Ai and its conjugate momentum Ei which contains the information on the tetrad. The constraints are all first class and take the form DiEi

=0

=0 Tr FijEiEi = 0 Tr Fi;Ei

(5) (6) (7)

Since Ai is complex, there is also a reality condition that must be imposed*. The first constraint is the standard Gauss law constraint of Yang-Mills theory. Thus every initial data set for general relativity is also an initial data set for an SU(2) Yang-Mills theory. The only difference is that it is also subject to four additional constraints which are related to reparameterization invariance. Note that the degrees of freedom match: SU(2) Yang-Mills theory has 3 x 2 = 6 degrees of freedom at each point which are reduced to 2 by the four additional constraints. It should be emphasized that even though the initial data is similar, the hamiltonian for general relativity is very different than Yang-Mills theory. As in the three dimensional case, reparameterization invariance ensures that the hamiltonian for general relativity is just a multiple of the constraints (up to a surface term at infinity).

* The momentum conjugate to Ai is al10 complex, but it tum• out that ita imaginary part commuta with Ai· Thus one can choose Ei to be real. The reality condition is simply that p;i and ita first time derivative - computed via PoiaiOn brackets wiUa the hamiltonian - be real.

157

Notice that all the constraints are simple polynomials in the basic fields. {The reality condition is also polynomial.) This is one of the main reasons that this approach initially attracted so much attention. But just from the form of the constraints it is difficult to tell how much of an advance this represents. Polynomial equations do not, of course, imply that the quantum theory is necessarily solvable (or even exists!) Although the standard constraints in terms of the spatial metric and conjugate momentum are not polynomial, they can be made so by simply multiplying by appropriate powers of the determinant of the metric. Furthermore, the constraint (7) is quadratic in the momenta, which means that the corresponding operator involves functional derivatives at the same point and must be regulated. This was also true in the old variables and was perhaps the main difficulty in finding solutions to the quantum constraints. To see the real advantage of this form of the constraints one must begin to construct the quantum theory. Classically, the Gauss law constraint {5) generates gauge transformations, just as in any gauge theory. One can show that the vector constraint (6) generates reparameterizations of the three dimensional surface and the scalar constraint {7) is related to reparameterizations of time, or motions of the spatial surface in the four dimensional solution. To construct the quantum theory, we begin by representing states by functionals of A,. We wish to tum the classical constraints into quantum operators by replacing E 1 by -i6/6Ai and define physical states to be those annihilated by the quantum constraints. This of course requires a choice of factor ordering. For the constraints linear in the momenta, the ordering given in {5) and {6) ensures that the quantum constraints have a similar action as their classical counterparts. However the quantum scalar constraint (7) has no direct interpretation since, as we have already mentioned, it must be regulated. Jacobson and Smolin have shown 10 that there exists a regulated form of this constraint C6 and a class of states ,p.., {parameterized by a loop 7) such that lim

6--+0

c6 ,p.., = 0

(8)

The regulator is a type of point splitting in which the functional derivatives are evaluated at dift'erent points separated by a distance 6. The states are just the familiar Wilson loops. Given a non-self-intersecting smooth closed curve 7, set

,P..,(A) = TrPeJ.,

A

(9)

Roughly speaking, the reason this satisfies the constraint is that each 6f6Ai brings down a term proportional to the tangent vector to the curve. Both of these tangent vectors are contracted into the antisymmetric Fa; and hence vanish. This type of solution is possible only if one uses variables like Ashteltar's in which the momentum E' has two type of indices {the tangent space index i and an internal index which we have suppressed). Although the constraint (7) is symmetric under interchange of the two momentum {as it must be), it is anti-symmetric under interchanging each type of index separately. There are several reasons why one might feel uneasy about this result. First, since one must introduce a notion of distance to regulate the constraint, the regulator breaks three dimensional reparameterization invariance. Formally, this invariance is restored as the regulator goes to zero, but there is always the possibility of

158 anomalies. A related difficulty is that 06'1/J-y 'I 0 for 6 'I 0. Thus in some sense the regulator breaks four dimensional reparameterization invariance as well. Finally, the regulated constraint is not unique. At the moment, there are several proposals for the regulated constraint which appear to be inequivalent 11 . Nevertheless, this is a significant achievement. Despite extensive work on the old canonical formalism for general relativity, no one has ever achieved an analogous result. The analog of the scalar constraint in the old variables is known as the Wheeler-DeWitt equation. Because of the difficulty in regulating and solving this equation, extensive work was done on simpler "minisuperspace" models in which one freezes out all but a finite number of degrees of freedom of the metric*. The full Wheeler-DeWitt equation has never been solved. Even more remarkable is the fact that the solutions to the (analog of the) Wheeler-DeWitt equation are just the simple Wilson loops. These have long been considered as natural gauge invariant variables for describing Yang-Mills theory both classically and quantum mechanically 12 • The fact that these same objects solve the scalar constraint of quantum general relativity is quite surprising. Although t/J-y solves the scalar and Gauss law constraint, it does not solve the vector constraint. In a key development, Rovelli and Smolin showed that one can obtain solutions to all quantum constraints by passing to a new representation in which states are functional& of loops 13 . This can be obtained formally by the integral transformation

.,P(-y) =

j VA W(-y, A) .,P(A)

(10)

where the kernel is again the Wilson loop W("Y,A)

= Tr PeJ..,A

(11)

This transforms functional& of A into functional& of loops. Alternatively, the loop representation can be introduced directly by starting with an algebra ofloop observable&, computing their Poisson bracket, and introducing operators on functional& of loops with the same commutation relations. One then expresses the constraints as operators in the loop representation. Since gauge invariance is automatic, there is no analog of the Gauss constraint. From the above discussion, it might appear that there should be no analog of the scalar constraint either. This would indeed be the case if one could restrict to only smooth non-self-intersecting loops. On the one hand this eounds reasonable since all gauge invariant information in the connection is contained in the Wilson loops for this class of "Y. On the other hand, to obtain a closed Poisson bracket algebra for the loop observables, it seems necessary to work with the larger space of all piecewise smooth loopst. Fortunately, even in this larger space, one can satisfy the scalar constraint by simply restricting the functional& to

* There ia a striking similarity between the motivation that used to be given for working on miniaut

perapac:e models and the motivation one currently hears for two dimensional gravity. The Poisson bracket ia nonsero only when two loops intersect, and involves loops with corners and intersections which result from combining the original loops.

159 have support on just the smooth non-self-intersecting loops. We can now impose the vector constraint. This says that the states are invanant under diffeomorphism• of the three surface. By definition, the diffeomorphism class of a smooth non-selfintersecting loop is called a "knot". Thus one is led to the remarkable result that function~ of knot clG,.el 111tilh Gil tlae con•trcainu of quGntum genercal re1otit~ity13 I This result probably represents the main achievement of Ashtekar's program 10 far*. In order to be sure that these knot states are physical we must check that they are normalisable. This is nontrivial since the inner product cannot be chosen arbitrarily but must be chosen 10 that physical observables are hermitian. (The classical reality condition will enter here.) Unfortunately, at the present time, very few observables are known and hence the inner product has not yet been determined. One might worry that since one starts with functions on the infinite dimensional space of loops (or connections) the inner product will necessarily be a functional integral which could only be evaluated perlurbatively. This would violate the whole spirit of this nonperturbative approach to quantization. However, one only needs the inner product on the solutions to the constraints which, like the knot states above, might well have a countable basis. In simpler models such as three dimensional gravity 14 and the weak field limit 15 , the loop representation and the inner product have been constructed with the result that the loop states are normalizable. This lends support to the idea that they will be normalisable in the full theory as well. We have not yet discussed the algebra of the quantum constraints. If one ignores regularization and formally calculates the commutator of the quantum constraints, one finds that there exists a choice of factor ordering- such that the algebra closes5 • However before the discovery of the knot states, there was little reason to trust this result since it was shown 18 that regularization has an important effect on the operator algebra. The calculation of the regulated constraint algebra has not yet been completed. But the existence of solutions to all the constraints shows that there can be no c-number central extension. Either the constraints will close, or the nonclosure will be a term which annihilates all the knot states. How might one physically interpret these knot states? Work on this question is currently in progress. One possibility is the following. Consider a collection of knots defined as follows. Take' a flat metric on R:' and draw three families of parallel nonintersecting lines separated by a distance 11 as shown in Fig. 1. Now connect the ends at infinity to form a knot. (This can be done in many inequivalent ways.) This collection of knots are called V1eGt1e1t. A state which is one on these weaves and zero for all other knots might be interpreted as representing the flat Euclidean metric on R:'. (One should state this more precisely since a knot is diffeomorphism invariant while a particular flat metric is not. The correct statement is that given a flat metric, one constructs a particular representative of a knot class to describe it. A diffeomorphism acting on the knot representative describes the new flat metric obtained by applying the same diffeomorphism to the original metric.) Preliminary

* There have allo been applicatiom o£ th- variabla to problema in claaicalgeneral relativity which t

- will not review here. Similar two dimeDaional weava were comidered by Witten17 in hia diacu•ion of the relation betChem-Simom theory and integrable modela in atatiatical mechanica.

160

calculations indicate that the physical spacing between the lines is determined by the theory to be the Planck length: Hone considers the operator representing the metric and averages it over scales much larger than a, then the weave state gives the best approximation to the flat metric for 11 equal to the Planck length. It is tempting to conjecture1 that other background metrics might correspond to topologically different weaves. Roughly speaking, given any metric on a three manifold, one might associate a weave consisting of fibers whose tangent vectors form an orthonormal basis for the given metric. It is intriguing to see discrete structure at the Planck scale emerge from the theory. In the past, many people have referred to the "fabric of spacetime". If these ideas are correct, this phrase may have literal and not just literary meaning!

I

I

I

I

I

I

•I•I•I•I•I•I•I •I•I•I•I•I•I•I • I ·I • I • I • I • I • I }a ·1·1·1·1·1·1·1 Fig. 1 A weave may be interpreted u repreeenting a flat metric on R 3 .

The loop representation has also been explored for electromagnetism 18 and linearized gravity 15 • It can be constructed either by traut~forming the connection representation or directly in terms of loop observables. In this case it suffices to consider simple, unknotted loops. The result is that for electromagnetism, a one photon state with momentum lc and polarization E is described by the following functional of loops: (12) This is simply t/J( 'Y) = JT A where A is the wave function for the one photon state. Notice that gauge invariance is automatically enforced by the line integral around the loop. The states of linear gravity are similar. The main difference is that for linearized gravity, there are essentially three copies of the electromagnetic states (since the self dual connection has three complex internal components). How does one incorporate these linear states into the full theory'! One possibility is the following. For each simple loop 'Y, one considers a knot consisting of the weave together with the loop 'Y attached (in a manner analogous to ordinary embroidery on fabric). One now defines a state by the condition that it equal t/J(-y)

161 on this knot and similarly for all possible positions of the loop in the weave. Notice that in this picture, gravitons do not make sense on scales less than the background scale a. Any loop smaller than this will be topologically disconnected. 3. Open Questions and New Directions Although the results that have been obtained so far are promising, there is much that remains to be done before one can claim to have a consistent quantum theory of gravity. This section is divided into three parts. In the first, I consider some open questions in the main program described above. The second includes a short discussion of other approaches to quantum gravity using Ashtekar variables. In the third I consider the question of whether there exists yet another set of canonical variables for general relativity (or a theory containing general relativity) such that the constraints are simplified even further. 3.1 Open que1tion1 in the moin program

We have already discussed two important unresolved issues in Ashtekar's approach to quantum gravity. One is to determine the physical inner product and show that the knot states are normalizable. The other is to understand better the regularization procedure. Are there principles which determine it uniquely? Does it lead to anomalies? One also needs to improve the physical interpretation of the knot states. For example, can a black hole be described in terms of functionals of knots? Since there is some indication how to interpret flat space and linearized gravitons in terms of knot states, one can now begin to consider graviton-graviton scattering. This will be an important test of this approach to quantum gravity. Uncontrollable divergences will show that this approach sufl'ers from the same difficulties of standard quantum field theory methods. On the other hand, finite answers will be an important confirmation of the basic principles underlying this approach. A related issue is whether there exist other solutions to the quantum constraints besides the knot states. If one can work with a loop representation consisting only of smooth nonself-intersecting loops, it would appear that the answer is no. If one works with the larger space of piecewise smooth loops, then additional solutions can be found 19 . So far, I have only considered pure general relativity without matter. The first step toward including matter is to show that the combined gravity matter system can be written in terms of the self dual connection in such a way that the constraints are still polynomial in the basic canonical variables. This requires that the metric and its inverse cannot both appear. Thia step has been carried out for scalar, spinor and gauge fields 20 . In particular, the action for supergravity has been written in terms of Ashtew variables 21 . The next step is to find solutions to the quantum constraints. In the presence of matter, this is not well understood. It should be kept in mind that unlike superstring theory, this approach does not at present provide a unified picture of all forces and matter. Its main advantage (assuming it is successful) is in staying as close as possible to the experimentally tested general relativity. If one considers general relativity with a cosmological constant A, then one

162 solution to all of the quantum constraints turns out to be (13) where Scs is the Chern-Simona action for the self dual connection Ai 22 . The calculation of the transform of this state into the loop representation is similar to the calculation performed by Witten23 which reproduced knot invariants. One might worry that this indicates that the knot states will not be normalizable. It' one considers the state (13) as a state in ordinary Yang-Mills theory, then for some choice of A, it turns out to be a zero energy eigenstate. But it is outside of the physical Hilbert space and hence appears to have no physical significance. Why should the situation for gravity be any better? The key point is that for gravity one is using self dual connections rather than real connections. Even for electromagnetism, one can show24 that if one uses the self dual representation (and Ei real) then the ChemSimon's state is just the vacuum for one helicity of the photon! (The vacuum for the other helicity is a constant*.) These states are normalizable with respect to the standard Poincare invariant inner product. This gives further evidence that the knot states are physical. As we have mentioned, Ashtekar's approach to quantum gravity is similar in spirit to the functional Schroedinger approach to ordinary field theory. However the reparameterization invariance leads to the technical simplification that one does not have to solve the time dependent Schroedinger equation since the Hamiltonian is proportional to the constraints. However this raises a conceptual problem: How does one recover time and make physical predictions? This is one of the deep issues that every (nonperturbative) approach to quantum gravity must address. It has been discussed extensively25 , but there is still no clear answer. Simple models of reparameterization invariant systems suggest that one part of the argument of the wave function should play the role of time. In Ashtekar's approach there has been some progress in identifying "time" in the connection representation26 but not much is known yet in the loop representation. Another possibility is that time will arise only when gravity is coupled to matter and "physical clocks" can be constructed.

3.! Other approa.chea to quantization with A1htekar variable• Although canonical quantization with constraint operators has been the main focus of work in this area, it may be worthwhile to examine other approaches. One alternative is to solve the constraints classically and then quantize the resulting "true degrees of freedom". (This was in fact the way Witten first quantized the 2+1 theory.) Remarkably enough, the general solution of the vector and scalar constraint can be exfressed in terms of an arbitrary symmetric, invertible, ~raceless 3 x 3 matrix ~GIJ( z )2 . Given an arbitrary self dual connection Af, define E! so that

Ff:: IJ

*

= e·IJII·LEbAI~GIJ 'I'

(14)

In the standard treatment one worb with positive frequency fields. Then self dual configuratiou describe one helicity and anti-self dual the other. Here one worb only with self dual fields but allows both positive and negative frequency. This explains how both helicitiea can be obtained and why there is an asymmetry.

163

Substituting this into the constraints, one sees immediately that (6) is satisfied since 41Gb is symmetric, and (7) is satisfied since 41Gb is tracefree. One can argue that this is the general solution since 41Gb has five independent components which is the number one expects after solving four equations for the nine components of E!. Gauss' law is the only remaining constraint on Ar and 41aJJ. Unfortunately, a simple solution to this equation is not yet available. Another possibility is to consider covariant approaches to quantization. This should be more conducive to answering a certain class of questions such as whether the topology of space can change in quantum gravity. Even classically, one has the following result. Both the action (4) and the one obtained by replacing R by the curvature of the self dual connection A do not involve the inverse of the tetrad. Thus the action and the resulting field equations remain well defined even in the limit that the metric becomes degenerate. In general relativity, it has been shown that any solution to the vacuum Einstein equation which interpolates between spaces of different topology must be singular. But the only "singularity" that is required is for the metric to become degenerate at one moment of time: There exist smooth solutions to the equations derived from (4) which change topology and have an invertible metric almost everywhere28 . Since Ashtekar's approach and the tetrad approach to general relativity both naturally include degenerate metrics, one is faced with the question of why the metric we see is invertible. In fact, it is not even clear how to formulate this question precisely. It is tempting to consider the expectation value of the metric < 9,w >,and one often hears speculation that< 9,w >= 0 may correspond to a diffeomorphism invariant phase of quantum gravity while < 9,w >= 'l,w corresponds to a state of broken symmetry. However, it is clear from the quantum constraints that the physical states of quantum general relativity are alwa111 diffeomorphism invariant. Moreover, the expectation value of any non-gauge invariant operator (such as the metric) must always be gauge invariant: If U denotes a general gauge transformation, then (15) since physical states are gauge invariant. As there are no nonvanishing diffeomorphism invariant tensor fields, this shows < '¢l9,wl1/1 >= 0 for all physical states*. Analogous ar~ents can be made for spontaneous symmetry breaking in ordinary gauge theory 9 • However in that cue, one can argue that even though the local symmetry is not spontaneously broken, the correspoading global symmetry is. It may be possible to extend this argument to gravity with asymptotically flat boundary conditions. But it certainly cannot apply to closed universes where there is no way to disentangle local and global diffeomorphisms. What is the appropriate gauge invariant operator which captures the notion of nondegenerate metrics? 3.3 Newer varioble1r Although the constraints (5-7) are considerably simpler than the usual form in terms of the old canonical variables, it is reasonable to ask whether this is the

* Tha -uma that the inner product a

defined not just on physical atata, but allo on ltata such < 1/1191111 1'¢ > a aimply not defined.

u 9~&1111/1 >which are unphyaical. OthenriM,

164

best one can do. Does there exist an even more clever choice of variables which will lead to further simplifications? As we have discussed, one of the constraints in Ashtekar variables is quadratic in momenta and must be regulated. Are there canonical variables for which all constraints are linear in momenta? To see that one's choice of variables can, in principle, change the structure of the constraints in this way, consider again three dimensional general relativity. In terms of the standard canonical variables (the spatial metric and extrinsic curvature) the constraints are very similar to the four dimensional case. In particular, they are quadratic in momenta (and nonpolynomial in the spatial metric). However in gauge theory variables, although the constraints are quadratic in the connection, they are linear in the triad which is its conjugate momentum. Thus there is a natural representation in which all constraints are linear in momentum. Comparing the actions for general relativity in three (1) and four (4) dimensions there are two obvious differences: the group is changed from S0(2,1) to S0(3,1) and there is an extra e~ in the action. One can actually separate these two effects. There are three dimensional theories which generalize (1) to any gauge group including S0{3,1 ). Let A be the gauge field for an arbitrary Lie group, F = dA + A A A the field strength, and e be a Lie algebra valued one form. Then one can consider the action 30

S

= j.Tr ei\F

(16)

The constraints are identical to {2) and (3) with R replaced by F. In particular, they are linear in the momentum conjugate to Ai· In four dimensions, there are theories with actions similar to (4) for any gauge group: {17)

where e is again a Lie algebra valued one form. For the case of 50(3) the canonical quantization of this theory has been carried out31 . Unlike the three dimensional examples, this theory has an infinite number of degrees of freedom. Nevertheless, once again all constraints are linear in momentum. As a final example, consider supergravity. In this theory, the scalar constraint can in fact be replaced by its "square root" - the supersymmetry constraints. Since the original constraint is quadratic in momentum, one might hope that the supersymmetry constraints would be at most linear in the momentum. Unfortunately, this is not the case. Although the supersymmetry constraints are linear in the momentum conjugate to the metric, they contain a term which is the product of the momentum conjugate to the metric and the momentum conjugate to the spin 3/2 fi.eld32,21. This is again the product of functional derivatives at the same point and must be regulated. In retrospect, it is clear that the supersymmetry constraints cannot be linear in all momenta: The (anti) commutator of two constraints linear in momenta is always linear in momenta and cannot yield the scalar constraint of general relativity. It is perhaps worth mentioning that going to higher dimensions does not seem very promising. In higher dimensions, general relativity can still be expressed

166 in terms of a Lorentz connection w'/f and collection of one forms e~ with the action

S

=

I

eo /\ ••• /\

e" /\ R_Cd Eo.. ·kd

(18)

However, there is no obvious analog of using the self-dual part of the connection. Thus it is not even clear how to mimic the simplification obtained by Ashtekar: Ashtekar's variables do not have a natural generalization to higher dimensions. In conclusion, I would say that I find the general ideas of Ashtekar's approach to quantum gravity attractive, and the results obtained 10 far intriguing. It is still far from clear whether this program (or some variation of it) can be completed, but it certainly seems worth pursuing.

Adtnowledgements I wish to thank the organizers of the Strings and Symmetries 1991 Conference for a stimulating meeting. I am grateful to A. Ashtekar, C. Rovelli, and L. Smolin for discussions of their work. I have also benefited from comments and conversations with S. Giddings, J. Hartle, T. Jacobson, M. Srednicki, and A. Strominger. This work was supported in part by NSF Grant PHY-9008502.

166

REFERENCES 1. For more complete recent reviews, see C. Rovelli, "Ashtekar formulation of general relativity and loop space non-perturbative quantum gravity: a report" University of Pittsburgh preprint (1991), to appear in Class. Quantum Grav.; A. Ashtekar, Lectures on NonperturbatJve Canonical Gravity, (notes prepared in collaboration with R.S. Tate) World Scientific, 1991.

2. M. Goroff and A. Sagnotti, Nucl. Phys. B266, 709 (1986). 3. See e.g. R. Feynman, Nucl. Phys. B188, 479 (1981); R. Jackiw in Field Theory and Particle Physics, World Scientific (1990). 4. E. Witten, Nucl. Phys. B311, 46 (1988). 5. A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986); Phys. Rev. D 36, 1587 (1987). 6. A. Ashtekar, A. Balachandran, and S. Jo, International Journal of Modern Physics A 4, 1493 (1989). 7. T. Jacobson and L. Smolin, Phys. Lett. B196, 39 (1987); Class. Quantum Grav. 5, 583 (1988); J. Samuel, Pramana 28, L429 (1987). 8. J. Charap and M. Duff, Phys. Lett. B69, 445 (1977); A. Ashtekar and J. Samuel, GR12 abstracts, U. of Colorado Press 1989. 9. J. Samuel, Class. Quantum Grav. 5, L123 (1988). 10. T. Jacobson and L. Smolin, Nucl. Phys. B299, 295 {1988). 11. See. e.g. M. Blencowe, Nucl. Phys. B341, 213 (1990). 12. S. Mandelstam, Ann. Phys. 19, 1 (1962); K. Wilson, Phys. Rev. D 10, 247 (1974); J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975); A. Polyakov, Phys. Lett. B82, 247 (1979); Nucl. Phys. B164, 171 (1979), Y. Makeenko and A. Migdal, Phys. tett. :888, 135 (1979); G. 'tHooft, Nucl. Phys. B153, 141 (1979). 13. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B331, 80 (1990). 14. A. Ashtekar, V. Husain. C. Rovelli, J. Samuel, and L. Smolin, Class. Quantum Grav. 6, L185 (1989). 15. A. Ashtekar, C. Rovelli, and L. Smolin, "Gravitons and Loops", Phys. Rev. D., to appear. 16. N. Tsamis and R. Woodard, Phys. Rev. D 36, 3641 (1987); J. Friedman and I. Jack, Phys. Rev. D 37, 3495 (1988). 17. E. Witten, Nucl. Phys. 322, 629 (1989). 18. A. Ashtekar and C. Rovelli "A Loop Representation for the Quantum Maxwell Field", Syracuse preprint (1991 ). 19. V. Hussain, Nucl. Phys. 8313, 711 (1989); B. Brugmann and J. Pullin, "Intersecting N Loop Solutions of the Hamiltonian Constraint of Quantum Gravity", Syracuse preprint (1990).

187

20. A. Ashtekar, J. Romano, and R. Tate, Phys. Rev. D 40, 2527 (1989). 21. T. Jacobson, Class. Quantum Grav. 5, 923 (1988). 22. H. Kodama, Phys. Rev. D 42, 2548 (1990). 23. E. Witten, Comm. Math. Phys. 121, 351 (1989). 24. A. Ashtekar, C. Rovelli, and L. Smolin, "Self Duality and Quantization", J. Geom. Phys. in press. 25. See e.g. Conceptual Problems of Quantum Gravity, eds. A. Ashtekar and J. Stachel, Birkhauser, 1991. 26. A. Ashtekar, in Conceptual Problems of Quantum Gravity (op. cit.). 27. R. Capovilla, J. Dell, and T. Jacobson, Phys. Rev. LeU. 63, 2325 (1989). 28. G. Horowitz, Class. Quantum Grav. 8, 587 (1991). 29. S. Elitzur, Phys. Rev. D 12, 3978 (1975). 30. G. Horowitz, Comm. Math. Phys. 125,417 (1989); M. Blau and G. Thompson, Ann. Phys. 205, 130 (1991). 31. V. Husain and K. Kuchar, Pbys. Rev. D 42, 4070 (1991). 32. P. D'Eath, Phys. Rev. D 29, 2199 (1984).

168

Non-Einsteinian Gravity with Torsion at d = 2

WOLFGANG KUMMER and DOMINIK J. SCHWARZ ln&t. f. Theor. Phy&ilc, Tech. Univ. Wien Wiedner Ho.v.pt&tr. 8-10, 1040- Wien, Av.&trio.

Abstract Classical and quantum solutions of non-Einsteinian gravity in 1 + 1 dimensions are obtained.

1. The perennial problem of quantizing gravity on the one hand and the renewed interest in 2-d space-time, stimulated by string-theories, leads to the consideration of gravity in 1 + 1 dimensions, based upon an action with is not of the EinsteinHilbert type. That such a theory may be purgated of ghosts and be renormalizeable even in the presence of nonvanishing torsion is an old idea in four space-time dimensions 1• In two dimensions the action is much simpler and essentially unique 2 L

1-R2 + .f!._T2 + e~] = - j (or the string coupling constant g =exp !l>vae ) should also transform under the duality

~

=4>-ln4

,

r

-..w

4=-

(2)

(we shall consider a one-dimensional torus of dimensionless radius 4 ). In particular, it is (2) that is the symmetry of the higher loop (genus n ~ 2 ) vacuum partition function,

Zn(g,4)

=Zn(i ,a),

(3)

The necessity of the constant dilaton shift for the duality invariance of the string partition function was also noted in [6]. The symmetry (3) was demonstrated explicitly e.g. in [7]. The reason why the dilaton should be shifted under the duality is easy to understand from a field theoretic point of view [5]. Consider a string field theory action, S Jd0 z e- 2• ( ••• ) ~ f dy f d0 - 1z (cpA cp + cp3 + ...), where 11 is the periodic coordinate. Integrating over 11 ~e get the factor of a in front of the action for the Fourier modes. The theory will depend on 4 only through the effective "D- 1 dimensional"' coupling g gf.Ji and the masses of fields. Since the latter are symmetric under 4- 4- 1 (t;f. (1)) j should not change under the duality in order for the theory to be duality invariant. The invariance of g implies that g should transform according to (3). Recently, an attempt was made to understand if there exists a generalization of the duality to the case of "non-static" string vacua for which the radius of the torus may depend on other coordinates zi (e.g. on time) (8). As was emphasized in [9) (see also [10]) one should expect to find duality as a symmetry between string vacua with "radii" 4 and 4- 1 since for any z-dependent perturbation of a "static" theory (compactified on a torus) there is a corresponding one in the dual theory (compactified on a "dual" torus). It was found [9) that the exact "plane wave" classical solutions [11) of the bosonic string theory discussed in [8) have dual analogs provided one transforms the dilaton according to (2) with a now being z-dependent. The transformation between the theories with "radii" 4{z) and 4- 1 (z) is related to the "u-model duality" transformation for the corresponding string u-models. The need to shift the dilaton field in order for the duality related u-models to have equivalent one loop Weyl anomaly coefficients ("P-functions") was, in fact, already pointed out in ref.[l2},

=

=

=

1 4> = 4>- 2lnGu,

Gn =Gil.

Gu = 4 2 (z).

(4)

It is important to understand the relation between the "full" string duality and the "reduced" or "u-model duality". Let us consider the conformal theory corresponding to the string compactified on D = 1 torus of radius ao and perturb it by various possible vertex operators. The perturbation which shifts the radius of the torus ("modulus" of compactification) is given by the following term in the string action

j ~z [~ + A(z,y,i)) IJyby

(5)

175 The (linearized) duality is generated by the following 2d duality transformation in the perturbation A8yBy : {)y = 8ij, By= -By , i.e. A(z,y,ij)- -A(z,y,y)

(6)

Expanding A in Fourier modes with respect to y and y we conclude that the string duality transformation (6) includes the changing of the sign of the z-dependent "zero mode" of A, ..\--..\,

a(z)=ao+..\(z)+ ... ,

a 2 (z)=a~+A(z,O,O),

(7)

as well as interchanging of the momentum and winding modes. Similar analysis of the dilaton vertex operator implies that the z-dependent perturbation of the dilaton should transform under the ( linearized ) string duality according to =

=2o'b

2/

(1 + 2a'b2 ),

a'b2

=4>~- !lnsinh 26r + !~ (r)

= ~ =4

(34)

As was checked in [21] the background (34) satisfies the Weyl invariance condition 0 in two-loop aproximation. The duality in the SL(2,R)/U(1) conformal theory (20,22] corresponds to the substitition 6r- 6r + ~. i.e. sinhbr- icosh6r (c£.(32), (33)). It is easy to see that the thus obtained dual background is related to (34) by the duality transformation (18) where 11 contains terms of all orders in o'. Let us now discuss some implication of duality (18) for string cosmology. The fact that the dilaton transforms under the duality transformation resolves some confusion which existed in the literature about duality (non) invariance of the string analog of the Einstein equation. The basic point is that one cannot a priori ignore the dilaton dynamics {i.e. cannot set the original dilaton 4> equal to a constant ). Let us consider the spacially flat isotropic homogeneous background (i.e. we take all the radii in (24) to be equal)

fjG

=

i

= 1, ... , D- 1 , 4> =1/>(t)

,

(35)

Introducing the redefined dilaton field (36)

180 which is invariant under the duality transformation~- -~. 41- 41- (D- 1) ~ represent the effective action in the manifestly duality invariant form (cf.(22))

we can

= j dte- 2

O,

a (t) ~

aoe"' ,

p< 1.

(40)

The dual solution corresponds to reversing the sign of the timet.

References (1]

[2]

[3]

K. Kikkawa and M. Yamanaka, Phys. Lett. B149 (1984) 357 N. Sakai and I. Senda, Progr. Theor. Phys. 75 (1986) 692 M. B. Green, J. H. Schwarz and L. Brink, Nucl.Phys. B198 (1982) 474 D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B19T (1987) 81; Int. J. Mod. Phys. Lett. 3A (1988) 1615 D. Gross and P. Mende, Phys.Lett. B19T (1987) 129; Nucl.Phys. B303 (1988) 407 V. Nair, A. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B28T {1987) 402

181 [4] [5] [6] [7) [8] [9] [10) (11)

[12) (13) [14] (15] (16] (17) (18) (19) (20) [21) (22] (23) (24) (25)

B. Sa.thia.palan, Phys. Rev. Lett. 58 ( 1987) 1597 P. Ginspa.rg and C. Va.fa, Nucl. Phys. 8289 (1987) 414 A. Giveon, N. Malkin and E. Ra.binovici, Phys. Lett. 8220 (1989) 551 E. Alvarez and M. Osorio, Phys. Rev. D40 (1989) 1150 D. Gross and I. Klebanov, Nucl. Phys. 8344 (1990) 475 G. Horowitz and A. R. Steif, Phys. Lett. 8250 (1990) 49 E. Smith and J. Polchinsld, Univ. Tezos preprint UTTG-07-91 (1991) T. Banks, M. Dine, F. Dijkstra and W. Fischler, Phys. Lett. 8212 (1988) 45 D. Amati and C. KUmcik, Phys. Lett. 8219 (1989) 443 G. Horowitz and A. R. Steif, Phys.Rev.Lett. 84 (1990) 260; Phys. Rev. D42 (1990) 1950 T. H. Buscher, Phys. Lett. 8149 (1987) 59; Phys. Lett. 8201 (1988) 466 A. A. Tseytlin, Mod. Phys. Lett. Al8 (1991) 1721 A. A. Tseytlin, Phys. Lett. 8194 (1987) 63 M. Mueller, Nucl. Phys. 8337 (1990) 37 L. Ibanez, D. Lust, F. Quevedo and S.Theisen, unpublished (1990) G. Veneziano, preprint CERN-TH-6077/91 (1991) S. Elitzur, A. Forge and E. Ra.binovici, &oah preprint Rl-143/90 (1990) M. Rocek, K. Schoutens and A. Sevrin, preprint IASSNS-HEP- 91/14 G. Ma.ndal, A.M. Sengupta and S. R. Wadia, preprint IASSNS-HEP-91/10 E. Witten, preprint IASSNS-HEP-91/12 R. Dijkgraa.f, H. Verllnde and E. Verlinde, preprint PUPT • 1252/91 (1991) A. A. Tseytlin, Johns Hopkins Univ. preprint JHU- TIPAC-91009 E. B. Kiritsis, Berkeley preprint LBL-30141 (1991) I. Bars, Univ. Southem Califomio preprint USC-91/HEP-B3 (1991) R. Brandenberger and C. Vafa, NucL Play•. 8318 (1988) 391 M. Hellmund and J. Kripfganz, Php•.Lett. 8241 (1990) 211 E. Guendelman, Clul. Quant. Grav. 23 (1991) 521

182

TWO-LOOP QUANTUM GRAVITY Anton E.M. van de Ven II. Institute for Theoretical Physics, University of Hamburg, Luruper Chaussee 149, £000 Hamburg 50, FRG

ABSTRACT We prove the existence of a nonrenormalizable infinity in the two-loop S-matrix of d = 4 quantum gravity based on the Einstein-Hilbert action. Finding a consistent quantum theory of gravity is one of the outstanding goals of theoretical physics. Merging conventional quantum field theory with classical general relativity is known to lead to a nonrenormalizable theory. This has been verified in an explicit twoloop calculation (1) (gravity with the Einstein-Hilbert action does have a finite one-loop S-matrix (2]). In part, it is this crisis that has led to the study of string theory. In view of the importance of the failure of perturbative quantum gravity, we have recently repeated the two-loop calculation of (1], but using rather different methods. Our final answer is in complete agreement with that obtained earlier. The Einstein-Hilbert action in d = 4 euclidean space is given by SEH

= : 2 j a'x ../9 R

,

K2

= 3211" 2 G

(1)

,

where R is the Ricci scalar, g is the determinant of the metric 9mn and G is Newton's constant. In two-loop order, the divergent part of the on-shell (R,.,.,. 0) background

=

field effective action must be of the form

(2) where

C~ctmn

denotes the Weyl tensor and

~

= 4 -d.

The residue c can be determined

only through an explicit calculation. Barring the existence of a hidden symmetry, we should expect it to be nonzero. We calculate c by means of the heat kernel method in coordinate space (3,4), which has the advantage that general covariance is manifestly preserved. We begin with the linear background-quantum splitting 9mn-+ 9mn

+ K.(hmn + 9mnV')

'

(3)

183 where the new g,.,. is the background metric, while cp and h,.,. are the quantum fields, the latter being traceless symmetric w.r.t. the background. We then consider the following class of nonlinear gauges (with h,. h,.,.;")

=

F,.

= h,.- ! (d- 2) 0 is the "black hole" singularity; being to the future of the physical region, it can absorb but not emit. The white hole violates the *

Research aupported in part by NSF Grant PHY86-20266.

185

predictability of the classical theory, as classically one cannot predict what it will emit. Relativists have dealt with the white hole by the cosmic censorship hypothesis, according to which white holes and more general "naked singularities" never form from acceptable initial data. (For instance, spherical stellar collapse from standard initial data gives a spacetime that coincides with the Schwarzschild solution only in the exterior of the collapsing star; the exterior contains the black hole singularity but not the white hole singularity.) Actually, there is not very much evidence for the cosmic censorship conjecture, and it may well be false. It is not clear that we should wish for the truth of the conjecture. If cosmic censorship is false, then in principle we would have the chance to observe new laws of physics that must take over near the would-be naked singularity. This might be more useful for physics than extending the scope of classical general relativity by proving cosmic censorship.

It is usually assumed that the new physical laws associated with a possible breakdown of cosmic censorship would involve quantum gravity, but this may be wrong, particularly in view of the fact that the length scale of string physics is probably a little larger than the Planck length. If indeed cosmic censorship is false in general relativity but its analog is true in string theory, then it may well be that it is classical string theory that becomes manifest near the would-be naked singularity. As for black holes, at the classical level they cause no breakdown in predictability for the outside observer. Quantum mechanically, though, the idea that black holes exist and white holes do not is paradoxical as the white hole is the CPT conjugate of the black hole. In any event, Stephen Hawking rendered the classical picture obsolete with his discovery of black hole radiation, showing that an isolated hole will radiate until (after Hawking's approximations break down) it reaches a quantum ground state or disappears completely. A key aspect of the problem of quantum mechanics of black holes is to describe this endpoint. More broadly, one would like to describe the Smatrix for matter interacting with the black hole if (as I suspect) there is one. If (as argued by Hawking) such an S-matrix does not exist, one would

186 like to get a precise account of the nature of the obstruction and to learn how to calculate in whatever framework (such as the density matrices advocated by Hawking) replaces that of S-matrices. I think most physicists expect that Hawking radiation leads to complete disappearance of neutral black holes. For charged black holes the situation is likely to be quite different, under appropriate conditions, since a charged hole might be lighter than any collection of "ordinary particles" of the same total charge. Let us call the electric and magnetic charges of the hole q and m. A charged black hole has a classical ground state with mass M = q2 + m2 in Planck units; this ground state is simply the extreme Reissner-Nordstrom solution of the classical field equations. One would imagine that for suitable values of the charges the classical ground state is some sort of approximation to the quantum ground state.

J

In this respect, the case of a magnetically charged black hole is especially interesting: (1) This case may be realized in nature if McuT "' MPlanc/c, since then the 't Hooft-Polyakov monopole is a black hole. In particular it is at least conceivable that the dark matter in our galactic halo could consist of magnetically charged black holes. This hypothesis is subject to experimental test; indeed recent experiments are relatively close to the sensitivity required to exclude it [2). Any type of discovery of galactic halo particles would give particle physics a big boost, of course, but magnetic black holes would be particularly exciting. {2) Fore 0, the surface is the upper half plane, moded out by a Fuchsian group of the first kind D: SL(2,R)/ U(l)xD

For p < 0, the surface is the sphere. We now extend Riemannian geometry, and introduce a U(1) connection w and a 2-D frame e4 (a = 1, 2), with torsion T 4 = de 4 + w A e4 and curvature R = dw. Putting both forms into an SL(2, R) gauge field, we get

A= -iwJ3 + e"J, + e1 J 1

(4.3)

where J1c are representation matrices of SL(2, R). The curvature of A is

Flatness of A can always be achieved by choosing it to be the Maurer-Cartan form on SL(2, R), pulled back to the surface. But ifF = 0, then the torsion form T 4 vanishes, and the curvature R9 is constant, i.e. we have a solution of Liouville's equation.

201 Flatness of A automatically guarantees integrability of the parallel transport equation

(d+ A)t/J = 0

(4.5)

But writing out this equation for J1: in the fundamental representation of SL(2, R) in terms of frame and U(1) connection, we get the Lax pair of Liouville, generalized to arbitrary surfaces. (4.6)

From the Lax pair, we may derive a Backlund transformation, valid again for general Riemann surfaces. Liouville theory is again completely integrable just in terms of a free field [20, 21]. These results simply extend to N=1 super Liouville theory as well [20]. V. Canonical Quantization and Closure of the Conformal Algebra Postulating the traceless quantum mechanical stress tensor, (5.1)

where II is conjugate to ¢>, and u is the "space" coordinate. We have included here as yet undetermined constants o and ~t, since canonical quantization is not in general conformal invariant. Thus we should allow for non-invariant counterterms in the action and also in the stress tensor. The normal ordering will here be performed with respect to free field theory singularities. Canonical quantization allows one to check closure without assuming anything further about the Liouville dynamics, and this closure is obeyed provided 1 + 3~t 2

= 26 -

c

2o2 -~to+1=0

(5.2)

The 1 in the first equation and 2o2 in the second are the quantum contributions. These results we first derived in [10]. The same calculation yields the conformal dimensions of exponential operators, which are all primary fields of Liouville dynamics [10)

so the dimension is

h({J) = -2{J2 + fjK..

202 Notice that in the derivation of these results, we only assume that the renormalization of the exponential does not involve the canonical conjugate momentum II. We do not really have to assume that the OPE is constructed with free field coefficient functions.

VI. Gravitational Dimensions We now discuss the construction of vertex operators for physical states in the combined conformal matter-gravity theory, following Polyakov and Knizhnik, Polyakov and Zamolodchikov [7]. We can only couple to gravity the spinless, primary fields of the underlying matter system, which we denote by b. Their dimensions are hb. Observables must be reparametrization invariant and "local"; so they must be integrals over the surface of dimension (1,1) primary fields. (6.1) The latter requires that (6.2)

In particular when b is the unit operator, hb

= 0 and we have the area operator (6.3)

where a satisfies (5.2). Now the normalization of tP is not physical. Hence the normalization of the operators is not physical either. KPZ introduced an invariant physical gravitational scaling dimension of any operator B with respect to the area operator A. Scaling up A, we have A--+ ..\A

(6.4)

and thus B scales as A= 1- (3/a

(6.5)

A is called the gravitational dimension; it is zero, by definition for the area operator itself.

203

For minimal models, we have for p

E

Q and m, mN :

h _ ( m(p + 1) - np) 2 mn4p(p + 1)

6

c=l---p(p+ 1)

-

1

(6.6)

and one obtains (7]. 6mn

1

1

= 2(m- n) + 2/m -1)

(6.7)

This spectrum of scaling dimensions is linearly spaced, suggesting a group theoretic foundation of this formula. The relevant dynamical symmetry was exhibited in (7], and is an SL(2, R) Kac Moody algebra.

VII. Equivalence of Weyl and Translation Invariant Quantization We have reduced the problem of quantizing 2-D gravity to performing a functional integral with the Liouville action and the Weyl-invariant measure on the Liouville field. As we pointed out earlier through, there is no conformally covariant regulator, so that effectively one is reduced to quantizing the Liouville action with another measure, which is not Weyl-invariant. It is most convenient to use the translation invariant measure instead (19]. Consistent quantization will require the inclusion of Weyl non-invariant counterterms. In general, their coefficients are to be determined by enforcing the full quantum mechanical Weyl Ward identities in the theory. Our basic program will start by establishing that the translation invariant quantization, order by order in a perturbation expansion in cx2 "' !c is equivalent with the Weyl invariant quantization (18]. To do so, we begin by considering the generating functional for the translation-invariant quantization: exp{- WT(D)}

=

J

D;4>exp{- ::

JJb[ ~4>6;4> + R;4> + +f

p : e20 ";

:;]

(7.1)

VbJ4>}

The following remarks are in order 1. For cx~e

> 0, the potential is bounded from below for sufficiently large J vgJ.

2. The normal ordered exponential : exp 2cx~e4J :; is defined perturbatively by normal ordering with respect to the metric g.

3. We have the renormalization group equation [2]

·• e201 "~(z) ·· - e4012"(z) •· e 201 "~(z) ·· ·ge2•g

x (zero mode)

(7.2)

where the last factor appears when the worldsheet is compact, due to the presence of a zero mode in the spectrum of the Laplacian. 4. The above functional integral formula is valid perturbatively for all genera.

5. Quantization with respect to a. Weyl invariant measure was defined earlier, and it is assumed that no extra conformal anomalies arise in this quantization. Using the renormaliza.tion group equations for Weyl rescaling the renormalized exponential, we easily derive a. set of Weyl Ward identities for the translation invariant quantization. For the Weyl invariant quantization, the Ward identities are the naive ones. We find [18].

WT[g,e 2"(J+ " 2 ~1r~tc)~gu)] =WT[ge2",J] +fr (7.3) Ww

[9, e

2"

J]

Here 111 , with p = T or W is given by

I,= 4~

j .f9 [-~c.,u~gu+dpRgu+e.,uJ]

and

• a a=--1 +2a2

(7.4)

with 1

eT = --.Ka cw

= "~

dw

= "~

ew

(7.5)

= -1

In order to avoid the inhomogeneous shift of J in WT, we first choose u= constant. Equating the contribution in d., and e.,, we readily find that

aK = 1

or

2a2

- aK

+ 1 =0

and

3K2

+ 1 = "~ = 26 -

c

(7.6)

As a consequence, we automatically also have CT = cw, and hence the Weyl Ward identities completely coincide, since also the inhomogeneous shift in J cancels.

205

Now that we know the Weyl Ward identities, we can control the Weyl noninvariant counterterms.

VIII. Heat-Kernel Regularization Arguments for Equivalence We now pass from the Weyl invariant measure fJq, to the translation invariant one. We change measures by changing metrics on function space [19, 18, 22] (8.1) This can be effected by a field-dependent linear operator: (8.2) where the "entries" of the operator are labeled by the points on the surface. Formally, we have

(8.3) where the Jacobian factor is given by

(8.4) Of course, this object is severely divergent, since we evaluate the determinant of an infinite-dimensional operator, whose diagonal elements are delta functions evaluated at coincident points. We shall also have to specify at which metric the delta function, and thus the whole operator are evaluated [22). The regularization of this Jacobian is expected to break conformal invariance, which will ultimately have to be enforced by the Ward identities. It is convenient to use the reparametrization invariant heat-kernel regularization. Of course, the heat-kernel of any reasonable operator tends towards a delta function in the zero time limit. On dimensional grounds, and preserving reparametrization invariance, the most general choice is as follows

(8.5) Here /3 is an arbitrary constant. Which metric should we pick to evaluate the Jacobian? There are several natural metrics: g or g = e2" g or any metric in between.

206 Really, one should be allowed to pick any metric, and then fix the ambiguity that results for the Jacobian factor by enforcing the Ward identities. As a result, we find that (18] (8.6)

Where the constants 'Yi depend upon the explicit choice of metric at which the operator is evaluated, as well as upon the choice of regulator as in (8.5). Their numerical value is to be fixed by Weyl invariance. Notice that now the Jacobian F is of the Liouville form, as indeed conjectured in [19), and the values are determined as in §VI I I. Thus we have shown completely the equivalence of the translation and Weyl invariant quantizations.

IX. Quantum Solutions Through Free Field Theory There are two major systematic attempts at completely solving the quantum Liouville theory that have emerged over the years. One is by Braaten, Ghandour, Curtright and Thorn [12] and another is by Gervais and Neveu [13]. We shall briefly discuss each of these approaches within the context of modern Liouville theory. Both approaches start from a rewriting of the classical solutions to Liouville theory, which are then quantized. In the approach of Gervais and Neveu, the solutions of the classical Liouville field equations are represented in terms of two solutions tP% of the same second order, linear differential equation

e-i•(z,z) = (l- A(z)A(z})i = (tP+(z)..P+(z)- ..P-(z)t/1-(z))j A'(z)A'(z)

V

(9.1)

where 1/1: satisfy a differential equation involving the stress tensor Ou (9.2)

The key observation is that ( ~~ ) transforms linearly under the action of the group SL(2, R), which leaves tP invariant. For j ~ 0 and integer, the corresponding field produces a finite dimensional representation of SL(2, R), which is easily studied. For j < 0, we get infinite dimensional representations, which are more intricate.

207

The basic assumption of (13] is that the classical free fields f/J± can also be quantized as free fields, inducing certain representations on the operators of (9.1). From the outset, all renormalizations are assumed to be those of free field theory. It is somewhat unclear from their work though whether from the operators of (9.1) with j;::: 0, and integer, one can construct a field tP that obeys the canonical commutation relations, as well as some version of the Liouville Heisenberg equations of motion. For certain values of c = 1, 7, 13, 19, 25, special truncations of the free field induced spectrum exist, and appear to have consistent fusion rules. The conjecture (13] was recently proven using the tools of quantum groups: there emerges a Kac determinant for quantum Virasoro algebras that exhibit special additional null states (23]. In the approach of Braaten, Curtright, Ghandour and Thorn (12], one starts with the classical Backlund transformation mapping classical solutions of Liouville theory onto classical solutions of the two dimensional Laplace equation. This is the Backlund transformation that can be derived from the Lax pair of (4.6). We shall need it only on the surface of a cylinder:

OztP- ozf/J = p.A eH~ OztP + OztP = -p./ ~ e•-~

(9.3)

where f/J is a free field. As always, the consistency of these equations, viewed as determining tP (resp. f/J) requires ilf/J = 0 (resp. tP obeying the Liouville equation). The classical correspondence, for fixed sign of A is 1 --+ 2. This is most easily seen by parametrizing the cylinder by a "space" coordinate u and a "time" coordinate T. Then the momenta of the tP and t/J fields are (9.4)

From integrating the Backlund transformation, we get (9.5)

In view of the positivity of the exponential function and for fix~ sign of~ (and P~ is fixed. Hence all solutions of the Liouville equation are mapped onto all free solutions, but with P~ > 0 for p.A > 0. This argument is easily generalized by using generalized Lax pair of (4.6).

p. of course), the sign of

208 Braaten, Curtright and Thorn define the quantum Liouville field through this Backlund transformation in terms of the free field ¢. Again, all renormalizations are carried out within the context of free field theory. However in [12], it is shown that the Liouville field operator 4> satisfies canonical commutation relations as well as the Heisenberg field equations of motion, provided that the underlying free field Hilbert space is restricted by the condition that P., > 0. On the full free field Hilbert space, the Heisenberg field equations are violated. From this point of view, Liouville and free field dynamics are different really only through this "constant collective mode" corresponding to global P., momentum. Actually, this is really to be expected from the restriction of the Liouville field (say on the cylinder) to 4> independent of u. We obtain the Liouville quantum mechanics problems with Hamiltonian HL = p2 + e29. Analogously, the free field Hamiltonian can also be reduced and we get HF = p 2 • The eigenstates of HF are labeled by -oo < p < oo : IP >, whereas those of HL are labeled by 0 < p < oo: IP >. All remaining oscillator modes of Liouville and free field theory on the other hand are in one to one correspondence [24, 12]. The restriction on the states may be translated into a restriction on the operators occurring in the theory, and has important implications on the structure of the operator product expansion [25]. The only exponential operators allowed in the theory are K.

/J j, the leading singularity and hence operator on the right hand side

If

would be absent. Thus the operator product expansion is in general less singular than in free field theory.

X. Correlation Functions from a Coulomb Gas Point of View The Dotsenko-Fateev approach to computing the correlations functions in minimal models using Coulomb gas technique, may be recast in the form of a Liouvilletype problem [26]. This is achieved by including the "charge at infinity term" and

209

the screening charges directly into the action. The corresponding action, in the background of a worldsheet metric g then reads

(10.1)

Charge conservation arranges the correlation functions of primary fields (exponential of with charges related to 'Y±) in such a way that a definite (positive integer) number of powers of I'+ and 1'- are retained only. This produces a calculus for correlation functions equivalent to the one of (26).

e

Vice-versa, Liouville correlation functions may be cast in a form very close to that of the Dotsenko-Fateev calculus. To see this, we start with (8.1) (i.e. the translation-invariant quantization) on a compact Riemann surface with no boundaries and metric g. Here the constant -mode always plays a special role, since it is annihilated by the Laplace operator. It has been noted for some time that upon carrying out the zero mode integration in the functional integral, one obtains expressions that seem very close to free field correlators [27]. The problem is that in general, the power 11 to which the area (or Liouville exponential) operator occurs is not a positive integer. Worse, the expression has a prefactor of r( -11), which diverges precisely at all positive integers. It seems rather hopeless to make any progress directly with the help of such expressions. Recently, it was nevertheless proposed to proceed as if 11 were an integer, and then attempt to formally continue in 11. This procedure initially seemed justified only by the fact that it reproduces the matrix model results for the 3-point function [28, 29]. However now, arguments are available that show that this formal continuation in 11 is indeed justified and holds as a mathematical identity for minimal models [30). We shall now summarize some of the key ingredients of these calculations for the case of general operators [30]. First, we restrict the matter to a minimal model, and we can of course only couple spinless to 2-D gravity. The operators in the full theory are 2 A •· -- A( ,.;,r;) -- ·'·( 'Y T!,Ta) e P•~

(10.2)

210

The dimension of t/J is given by the Kac tormula 6 -:---:p(p+ 1)

(10.3)

r~ and r; integer

(10.4)

c

=1 -

and

/l -

+ l)r;l- 2p- 1 .j8p(p + 1)

lpr~- (p

.::--l·t..--~==#===:~~-

1-

We shall only consider the three point function, since that is the only quantity that has been completely calculated so far. It is given by

We integrate out the constant t/>-mode first, in the Liouville sctor and are left with

where

(10.6b) II=

3

L P·a

~

-(1 -h)2a .

__!

•=1

and the prime on the I/>- integration stand for the omission of the t/>-zero mode. To evaluate this expression, we restrict to the case of the sphere h = 0. S L(2, R) invariance allows us to pick e.g. z1 = 0, z2 = 1, Z3 = oo, and using free field tf>-correlators, we get

(A1A2A3)

= Df2(~)" r(:ll)

J•=1fi u d2 w;

lw;-wii-W

•( w) :-

lz - wi-SQ, + ...

(11.2)

As a consequence, there is a new ultraviolet divergence at 8a2 = 2, which however only affects the vacuum energy. There is a further ultraviolet divergence at 8a 2 = 4, which is easily seen to be compensated by mere -field renormalization [33]. A more instructive way to appreciate the appearance of these divergences is by remarking that : cos 2a4> : has dimension 4a2 , and becomes marginal as 4a2 = 2. Clearly, these U. V. divergences cannot be detected in perturbation theory to any finite order. Liouville theory is formally equivalent to the dynamics of a gas of gravitational particles, with equal masses that all attract one another. To see this, one may expand in the parameter Jl in (7.1). However, since the exponential interaction is not bounded, this expansion is not well defined. This is clear from the fact that the dependence on Jl is not analytic around Jl = 0. An improved expansion may be devised by treating the constant Liouville mode exactly [12, IS, 24]. After that, one may expand in the exponential depending on the non-constant Liouville modes only, and it is in the sense that Liouville dynamics is still equivalent to gravitational dynamics. The short distance expansion of the exponential interaction is radically different from that of the Sine-Gordon interaction ( 11.3) Again, there are U. V. singularities at Sa~ = 2, 4, .... But now they cannot be renormalized by field renormalization alone, as is signaled by the fact that the leading singularity is governed by a non-trivial operator. To higher order in the exponential expansion, the first U. V. singularity furthermore moves closer and closer into the origin

(11.4)

The first singularity occurs at Sa~ = 4/3 < 2. To order n exponentials, the first singularity occurs at Sa~ = 4/n, which accumulates into the origin.

214

Some of these singularities are reproduced by the screening charge approach to Liouville theory, as can be seen from (10.11). The remaining singularities require counterterms that do not occur in the original Lagrangian. They would be of the form 8a2

1 -

J

e2na•

4/n

(11.5)

which is not conformally invariant. We believe (conjecture if you wish) that they only appear because our regulator scheme is not manifestly conformally covariant. They would be absent in a manifestly conformal scheme. As a result, we believe they can be ignored, and do not upset the screening charge approach as outlines in §X.

XII. Duality of Matter and Gravity Matter, in the form of minimal models may be described by a complex action for a scalar field

e:

1[9, e) = 4~

J..fi [~e6;e +

iQ R;e + P+ e2i'T+( + P- e2i'T-(]

(12.1)

Here, an expansion in (integer) powers of P+ and P- generates the correlation functions in minimal models evaluated with screening charges (see §X). Minimal models are obtained by requiring the existence of a discrete translation invariance symmetry [18]. 7(

e---+ e + - k

keZ (12.2) "Y+ By construction, the first exponential is left invariant for any integer k. The QR1-term will of course never be classically invariant, but in quantum mechanics, it suffices that the action shift by an integer multiple of 27f'. Thus, to have invariance, Q/2n must be a rational number. This happens if and only if we have a minimal model, with

Q p-rf -=2--

"Y+= p "Y- rl and it suffices to take k a multiple of p. The shift in S is "Y+

p

(p,p')

=1

iQ .< 1-h>I p-p'lk 68= -x1f'k=21f'a 2-y+ p For

IP- r/1 = 1, these models are also unitary.

(12.3)

(12.4)

218

REFERENCES (1] A.M. Polyakov, Phys. Lett. B103 (1981) 207, 211. (2] E. D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917. (3] S. Weinberg, in General Relativity, an Einstein Centenary Survey, ed. S. Hawking and W. Israel, Cambridge U. Press 1979. A. Duncan, Phys. Lett. B66 (1977) 170. L. Brown, Phys. Rev. D15 (1977) 1469. (4] D. Weingarten, Phys. Lett. B90 (1980) 285. V. A. Kazakov, Phys. Lett Bl50 (1985) 282. F. David, Nucl. Phys. B257 (1985) 45. J. Ambjorn, B. Durkuus, J. Frohlich, Nucl. Phys. B257 (1985) 433. V. A. Kazakov, I. K. Kostov, A. A. Migdal, Phys. Lett. B157 (1985) 295. J. Drouffe and C. ltzykson, Statistical Mechanics and Quantum Field Theory, Cambridge 1990. (5] V. A. Kazakov, Phys. Lett. B150 (1985) 282. F. David, Nucl. Phys. B257 (1985) 45. (6] E. Brezin, V. A. Kazakov, Phys. Lett. B236 (1990) 144. M. Douglas and S. Skenker, Nucl. Phys. B335 (1990) 635. D. J. Gross and A. A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [7] A.M. Polyakov, Mod. Phys. Lett. A2 (1987) 893. V. Knizhnik, A.M. Polyakov and A. B. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819. A. M. Polyakov, in Les Bouches XLIX, 1988, Fields, Strings and Critical Phenomena eds. E. Brezin and J. Zinn-Justin (Elsevier 1989). (8] A.M. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 1213. [9] Y. Matsuo, Phys. Lett. B241 (1990) 503. (10] T. Curtright and C. Thorn, Phys. Rev. Lett. 48 (1982) 1309. E. D'Hoker and R. Jackiw, Phys. Rev. D26 (1982) 3517. J.-L. Gervais and A. Neveu, Nucl. Phys. B209 (1982) 125, 224 (1983) 329. [11] E. D'Hoker, D. Z. Freedman and R. Jackiw, Phys. Rev. D28 (1983) 2583. (12] E. Braaten, T. Curtright, G. Ghandour C. Thorn, Ann. Phys. (N.Y.) 147 (1983) 365; Ann. Phys. (N.Y.) 153 (1984) 147. [13] J.-L. Gervais, A. Neveu, Nucl. Phys. B257 (1985) 59; Nucl. Phys. B264 (1986) 557.

217

[14] E. Witten, Nucl. Phys. 8340 (1990) 280. J. Distler, Nucl. Phys. 8342 (1990) 523. R. Dijkgraaf and E. Witten, Nucl. Phys. 8342 (1990) 486. E. Verlinde and H. Verlinde, Nucl. Phys. 8348 (1991) 457. M. Fukuma, H. Kawai, R. Nakayama, Int. J. Mod. Phys. A6 (1991) 1385. R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. 8348 (1991) 435. [15] K. Aoki, D. Montano, J. Sonneschein, Phys. Lett. 8247 {1990) 64. J. Hughes, K. Li, Phys. Lett. 8264 (1991) 261. H. Yoshii, Phys. Lett. 8259 (1991) 279. (16] H. Kawai and M. Ninomiya, Nucl. Phys. 8336 (1990) 115. S. lchinose, YITP /K-876 Kyoto Preprint {1990). (17] M. Douglas, Phys. Lett. 8238 (1990) 176. M.A. Awada and Z. Qiu, UFIFT-HEP-90-4 preprint. F. Fucito, M. Martellini and A. Gamba, Phys. Lett. 8248 (1990) 57. M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A6 (1991) 1385. E. Martinec, Comm. Math. Phys. 138 (1991) 437. P. Di Francesco and D. Kutasov, Nucl. Phys. 8342 (1990) 589; Phys. Lett. 8261 (1991) 385. (18] E. D'Hoker, Mod. Phys. Lett. A6 (1991) 745. E. D'Hoker, Lecture Notes on !-D Quantum Gravity and Liouville Theory, VI-th Swieca. Summer School Lecture Notes (1991), UCLA/91/TEP/35 preprint. (19] F. David, E. Guitter, Euro. Phys. Lett. 3 (1987) 1169. F. David, Mod. Phys. Lett. A3 (1988) 1651. J. Distler and H. Kawai, Nucl. Phys. 8321 (1988) 509. J. Distler, H. Kawai and Z. Hlousek, Int. J. Mod. Phys. A5 (1990) 391, 1093. [20] E. D'Hoker, Phys. Lett. 8264 (1991) 101. [21] C. Preitschopf and C. Thorn, Phys. lett. 8250 (1990) 79. (22] E. D'Hoker and P. S. Kurzepa, Mod. Phys. Lett. A5 (1990) 1411. N. Mavromatos and J. Miramontes, Mod. Phys. Lett. A4 (1989) 1849. [23] J.-L. Gervais, Phys. Lett. 8243 {1990)85; Comm. Math. Phys. 130 (1990) 257. J.-L. Gervais, NSF-ITP-90-182 Santa. Barbara preprint (1990). (24] E. D'Hoker and R. Ja.ckiw, Phys. Rev. D26 (1982) 3517.

218 (25) J. Polchinski, Texas Preprint: UTTG-19-90 (1990); UTTG-39-90. N. Seiberg, Rutgers Preprint RU-90-29 (1990) (26) VI. S. Dotsenko, V. A. Fateev, Nucl. Phys. B240 (1984) 312, Nucl. Phys. B251 (1985) 691. [27] A. Gupta, S. P. Trivedi and M. B. Wise, Nucl. Phys. B340 (1990) 475. H. Dorn and H. J. Otto, Phys. Lett. B232 (1989) 327. (28) M. Goulian and M. Li, Phys. Rev. Lett. 66 (1991)2051. (29) Y. Kitazawa, HUTP-91/ A013 (1991) preprint. Vl.I. Dotsenko, PAR-LPTHE 91-18 (1991) preprint. P. DiFrancesco and D. Kutasov, Phys. Lett. B261 (1991) 385. (30] K. Aoki and E. D'Hoker, UCLA/91/TEP/32 (1991) preprint. [31] K. Aoki and E. D'Hoker, UCLA/91/TEP/33 (1991) preprint. [32] S. Coleman, Phys. Rev. Dll (1975) 2088. (33) D. J. Amit, Y. Y. Goldschmidt and G. Grinstein, J. Phys. A13 (1980) 585. M. T. Grisaru, A. Lerda, S. Porrati, D. Zanon, Nucl. Phys. B342 (1990) 564.

219

GAUGE-INDEPENDENT ANALYSIS OF 2D-GRAVITY E. Abdalla Univ. of Sio Paulo, Cep:20516, Sio Paulo, Brazil M.C.B. Abdalla IFT/UNESP, Cep:01405, Sio Paulo, Brazil J. Gamboa Univ. de Zaragoza, Dep. of Physics, Zaragoza 50009, Spain A. Zadra CERN-TH, CH-1211, Geneve 23, Switzerland In this note we show that the induced 2D-gravity SL(2, R) currents can be defined in a gauge-independent way although they manifest themselves as generators of residual symmetries only in some special gauges 1 •2 • In the light cone gauge, we show that the Poisson algebra obtained from the naive canonical formulation leads to the above symmetry, and the generators may be easily computed. In the Coulomb gas representation we investigate two approaches, namely one resembling string field theory and another that privileges the SL(2, R) structure of the theory. We have considered the usual 2D gravity Lagrangian 1 as a starting point for the Hamiltonian formulation of 2D-gravity before fixing the gauge. The results are (a) the momenta conjugate to g00 and g01 (1r00 and 1r01 respectively) are constrained (a pair of abelian primary constraints); (b) the primary constraints generate two secondary constraints; (c) altogether the constraints form a consistent first-class system, which generates local coordinate transformations (gauge symmetry); (d) the light-cone and conformal gauges can then be worked out by fixing goo and go 1 as appropriate functions of gu. All these results are consistent with previous analysis; however it turns out that the Hamiltonian formulation simplifies if we add a surface term 3 ; we obtain the simplified constraints 1r00 ::::l 0 ::::l 1r01 , and the Hamiltonian is a linear combination of the secondary constraints, which are given by

The constraint algebra generated by After a change of variables

~•

•

is classically equivalent to the Virasoro algebra.

Q

v>-+ X= v>- 2ln lg11l , 7r ,..., -+

1fx = ,..., , 911

-+ ~

=

11 Q

-+

7r•

2 11 = c;gu7r + 1r., ,

2ln lg11l ,

the constraints turn out to be a combination of quadratic functions of these above defined fields, as follows

where h± = b ± "'' J± = 4'P ± x'. In this case, a quantization procedure similar to that used in the case of string theory can be envisaged, since we obtain the Virasoro algebra

The Hamiltonian can be written as

where U% play the role of Lagrange multipliers, so that fixing U± corresponds to choosing a gauge in this formulation. In the conformal gauge we propose a solution of the Liouville theory in terms of the SL(2, R) currents. It is interesting to speculate about the consequences of the previous discussion for the Liouville theory. The general solution for the Liouville field is usually given as c/> = f (f_AJ:B~] , where A and B satisfy simple linear first order equations, (LA = 0 and o+B = 0. Now we can use (26) and (41) to express the Liouville field in terms of the currents,

ln [

cf>

=

iln [1- ~2 (r +2x- J

0

+(.:~:-) 2 J+)]

where the J -components non-linear equations". We have then obtained the essential features of two dimensional gravity in a gauge independent way". Although the SL(2,R) currents can be defined in any gauge, they are residual symmetry generators only in some gauges. In the Coulomb gas representation we deal with a local action. We obtain a description very similar to string theory. In particular, the constraints algebra is a Virasoro algebra in that case. Alternatively, one can switch to a description that emphasizes the SL(2, R) structure, obtaining the corresponding KacMoody algebra. In the conformal gauge, we can write a general solution of the Liouville equation in terms of SL(2, R) currents which satisfy a certain set of non-linear first-order equations of motion. This might give us a hint for quantizing Liouville theory. (1} (2} (3] (4)

References A.M. Polyakov, Mod. Phya. Lett. A2(1987)893. E. Abdalla, M.C.B. Abdalla, A. Zadra ICTP prep 89/56. Ed.Sh. Egorian, R.P. Manvelian, Mod. Phya. Lett. A5 (1990)2371. E. Abdalla, M.C.B. Abdalla, J. Gamboa, A. Zadra, to appear.

221 The Discrete Integrability of Matrix Models of 2D-Gravity Mark J. Bowickt Institute for Theoretical Physics University of California at Santa Barbara Santa Barbara, CA 93106

Abstract Matrix models of random surfaces and 2D-gravity already exhibit at the discrete level many of the features of the double-scaling continuum limit. I discuss the one-matrix Hermitian and Unitary matrix models. The partition functions of these models are generically r-functions of an associated integrable hierarchy. They are not, however, arbitrary r-functions. They are those r-functions annihilated by an infinite dosed set of constraints which form a Virasoro algebra. The compatibility of the integrable flows and the constraints yields the string equation.

1. Summary The classical solutions of string theory are described by conformal field theories on the sphere. Higher genus effects may be accounted for by integrating over the location of all possible handles on this sphere. In this way one obtains an effective theory on the sphere which represents the complete sum over all genera. The action of the original conformal field theory Scft is replaced by Self = Scft + I::I.S. Although Self no longer describes a conformal field theory we may hope that in certain cases it still describes an integrable system. Zamolodchikov (1) has demonstrated that one can perturb the minimal series of models by certain operators and still have an integrable system. This also seems to be the case for string theories embedded in less than one target space dimension. In this talk I reviewed the results of [2) which illustrate how matrix models provide a concrete realization of integrable but not conformal theories, even at the discrete level. Matrix models arise as a simplified discrete approximation to the full functional integral over metrics in 2D quantum gravity. In the double scaling limit (3-5) the scaling function f(x), the second derivative of the partition function, is determined by a specific nonlinear differential equation, called the string equation, of a single scaling variable x. In addition the flows from one multicritical point of the theory to another are the KdV flows

(6). At the discrete level there are analogs of all these beautiful features of the double scaling continuum limit (7-11). For Hermitian models with a generic non-even potential t Permanent address: Physics Department, Syracuse University, Syracuse NY 132441130: E-mail: [email protected].

222 the weights of the appropriate orthogonal polynomials satisfy Toda-chain flows on the halfline with respect to couplings in the potential (8,9). In the continuum limit these flows become KdV flows. For even potentials the flows are described by the Volterra hierarchy (8,9,12). These are natural discretized versions of KdV flows to which they tend in the continuum limit. For unitary matrix models the flows are given by the modified Volterra hierarchy in the conventional basis of orthogonal polynomials on the circle [8,2) and by the Toda-chain in the trigonometric orthogonal polynomial basis (9,2). The partition functions at the discrete level are products of powers ofT-functions of the appropriate hierarchy. They are not, however, arbitrary T-functions but rather those that are annihilated by an infinite set of closed constraints which follow from simple invariances of the matrix model partition function and which satisfy a Virasoro algebra [13-15). The string equation may be viewed as a compatibility condition between the constraints and the integrable flows.

2. Acknowledgements This research was supported by DOE Grant DE-FG02-85ER40231, by funds from the Office of Sponsored Research of Syracuse University and by NSF grant PHY 89-04035. References [1) A. B. Zamolodchikov, "Integrable Field Theory from Conformal Field Theory," in Integrable Systems in Quantum Field Theory and Statistical Mechanics, Advanced Studies in Pure Mathematics 19 (1989), eds. M. Jimbo, T. Miwa and A. Tsuchiya. (2) M. J. Bowick, A. Morozov and D. Shevitz, Nucl. Pbys. B354 (1991) 496. (3) E. Brezin and V. Kazakov, Pbys. Lett. B238 (1990) 144. (4) M. Douglas and S. Shenker, Nucl. Pbys. B335 (1990) 635. (5) D. Gross and A. Migdal, Pbys. Rev. Lett. 84 (1990) 127; Nucl. Pbys. B340 (1990) 333. (6) M. Douglas, Pbys. Lett. B238 (1990) 176. (7) E. Witten, lAS preprint IASSNS-HEP-90/45 (1990). (8] A. Gerasimov et al., Nucl. Pbys. B357 (1991) 565. (9) E. Martinec, Chicago preprint EFI-90-67 (1990). (10) H. Itoyama andY. Matsuo, Pbys. Lett. B255 (1991) 202. [11) G. Moore, Commun. Math. Pbys. 133 (1990) 261. [12] L. Alvarez-Gaume, C. Gomez and J. Lacki, Pbys. Lett. B253 (1991) 56. (13) M. Fukuma, H. Kawai and R. Nakayama, Int. Jour. Mod. Pbys. A8 (1991) 1385. (14] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Pbys. B348 (1991) 435. (15) A. Mironov and A. Morozov, Pbys. Lett. B252 (1990) 47.

223

MULTICUT CRITICA.LITY IN THE PENNER MODEL .AND C=l STRINGS S. CBAUDBURI Tl&eorJ Group, JIS106 Fermi National Accelen~tor Labon~torJ P.O. Boa 500, Batot~ia, IL 60510

ABSTRACT The •ieepe~~t daeeat 80lution of the m-th critical point of the Penner matrix model hu an m-component eicen...Jue nppori, colllliiiW.. of •rmmetricallr placed in the complu eicen...Jue plane. Criticalitr rwult. when the branch point• of tiWI •uppori collleKe in plli.n to fol'Dl a elOMd contour. We deriYe the •trins eq-tiou of th- matrix modu for: Kbit:rv)" m 1 uiq the orihopnal pol;rnomial method. The double-.caled continuum 80lutiou Me dCIICribecl br non-lineR ftnitecWI'erence eq-tiou. The 6ee enersr of the mth model ia QOWD to be the Lepndre trauform or the 6ee enersr or the c=l •trias comp.ctifted to • cirele or radiu equal to an inteser multiple, m, of the llelf dual radiu.

In this talk I give a brief overriew of recent work done in collaboration with Bans Dykstra and Joe Lykken at Fermilab on multicut criticality in the Penner matrix model. First introduced as a means of computing the orbifold Euler characteristic of the moduli space of punctured Riemann sur!aces, the ( -k) expansion of the Penner Model can also be made critical, and thereby identified with the continuum free energy of a theory of two-dimensional gravity [1]. The double-scaled free energy coincides with the Legendre transformed free energy (the generating !unction of lPI amplitudes) of the e = 1 string with compact target space at the sel£-dual radius [2]. Recently, we discovered that this property extends to an infinite series of matrix model solutions that are polynomial perturbations o£ the Penner model [3)[4). Their free energy coincides with the generating !unction o£ lPI amplitudes of the c=l string with compact target space o£ radius equal to some integer, m, o£ the self-dual radius. The matrix integral eF

=

1

tlMeNt Tr(U(M)+Io- 25 scalar matter fields has all the ingredients to make Coleman's argument. The main requirements are: • a sum over geometries which includes non-trivial topologies • an action for the conformal mode which is unbounded from below • a Euclidean saddle point consisting of a sphere with radius "' the two-dimensional cosmological constant.

jr, where ~ is

l Work supported by the Department of Energy, contract DE-AC03-76SF00515. 2 Supported in part by NSF grant PHY89-17438

226 The leading order semi-classical approximation to the Euclidean action for such a spherical geometry is given by S E = log ~ for large D 3 • The power law behavior in four dimensions is replaced by a logarithm in the two-dimensional theory. As a result the Baum-Hawking amplitude [4,5] becomes ~- 0 16 , and after performing the sum over wormholes Coleman's 'exponential of exponential' [1] reduces to a single exponential exp ~-D/ 6 • Nevertheless, if, as suggested by Coleman, this expression can be regarded as a probability distribution for the cosmological constant, then it implies the vanishing of A in two dimensions.

f

Coleman's argument consists of two parts. The first says that the sum over wormhole topologies converts the constants of nature into probabilistic quantum variables governed by a wave-function on superspace. The second part assumes that the wavefunction is proportional to the Euclidean path integral and that this ill defined path integral can be evaluated by formally summing over saddle point configurations. This is to be compared with the conclusions that string theory leads to. First of all, it appears to be correct that topology change makes the couplings of the twodimensional worldsheet theory into quantum variables in target space. However, the second part of Coleman's argument is not supported by string theory. The wavefunction for coupling constants appears to be controlled by phenomena which know nothing of the large scale structure of space-time and have no reason to prefer A= 0. The remainder of the paper is organized as follows. In section 2 we briefly review the formulation of two-dimensional quantum gravity in conformal gauge and establish the connection with string theory in background fields. In section 3 we study "cosmological" solutions and derive the Wheeler-DeWitt equation which governs the propagation of a one-dimensional universe in a background condensate of baby universes. In section 4 we examine the relation between the target space equations of motion and the renormalization group, and consider the evolution of couplings with scale. Section 5 deals with the question of the two-dimensional cosmological constant in this framework. We present an explicit calculation of the required string theory beta-function, using an appropriate renormalization procedure. Finally we conclude with a discussion of our results.

3 Gravitational fluctuations are suppressed as D - oo. The existence of this semi-classical limit will be crucial for some of our arguments later on.

227

2. Two-dimensional Quantum Gravity and String Theory Let us begin with a theory of quantum gravity, defined on a two-dimensional spacetime (u 0 ,u 1 ), involving a metric '"fa& and D scalar matter fields Xi(u•). Most treatments have focused on the case D ~ 1 but for the purpose of modelling quantum cosmology it is appropriate to consider D > 25. The action is taken to be local and coordinate invariant, but can otherwise be quite general. It is also assumed that some covariant non-perturbative method of regularizing the theory exists. Unfortunately no such method is known at present for carrying out the continuum theory path integral in a manifestly covariant manner. Instead we shall have to rely on a prescription which is the analogue of old-fashioned methods of regularization and renormalization in gauge theories. Before the invention of gauge-invariant regulators, a procedure which worked was to regularize the theory in a non-covariant way and then compensate for the resulting non-invariance by allowing the Lagrangian to contain non-gaugeinvariant terms, such as a photon mass. At the end of the day, the gauge symmetry is re-imposed through Ward identities, which place constraints on the values of the added terms. A particular version of this method for two-dimensional gravity follows: • Gauge fixing: The first step is to remove the over-counting of metrics due to general coordinate invariance. For each worldsheet topology introduce some fixed fiducial, metric .y,.,. Then choose coordinates such that the physical metric is conformal to the fiducial metric, ~

.

'"fa&= e 'i'a6 •

(2.1)

The original path integral is replaced by an integral over the matter fields Xi(u) and the one remaining degree of freedom of the metric 4>( u ), which is called the Liouville field. • Regularization: In order to define the gauge-fixed path integral the ultr• violet divergences of the theory need to be regularized. For example, a nonperturbative regulator can be introduced by discretizing the worldsheet. This discretization is to remain fixed and not be summed over as in matrix models. The regulator involves a shortest fiducial length defined by,

(2.2) where f.4 is the line element connecting the nearest lattice points and c tends to zero as the cut-off is removed. A more covariant definition would refer the cutoff scale to the physical metric '"fa&, but then the regularization would depend on the Liouville field which is being integrated over. Thus we are obliged to use a non-covariant regularization procedure in order to have a concrete definition of the continuum theory path integral.

228

• Renormalization: Performing the path integral over short distance fluctuations of both the matter and gravitational fields generates various interaction terms, involving tP and Xi, in the effective Lagrangian. These terms will in general depend on the arbitrarily chosen fiducial metric 'tab and therefore the effective theory will not be manifestly covariant. On the other hand, the original theory is assumed to be invariant under general coordinate transformations so no such dependence on 'tab should occur. Thus we must impose upon the renormalized theory that the value of the path integral is not affected by the choice of fiducial metric. This can be achieved by, first of all, arranging the terms in the effective action to be covariant with respect to 'tab· This does not restrict the possible couplings but merely labels them according to their transformation properties under fiducial reparametrizations. The condition that the path integral does not depend on the determinant of 7 requires that the beta-functions of all couplings in the theory vanish. This means that the gauge fixed theory must be an exact fixed point of the renormalization group in order to maintain the original general covariance. To summarize: We start with a generally covariant theory of gravity coupled to scalar fields, Xi. In order to define the path integral we fix a gauge and regularize in a non-covariant manner. The resulting theory involves a scalar field, q,, in addition to the matter fields, and is in general quite complicated. The original covariance appears as a set of restrictions on the couplings, which include the requirement that all the beta-functions vanish. Notice that in this way of stating things tP and Xi are placed on equal footing. The Liouville field has been promoted to an additional target space dimension. This approach to the quantization of two-dimensional gravity has been advocated by a number of authors [3,6,7,8,9]. We are thus led to consider reparametrization invariant scalar field theory in two dimensions. The action can in general include terms with arbitrary functional dependence on q, and Xi, and with any number of derivatives acting on the fields. For convenience, let us define X 0 = ttl> where q 2 = D325 • This rescaling leads to standard normalization for the Liouville kinetic term. The two-dimensional action can then be written,

with p = 0, 1, ... , D. We have written down the terms of scaling dimensions zero and two~ but there is an infinite sequence of possible couplings involving more derivatives on the X" and higher powers of the two-dimensional curvature R. 4 For simplicity, we have not included the anti-symmetric tensor field. Its presence would not qualitatively alter our conclusions.

229 This class of theories has been extensively investigated in string theory, where the action (2.3) describes strings in background fields in D+ 1 spacetime dimensions. The beta-function equations, implementing the conformal invariance of the two-dimensional theory, have the form of field equations in target-space for T(X), ~(X) and Gp.,(X) (tachyon, dilaton and graviton fields respectively), along with additional fields representing higher order couplings. These field equations describe the propagation and creation and annihilation of the particle-like eigenmodes of strings in spacetime, or more to the point of this paper, one-dimensional universes containing matter fields~ Because of the identification X 0 = !4>, the role of time in target space is played by the two-dimensional scale. The tachyon field, T(X), is of primary interest because it controls the two-dimensional cosmological constant. A cosmological term in the original classical action corresponds to a tachyon background which grows exponentially with increasing two-dimensional scale, (2.4)

As we shall see, this remains qualitatively true in the quantum theory, as long as the tachyon background remains weak, but the rate of the exponential growth is modified by quantum fluctuations. The exponentially growing background will eventually become strong and then non-linear effects in the target space theory can no longer be ignored. The two-dimensional cosmological constant will still be governed by the behavior of the tachyon background in the non-linear regime, but the connection between the two is more subtle. The string theory equations of motion, obtained by setting beta-functions to zero, are derivable from a target-space action. For simplicity, we will work within a truncated theory, containing only the lowest order couplings, T(X), ~(X) and Gp.,(X). To leading order in derivatives, the target-space action for these fields is

[10]

where V(T) = -T 2 + ... is the tachyon effective potential. Since renormalization group beta-functions are not universal, the detailed form of V(T) will depend on the regularization and renormalization prescription used. This is believed to correspond to field redefinition ambiguities in the target space equations. In fact all higher order terms in the tachyon beta-function can be arranged to involve target space derivatives, and therefore be removed from the potential leaving only -T 2 [11,12]. It should be 5 We will use the string theory names for the target-space fields, but the reader should keep in mind their cosmological interpretation.

230 strened that using such a prescription in no way alters the fact that the target space equations are non-linear and are in general not exactly satisfied by a simple exponential tachyon background. The important question to ask is whether there exists a renormalization scheme in which the target space fields can be identified with Wheeler-DeWitt amplitudes of a one-dimensional universe. We will return to this point in section 5, where we propose an appropriate scheme and present a calculation of V(T) to all orders in T.

3. Quantum Cosmology in Two Dimensions The equations of motion which follow from the target space action (2.5) are

v 2 T- 2v~ · vT v

2

~- 2(v~)

1

2

=

V'(T), 25-D

=- - 6 -

+ V(T),

R,.v- 2G,.vR =- 2v,.v"~

(3.1)

+ G,."v 2 ~ + v,.Tv"T- 21 a,."(vT) 2 .

These equations have a simple solution, the so called linear dilaton background (13], which for D > 25 is given by T=O,

G,." = TJ"", ~

(3.2)

= - g_xo. 2

The target space is Lorentzian and it is the conformal mode, X 0 , which is timelike. ;This means that the kinetic term of X 0 in (2.3) has the "wrong" sign and the Euclidean action of the two-dimensional theory is unbounded from below. This is analogous to the instability of the Euclidean path integral in four-dimensional gravity, wHich lies at the heart of Coleman's argument for the vanishing of the cosmological constant. On the other hand, it means that Euclidean two-dimensional gravity coupled to D > 25 matter is ill-defined and the renormalization group computation, . which led to the target space equations (3.1), can only be viewed as a formal argument. Ideally the theory should be reformulated on a worldsheet of Lorentzian signature but it is unclear at present how to perform the steps involved in the quantization of such a theory (regularization, renormalization, etc.). In four-dimensional gravity people have sought to circumvent this problem by formally rotating the contour of path integration over the conformal factor into the complex plane to obtain a well defined integral [14]. While this formal procedure can also be applied in the two-dimensional theory, its validity has been called into question [15].

231 We will use the target space picture to define the two-dimensional theory for D > 25. The equations of motion (3.1 ), which were arrived at via a formal derivation based on a Euclidean worldsheet, lend themselves to an interpretation as a Lorentzian field theory of strings. Our assumption, which may be unwarranted, is that a consistent Lorentzian worldsheet formulation would lead to the same target space field theOry. Since the equations are non-linear, singular geometries which describe splitting and joining strings will have to be included in the Lorentzian two-dimensional path integral. In addition, the path integral will receive contributions from universes being absorbed or emitted from the background, which also involves two-dimensional singularities. By contrast, in Euclidean space the metric can be chosen with no singularities. We only use Euclidean methods to compute renormalization group betafunctions, but our subsequent discussion of the two-dimensional cosmology takes place with Lorentzian signature. An important difference between the worldsheet theory and four-dimensional gravity is that the gravitational coupling in two dimensions is dimensionless, so there is no proper Planck scale. However, as is well known, the strength of the string coupling depends on the dilaton field in target space. A key feature of the linear dilaton background (3.2) is that the string loop coupling constant is related to the two-dimensional scale, (3.3)

We can only expect the effective field theory to be simple where this coupling is weak. For D > 25 the theory is strongly coupled for sufficiently small strings and target space quantum mechanics (string loops) are important in the ultraviolet on the world sheet 6 • One can say that a Planck scale is spontaneously induced, and define it by the point at which goe• 1. The factor of go can be absorbed by a constant shift of the dilaton. The effective Planck-scale is then set by q- 1 . It depends on the number of scalar fields in the theory, and in particular, D --+ oo is a semi-classical limit for gravitational fluctuations 7 • Another way to see that q- 1 defines the Planck scale in this theory is to consider the relation between the classical conformal mode and the quantum variable, !

~ :., Tl e,

cua)

-J$.N$.J; -J'$.N1$.J'

where we have let (1.14)

Such is the Poisson bracket structure which is naturally associated with the standard Lie group sl(2).

It is the purpose of the present notes to summarize how such objects arise in twodimensional gravity. We shall see that the Poisson bracket structure (1.13) comes out from the canonical Poisson bracket associated with the Liouville action. Its quantization will thus exhibit the Uq(sl(2)) symmetry, where the non-commutativity associated with the deformation of the group is due to the non-commutation of the Heisenberg quantum-field-operators. 2. THE CLASSICAL LIOUVILLE DYNAMICS. The problem of quantum gravity in the conformal gauge is equivalent to the quantization of the Liouville action which may be written as (2.1) and r are the local coordinates such that the metric tensor takes the form The complex structure is assumed to be such that the curves with constant q and r are everywhere tangent to the local imaginary and real axis respectively. In this section we discuss the resolution of the corresponding field equations at the classical level. The action corresponds to a conformal theory such that exp(2cl) is a conformally covariant field 14 with weights (1,1) as we shall recall below. The corresponding field equation where

Dab

q

= 6a,b e2c) (We use Euclidean coordinates).

(2.2) is conveniently solved using the following theorem. The field c)(q, r) satisfies (2.2) if and only ift i "" (2.3) e -cl = y'2 LJ /j(z+)9j(z-); 2 j=l,2

where /j (resp.(gj), which are functions of a single variable, are solutions of the same Schrodinger equation

, +-T(z_)gj ).

( resp. - 9j

(2.4)

The solutions are normalized such that their Wronskians fiJi- fih and 919~- g~g2 are equal to one. The proof goes as follows. 1) First check that (2.3) is indeed solution. Taking the Laplacian of the logarithm of the right-hand side gives

t The factor i means that these solutions should be considered in Minkowski space-time

251 where 8± = (8/8u ± i8/8r)/2 . The numerator has been simplified by means of the Wronskian condition. This is equivalent to (2.2). 2) Conversely check that any solution of (2.2) may be put under the form (2.3). If (2.2) holds one deduces (2.5) r(±) are thus functions of a single variable. Next the equation involving r(+) may

be rewritten as

(2.6) with solution with -

tj' + r(+) /j = 0

where the 9j are arbitrary functions of z_. Using the equation (1.4) that involves + r(-)9j = 0. Thus the

r(-), one finally derives the Schrodinger equation

-g'J

theorem holds with T = r(+) and T = r(-) q.e.d. o One may deduce from (2.5) that the potentials of the two Schrodinger equations coincide with the two chiral components of the stress-energy tensor. ( Thus these equations are the classical equivalent of the Ward identities that ensure the decoupling of Virasoro null vectors). For the time being we shall concentrate on one of the two chiral components. Consider for instance the - chiral components which are analytic functions of z = T + iu. In a typical situation, u and T may be taken as coordinates of a cylinder obtained by conformal mapping from a particular handle of the Riemann surface considered. T plays the role of imaginary time and u is a space variable. One may work at T = 0 without loss of generality. The potential T( u) is periodic with period say 211" and we are working on the unit circle. Any two independent solutions of the Schrodinger equation is suitable. It seems natural at first sight to diagonalize the monodromy matrix, that is to choose two solutions noted 1/lj, j = 1, 2, that are periodic up to a multiplicative constantt. It is convenient to introduce tPj(u) :=ln('l/lj)/~-lndj, dj are suitable normalization constants. The fields constants and have the expansion

tPj(u)

t/>j are periodic up to additive

= q~j) + p~>u + i Le-ino- p~) /n,

j

= 1, 2,

(2.7)

n;to

As shown in Ref. 2,3, the canonical P.B. structure of the action (2.1) is such that the chiral fields tPj satisfy

t

We assume that the monodromy matriz is diagonalizable

252

{q~i),p~)}P.B. = 1

(2.9)

Th = ( tn, i = 1, 2,

L:

n~O

such that

p~l)

= -p~2),

(3.2) (3.3)

N( 1) (resp. N( 2)) denote normal orderings with respect to the modes of t/>1 (resp.

t/>2).

Eq. (3.3) defines the stress-energy tensor and the coupling constant -y of the quantum theory. The former generates a representation of the Virasoro algebra with central charge C = 3 + 1h. The chiral family is built up4 •5•7•9 from the following operators

1/Jj=djN(j)(e./hi"fi~i), ;ji=~N(j)(eJ'{;i2;¢;), h=

j=1,2,

(3.4)

~(G-13-y'(G-25)(C-l)),

'h = ~ (c- 13 + y'(c- 25)(0 -1)),

(3.5)

where dj and d; are normalization constants. The parameters h and h are determined as the two solutions of the quadratic equation :z:2 - 11"( c

- 13):z:/6 + 11" 2 = 0

(3.6)

254 which ensures 4 that the fields so defined are solutions of the equations

h ("' Lo h C- 1 ) h ("' ino' Lo) -1/J;" +(;) L..J Lne -ino' +2+( +2 = 0 16"' -24) 1/J;+(;)t/J; L..J LnenO

(3.7)

.iJ' + ("' Lo ( h"' -24 C -1>)Lo) -,_; ;: L..J Lne -ino' +2+ 1/J;+ (h);: 1/J; ("' L..J Lne-ino' +2 16 nO

=0

(3.8) These are operator Schrodinger-equations equivalent to the decoupling of Virasoro null-vectors4 •5•7 . They are the quantum versions of Eq. (2.4). Since there are two possible quantum modifications h and h, there are four solutions. By operator product t/1;, j = 1, 2, and ;j;;, j = 1, 2, generate two infinite families of chiral fields which are denoted

"'~), -J :5 m

:5 J, and

~, _J :5 m:5 J; respectively, with 1/J~{;J = tPlo

(1/2) :i.(l/2) :i.(1/2) tP 112 = tP2, and tP -l/2 = tP1 , 1/J112 = tP2·

(J) :t(J) tPm , tP;n , are of the type {1, 2J + 1)

and ( 2J + 1,1), respectively, in the BPZ classification. For the zero-modes, it is simpler9 to define the rescaled variables . (1)

w='Po

{?;; . (1)ff"' -;:::-; VT; w- ='Po h

h w=w-; 1r

- h w=w-. .,..

(3.9)

At this point a pedagogical parenthesis may be in order: the hatted and unhatted fields have the same chirality; if we go to r ¥:- 0 they are both functions of z_;

1/J

-(J)

.:::::.(1)

there are two counterparts 1/Jm (Z+) and 1/Jm (Z+) which will be discussed below. Returning to our main line, we recall that the Hilbert space in which the operators 1/J and ;j; live, is a direct sum9•10•11 of Fock spaces F(w) spanned by the harmonic excitations of highest-weight Virasoro states noted lw, 0 >. The harmonic excitations are created by the negative modes of the fields t/>; of Eq. (3.1). The highest-weight states are eigenvectors of the quasi momentum w, and satisfy Lnlw,O >= 0, n > 0; (Lo - .:1( w) )lw, 0 > = 0. The corresponding highest weights .:1(w) may be rewritten as (3.10) The commutation relations (3.2) are to be supplemented by the zero-mode ones:

[q~1), p~1)] = [q~2), p~2)] = i. The fields 1/J and ;j; shift the quasi momentum p~l) an arbitrary c-number function f one has

1/J~) f(w)=/(w+2m)1/J~),

= -p12) by a fixed amount.

;p0 f(w)=f(w+2m?r/h)i)!>. m m

For

(3.11)

255 The fields 1/J and ~together with their products may be naturally restricted to discrete values of w. They thus live in Hilbert apacest- of the form

+oo

E9

'H(wo)=

(3.12)

F(wo+n+n1rjh).

n,fi=-oo

w 0 is a constant which is arbitrary so far. The .s1(2, C)-invariant vacnum (witll highest weight ~( wo = 0)) corresponds to wo = 1 + 1r / h, 9 but other choices are also appropriate, as we shall see. Next we display the quantum-group structure of the chiral fields. The operators 1/J and ~ are closed under O.P.E. and braiding. Each family obeys a quantum group symmetry of the U9 (.s1(2)) type. However, the fusion coefficients and R-matrix elements depend upon w and thus do not commute with the '1/J's and ~·s. Their explicit form is unusual, therefore. One may exhibit the standard U9 (.s1(2))-quantum-group structure by changing basis to new families. Following my recent work,9 let us introduce (3.13) ei:>(u) := IJ,w)A( .,p~>(u), -J $ M $ J;

L

-J~m~J

jJ,w)l}

= j(,!~)

"'

LJ. e e-Mtm-1) mteger

eihmf2x

iht(w+m) (

J- M

(J-M+m-t)/2

) (

J

+M

) .

(J+M+m+t)/2'

(3.14)

lPJ! (QP) =lQJ!lPQJ!

lr j

= sin(hr) - sinh ·

(3.15)

r=l

The last equation introduces q-deformed factorials and binomial coefficients. The other fields ~ are defined in exactly the same way replacing h by The symbols are the same with hats, e.g. n

lnj! =IT irj,

• • _ sin(hr)

lrJ = sinh ,

1i. everywhere.

and so on.

(3.16)

r=l

e

In 9•11 the operator algebra of the ields was completely determined. In particular it was shown that, for 1r > u > u' > 0, these operators obey the exchange algebra

(J, J')~ ,, e~:>(u') e~>(u),

t

(3.17)

Mathematically they are not really Hilbert apacea Iince their metria are not positive definite

256 which exactly coincides with the braiding relations of U9 (sl(2)) recalled in section 1 Eqs (1.7)-(1.10). In Ref. 11, the short-distance operator-product expansion of thee fields was shown to be of the form

e~:>(u) e~:>(o-1 )

=

Jl+J2

'E 9h, J2 { (d(u- u )).6.(J)-.6.(Jl)-.6.(J 1

2)

J=lh-121

(Jt,Mt; J2, M2!J1,J2; J,Mt + M2)

=

(e~!+M2 (u) +descendants)}.

(3.18)

where d(u- u1 ) 1- e-i(u-u'), and where the Clebch-Gordan coefficients appear, A(J) := -hJ(J + 1)/7r- J is the Virasoro-weight of e~>(u). Define the quantum group action on the fields by

e

(3.19) Then the operator-product e~:>(u) e~:>(u') gives a representation of the quantum group algebra (1.2) with the co-product generators of the type (2.3) (3.20) where the tensor product is defined so that (3.21) and where each term in the expansion over J transforms according to a representation of spin J. The family of operators just described is what I call the universal conformal family (UCF) associated with U9 (sl(2)). Its fusion and braiding are explicit realisations of the abstract structure of the non-commuting symbols e~> '7~) introduced in section 1. The hatted operators similarly form another UCF associated with Uq(sl(2)). The two families do not commute, eventhough their mixed braiding relations are simple9 . They are noted Uq(sl(2)) 0 Uq(sl(2)) where the 0 is not a simple tensor-product. 4. THE QUANTUM METRIC FIELD OPERATOR. If C > 25 h and h are real and the structure recalled above is directly handy. This is the weak coupling regime which is connected with the classical limit (-y-+ 0). Let us discuss how the powers of the metric are reconstructed. The standard screening charges -a± of the Liouville theory 4•6 are such that

257

{C-1

ao

= ~JG- 25

(4.1) 3 2 Q, and ao are introduced so that they will agree with the standard notation, when we couple with matter. Ka.c's formula. may be written as Q=y~)

~

( ~ 1 ~ t:.xac(J, J; G)= -2f3(J, J; G) f3(J, J; G)+ Q , )

f3(J,J; G)= Ja_

+ Ja+,

(4.2)

where 2J and 2J are positive integers. Thus the most general Liouville field is to be written as exp( -(Ja_ + Ja+)~).At first we consider the operators exp( -Ja-~). which are direct quantum-analogues of Eq. (2.13). There are two cases to distinguish. 1) One may consider, as is most usual, closed surfaces without boundary. Then the natural region is the whole circle 0 ~ u ~ 211". The quantum version of Eq. (2.13) will involve the fields {~), together with their counterparts~) ( Z+) whose exchange properties are similar. Concerning the latter one should remember that they are functions of z*, that is, are anti-analytic functions. so that the orientation of the complex plane is reversed. This may be taken into account simply by replacing i by -i in the above formulae for the {-fields, that is by taking the complex conjugate of all the c-numbers without taking the Hermitian conjugate of the operators.

(

The appropriate definition of Z~~? u) is

~)(u)

L

:=

(IJ,w)M )* ~!,;)(u),

-J

~

M

~ J;

(4.3)

-J(u)e~1(u).}ro

(4.4)

M=-J where CJ is a normalisation constant. The transformation of thee fields is similar to the one of the fields:

e

258 Thus if we define,

J± = J±e-ihJ3 + eihJ3 ® J±,

J3 = J3 + J3,

(4.6)

which does give a representation of (1.2). Then one easily checks that

(4.7) so that the quantized Liouville field is a quantum-group invariant. Clearly, the quantum case and the classical case recalled in section 2 are remarkably similar. 2) One may also consider gravity with boundary, following Ref. 1,2,13. A typical situation is the half circle 0 ~ u ~ 1r. One may set up boundary conditions such that the system remains conformal, albeit with one type of Virasoro generators only. The left- and right-movers become related as is the case for open strings. The appropriate definition of the metric becomes1 3 :

e-Ja-~(u)

= CJ

I: A~~N e~)(u)et')(27r- u)

(4.8)

M,N

where

These fields may be shown to be mutually local and closed by fusion. 5. THE DRESSING BY GRAVITY. In this section we study the dressing of conformal models with central chargeD by the Liouville field with central charge C so that

C+D= 26.

(5.1)

We shall be concerned with the case D < 1, where the Liouville theory is in its weakly coupled regime C > 25. As is recalled in the appendix A, the existence of the UCF's is basically a consequence of the operator differential equations (3. 7), (3.8). The latter are equivalent to the Virasoro Ward-identities that describe the decoupling of null vectors. Thus the UCF's, with appropriate quantum deformation parameters also describe the matter with D < 1. We will thus have another copy of the quantum-group structure recalled above. li will be distinguished by primes. Thus we let h' 11" ii.' 11" D=1+6(-+-+2)=1+6(-+:==-+2), 11"

h1

h'

1r

h'

with

h'h'=,.. 2 ,

(5.2)

= ~(D-13-J(D-25)(D-1)), h'= ~(D-13+J(D-25)(D-l)), (5.3)

259 and (5.1) gives h + h' =

h + h' =

(5.4)

0

Of course we choose the matter and gravity fields to commute. Using the notation of last section, the complete quantum group structure is thus of the type:

{ [Uq(sl(2)) 0 Uq(sl(2))]®[Uq(sl(2)) 0 Uq(sl(2))]}

®

{ fUq-- 1 (sl(2)) 0 Uq-l(sl(2))]®[Uq-_ 1 (sl(2)) 0 Uq-l(sl(2))) },

(5.5)

where the first (second) line displays the quantum-group structure of gravity ( matter). According to the above results, the spectrum of weights of the gravity and matter are respectively given by

Ag(J,J)

= 0 2~ 1 - 2~

AM(J',J') =

((J + J + 1)v'c -1- (J- J)v'c- 25) 2 ,

D2~ 1 + 2~ ((J' + J' + 1)V1- D + (J'- J')\1'25- n) 2 .

(5.6) (5.7)

The connection with Kac's table will be spelled out in section 5. As we shall see in section 5, this last formulae is consistent with the usual formulae for minimal models. From the standpoint we are taking, the most general matter field is decribed by an operator of the form exp( -(J1 a:~ + .Ya:~)X), where X(o-, T) is a local field that commutes with the Liouville-field and whose properties are derived from those oft by continuation to central charges smaller than one. Following our general conventions we let h1 = 1r( a:~ )2 /2, and h' = 1r( a:~ )2 /2. The correct screening operator is the field exp(-(Ja:_ +Ja:+)t) with spins J and Jthat Ag(J,J)+AM(J1,J1) = 1. In this connection, it is an easy consequence of (4.1) that

Ag(Y, -J'- 1) + AM(J', Y)

=1

(5.8)

Ag(-Y -1,J')+AM(J',J1)

=1

(5.9)

These two choices correspond to the existence of two cosmological terms. Indeed, the unity-operator of matter (J' = J1 = 0) is dressed by exp(a:+t) and exp(a:_t) respectively with the choices (5.8), (5.9). As is usual, we choose the latter as cosmological term so that the spins of gravity and matter fields will be related by Eq. (5.9). Thus we shall be concerned with matrix elements of the operators

(5.10) Concerning matter, choosing J 1 > 0 and J1 > 0 gives AM > 0 and the formulae of last section may be directly used. Concerning gravity, it has been already emphasized 10•11 , that the dressing o£ matter-operators with positive spins requires the use of gravity fields with negative

260 spins. For example, the cosmological term has J = -1, J = 0. Thus the gravity-UCF with parameter h must be extended. This problem was overcome in Ref. 10,11 as follows. First, the quantum-group structure has an obvious symmetry in J --+ -J -1 with fixed M, so that the universal R-matrix and Clebsch-Gordan coefficients are left unchanged. Second, the coefficients jJ, w)~ also have a natural continuation to negative spin. For postive J, one has (~!:)(w) is defined in Ref. 10)

1- J -1,w)~ = IJ,w)~ (-1)J+m j[(2isin(h))1+2J ~!:)(w)] It follows that there exist fields fields ,P~J- 1 ) similar to

(5.11)

et'J - 1) with fusion and braiding similar to e~)

I

and

,p~) so that J

et"J-1)

(5.12)

m=-J

Next, the appropriate definition of positive powers of the metric is, by extension of the discussion of the previous section,

J

_1__ " (- 1)J-M eih(J-M) ..e(-J-1)( . t= ..jiiic_J-1 L.i M Z+ )e(-J-1)( .. _M z_ ) yW.

(5.13)

M=-J

Indeed, first this field is local since the exchage properties of the fields

et'-

et"J-

1)

and

1) are the same as those ofthe fields e~> I and~) respectively. Second, it may also be re-expressed in the Bloch-wave basis, and this shows that this field leaves the condition w = w invariant. The symmetry between J and -J- 1 is the explanation of the continuation performed in Ref. 16. Next consider the Hilbert space in which we are working. The UCF's which appear in (4.5) live in spaces with highest-weight vectors of the form

lw, 0 > ®!w, 0 > ®lw', 0 > ®jw', o > .

(5.14)

(The two UCF's of a product of the type Uq(sl(2)) 0 Uq(sl(2)) are realized in the same Hilbert space). In the restricted Hilbert space, one has w = w, and w' = fii'. Thus we introduce states of the form jw,

w1 >=

jw, 0 > ®j,w, 0 >

®!w', 0 > ®lw', 0 >.

(5.15)

The next point concerns the physical on-shell states. They should satisfy the condition

(Lo

l I + -Lo + Lof -+: rLo2)jw, w >= o,

(5.16)

261 where the notation is self-explanatory. According to Eq. (3.10), this is satisfied if

h 11" 2 h 2 h1 11" 2 h1 1 2 411" (1 + h) - 411" aT + 411" (1 + h') - 411" aT

= 1.

(5.17)

It follows from Eqs. (3.6), (5.1), that

!:_( 1 ~)2 !!_( 1 ~)2 = C + D- 2 = 1_ 411"

+h +411"

+h,

(5.18)

24

Moreover, according to Eqs. (3.9), (5.4),

h'w12 = h'w' 2 = -hw' 2 so that, according to Eq. (5.4), the on-shell condition is w 2 = w1 2. The sign ambibuity is related with the two possibilities of cosmological term. It will be shown below that the choice, which is consistent with the above definition of gravity-dressing is _, (5.19) or, equivalently, aT= -aT, The coupling constants C1,2,3 appear in the three-point function as follows (5.20) It was computed in Ref. 15. The result takes the form c1,2,3 =-

11" ( xr(2 + h/1r)r(2- h/11")

)2 II

- BJ,,. Jl I

2

0

(5.21)

z [r(1+2J1 +(1+2Jz)h/11")] where B J' I'

Ji are normalisation-dependant constants. The calculation is somewhat I

involved, but the result is remarkably simple. Next, we make contact with other approaches to the same problem. For this we first explictly connect the present group-theoretic notations with more standard conventions. Eqs (4.1), and (5.2), correspond to the usual notations C = 1 + 3Q 2, and D = 1-12~, for the gravity and matter central charges. The screening charges are given by (we chose ao > 0)

(5.22) The dressed vertex operator (5.10) may be rewritten as

262 where

{3

= (J' + 1)a-- J'a+,

ik

= a-(J'a'_ + J'a~).

(5.24)

A simple calculation, using the formulae just given leads to

= 21'- 21'hj1r = 2J + 2(J + 1)h/7r, {J(k) + Q/2 = (k + ko)fa_, ko = aoa- = 1- hj1r k

(5.25) (5.26)

According to (4.2) (4.6), and (4.7), the weights are given by ilg

For rational theories ( D h

1

= - 2!3(/3 + Q), =1-

ilM

6(p - p1)2/ pp1, p

7r

= 4h k(k + 2ko).

(5.27)

> p1 > 0),

= -h', = 1rp1fp, h = -h', = 1rpjp1,

ko

= (p- p')fp,

(5.28)

ilM reduces to Kac's table ilM

= __!_., [(rp- sp1) 2 4pp

(p- p 1) 2],

r

= 211 +1, s = 21' +1,

k+ko

= r-shj1r.

(5.29) The momentum k is defined so that it takes rational values for rational theories. For critical bosonic string with Regge slope a'. the tachyon vertex is exp(ikXH). Eq. (5.24) corresponds to a 1 = 1/(a-)2. With the conventions just introduced, the result Eq. (5.21) takes the form Ct,2,3

-(7rr(2+h/7r)r(2-h/7r)/h) 2

= A(k -

k k )-

1' 2' 3 -

II 1

B(kz)

[r(kz+ko)] 2 '

(5 30) ·

where we left out an overall factor which does not depend upon the momenta. Our last task is to re-establish the cosmological constant. In the present approach, it comes out as follows. Our basic quideline was to write down the most general local operators, as was discussed in section 2. We have not yet really done so, since we may multiply the right-hand sides of (4.4) and (5.13)) by p.;fZI/2 on the left, and by

P.":/2 on the right, without breaking locality. This constant

P.c is arbitrary, and will play the role of the cosmological constant. According to (3.11) one has

(5.31) In the Coulomb gas picture, the power in P.c is equal to the total number of screening charges. We shall agree with this definition if we let

Thus we have, according to Eq. (5.24),

V (~c)~ ( tT,T ) = P.c-(1'+1)+J'1r/h P.c-w/2V1 , J',J'

(

•

) w/2

Jl tT,T P.c

.

(5.33)

The first factor coincides with the KPZ-DDK scaling factor 17 •18 •19 . In particular, one has ) -1 -w/2 (5.34) V0(~c)( Vo,o ( tr,T ) l'cw/2 , , 0 tr,T = l'c l'c which is the expected scaling behaviour of the cosmological constant. The operators

,.,.;w/2 induce a translation on the variable conjugate to the Liouville momentum.

This Liouville position-operator is proportional to q~l) which the is the zero mode of the free field tPl whose properties were summarized in the section 3. This translation, which is actually a global Weyl transformation, is analogous to the translation of the Liouville field in the work of DDK. This is seen by computing next the l'c-dependence of the three-point function. One gets immediately (5.35)

,.,.:1

2 hit the left and right vacua respectively. The term p.~0 arises when ,.,.;w/2 , and In the DDK discussion 20 •19 , it comes from the term of the effective action which is linear in the field t. One may verify that the power of l'c is equal to "'2 + fi.J'If / h where ~'2 and i/2 are the gravity screening-numbers. This agrees with the usual definition (see, e.g. Ref. 18). Finally we compare (5.30) with the result of the matrix model. This part follows Ref. 18 closely. The two-point function .A(~c)(k,k) is determined by starting from the three-point function with one cosmological term .A(~c)(O, k, k), and writing

(5.36)

According to (5.34), this gives (5.37)

Similarly, the partition function satisfies

d3 dp.~z(~c)

= .Ac~c)(O,O,O),

(Pc) 3

z(~c) = (wo)(wo -l)(wo- 2)-Ac~c)(O,O,O).

(5.38) (5.39)

It would not be there on the torus, since one would take a trace. This agrees with the DDK result where its contribution is shown to be proportional to the Euler characteristic

264 We finally obtain the rescaled three point function

= (7!/h + 1)(7r/h)(7r/h -1)'

(5.40)

which agrees with the results of the matrix models. Clearly, the key point in this final verification is that the final expression (5.30) for the three-point function factorises. On the other hand, (5.30) vanishes whenever kz + ko = 0 for any of the three legs. According to (5.29), this happens for rational theories, at the border of Kac's table, where r = p1 , and s = p. Thus formula (5.40) holds only when the branching rules are satisfied.

REFERENCES 1. J.-L. Gervais, A. Neveu, Nucl. Phys. B199, 59 (1982).

2. J.-L. Gervais, A. Neveu, Nucl. Phys. B202, 125 (1982). 3. J.-L. Gervais, A. Neveu, Nucl. Phys. B224, 329 (1983). 4. J.-L. Gervais, A. Neveu Nucl. Phys. B238, 125 (184.) Nucl. Phys. B238, 396 (1984). 5. J.-L. Gervais, A. Neveu, Nucl. Phys. B257[FS14], 59 (1985). 6. J.-L. Gervais, A. Neveu, Phys. Lett. 151B, 271 (1985). 7. J.-L. Gervais, A. Neveu, Nucl. Phys. B264, 557 (1986). 8. 0. Babelon, Phys. Lett. B215, 523 (1988). 9. J.-L. Gervais, Comm. Math. Phys. 130, 257 (1990). 10. J.-L. Gervais, Phys. Lett. B243, 85 (1990). 11. J.-L. Gervais, Comm. Math. Phys. 138, 301 (1991). 12. J.-L. Gervais, "On the algebraic structure of quantum gravity in two dimensions" ICTP preprint NSF-ITP90-176, Sept. 1990, Proceedings of the Trieste Conference on topological Methods in quantum field theories, World Scientific. 13. E. Cremmer, J.-L. Gervais, "The quantum strip: Liouville theory for open strings" LPTENS preprint 90/32. Comm. Math. Phys. to be published. 14. J.-L. Gervais, B. Sakita, Nucl. Phys. B34, 477 (1971).

265 15. J.-L. Gervais, "Gravity-matter coupling from Liouville theory", LPTENS preprint 91/22. 16. M. Goulian, M. Li, "Correlation functions in Liouville theory" Santa Barbara preprint, UCBTH-90-61, 1990. 17. V. Knizhnik, A. Polyakov, A.A. Zamolodchikov, Mod. Phys. Lett. A 3, 819, 1988. 18. Vl.S. Dotsenko, "Three-point correlation functions of minimal theories coupled to 2D gravity" PAR-LPTHE preprint 91-18, 1991. 19. J. Distler, H. Kawai, Nucl. Phys. B321, 509 (1651). 20. F. David, C. R. Acad. Sci. Paris 307, 1051,1988; Mod. Phys. Lett. A 17, 1651, 1988.

266

Toplogical Strings from Liouville Gravity Nobuyuki Ishibashi University of California Department of Physics Santa Barbara, CA 93106 U.S.A.

Abstract

We study constrained SU(2) WZW models, which realize a class of two dimensional conformal field theories. We show that they give rise to topological gravity coupled to the topological minimal models when they are coupled to Liouville gravity.

Two dimensional quantum gravity has been studied via, roughly speaking, three aproaches, i.e. matrix models, Liouville gravity and topological gravity. Although the matrix models are the most powerful in calculating various quantities, it is of great significance to have a continuum field theory formulation of them. In continuum approaches, we can understand the geometrical meaning of various results more clearly. Also we can construct variants of two dimensional gravity, e.g. supergravity, W gravity, etc., which are not available thus far in the matrix model approach. Liouville gravity is supposed to be the continuum field theory describing two dimensional quantum gravity, but it is still very difficult to calculate correlation functions on a general Riemann surface. On the other hand, one can calculate correlation functions in the topological gravity approach and show that they coincide with the results of the matrix models. Therefore it is very important to show the equivalence of the two continuum approaches, Liouville gravity and topological gravity. In [1 ), Liouville gravity coupled to a c = -2 matter conformal field theory was considered and was shown to be equivalent to pure topological gravity. This model is supposed to correspond to the first critical point of the one matrix model. Here we generalize this work to the continuum theories corresponding to the first critical points of multimatrix models (2]. The first critical point of k chain matrix model is supposed to correspond to a Liouville matter conformal field theory. On the other hand, this gravity coupled to a c = 1 model was shown to be equivalent to topological gravity coupled to the topological minimal matter, which was obtained by twisting the minimal N = 2 superconformal field theory. (3). We have constructed a conformal field theory with c = 1 and coupled it to Liouville gravity. This model is shown to be equivalent to toplogical gravity coupled to the topological minimal model.

&(!!!)'

s(!!!l'

The construction of the matter theory goes as follows. It was shown in [4] that SL(2, 'R)A: WZW model with a constraint

(1)

1- sc:!!l•.

gives rise to a conformal field theory with c = Here J. is the holomorphic component of the SL(2, 'R) currents. We consider a similar procedure in SU(2) WZW model, which is SU(2)A: WZW model with constraints

(2) A gauged WZW model corrnponding to such a constraining can be given easily. This gauged WZW model is exactly soluble and is shown to be a conformal field theory with c = (k; integer). Notice that the level k of SU(2) WZW model should be an integer, contrary to SL(2, 'R) case. There are k + 1 primary fields in this theory which are made by dressing the k + 1 primary fields in WZW model by the gauge field. We then couple this matter theory to Liouville theory. Exploiting the fact that SU(2) WZW model can be expressed by a free boson and ZA: parafermion, it is possible to realize the whole system by ZA: parafermion and a bunch of free bosons and free fermions. By reshuffling these free fields, we can show that this system has the same field content as topological gravity coupled to the topological minimal model. Moreover, the stress tensors of the Liouville gravity and the topological theory are mapped to each other by this reshuffling. Also the observables in the Liouville theory made by dressing the k + 1 primaries mentioned in the previous paragraph, correspond to k + 1 gravitational primaries in the topological theory.

1- s(:!!>',

Acknowledgements I would like to thank my collaborator Miao Li for stimulating discussions.

References (1] J. Distler, Nucl. Phys. B342 (1990) 532. [2] N. Ishibashi and M. Li Phys. Lett. B262 (1991) 398. [3] T. Eguchi and S.-K. Yang, Mod. Phys. Lett. A21 (1990) 1693; K. Li, Nucl. Phys. B354 (1991) 711, 725. [4) N. Bershadskii and H. Ooguri, Comm. Math. Phys. 126 (1989) 49.

268

Matrix Models at Finite N H. Itoyama Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840 Abstract We summarize some aspects of matrix models from the appro~Khes directly based on their properties at finite N.

Current interest in string theory in less than or equal to one dimension has arisen from the discovery of the existence of the continuum limii called double scaling limit [1] which sums string perturbation theory. At the same time, it has become clear that the universal equations which govern the correlation functions near the critical points are given by hierarchical integrable differential equations [1, 2, 3). By some time last year, however, more than several people had come to realize that many of the results obtained in the limit are already visible when N- the size of the matrix- is kept finite. This provides an opportunity to study the system in its original definition and as a solvable model. The progress from that direction is what we will discuss below. We will summarize some aspects of zero dimensional matrix models from the point of view based directly on results obtained at finite N. Such analyses appear to be imperative in view of the recent status of nonperturbative 2d gravity: no limiting procedure has been found, up to now, which maintains both the reality of the partition function and the original combinatorial correspondence of a matrix model with a triangulated two dimensional surface (4, 5]. No satisfactory definition of continuum 2d gravity ( nonperturbative) has, therefore, been given. In the one matrix model, we will cover the following items: Virasoro constraints at finite N fully characterize the space of correlators: the equivalence of the two approaches, i.e. Dyson-Schwinger approach and the one based on orthogonal polynomials; how to take the double scaling limit of the Virasoro constraints at finite N near the (2, 2k - 1) critical points (6] to obtain the Virasoro constraints of a twisted

269 boson [7, 8]; correspondence between the one matrix model and the classical Toda lattice equation [9]: the derivation of Kazakov's loop equation [10] directly from the Virasoro constraints (See, for instance, [11]). See also [12] for these items. In the case of the two matrix model, the situation has been clarified since the original developments and proposals [2, 13]. By now, we agree that the two matrix models capture the essential feature of the 2d gravity coupled to a general (p, p') conformal matter [14, 15]: it contains all the critical points indexed by a set of coprime integers (p, p'). We will discuss the new constraints of w 00 type derived in

[16].

A.

One matrix model

Recall the partition function of the one matrix model:

N -LV(,\;;gt)

N

j'fi.d>.;D. (>..1 · · · >w) D. (>..1 · · · >.N) e i=1

(1)

• lmas

V(>.) =

_Egt>.t l=l

The Dyson-Schwinger equation is a set of consistency conditions on the space of correlators which is derived under general variations. It can be regarded as an equation for the space of string theories of a particular class ( in this case, the ones described by the one matrix model). An efficient way which leads to the Virasoro constraints is the following: insert-E >..?+1 dd orE _l_ _j_ and express the results i=l ' - >.; d>.; >..; i=l in terms of the couplings and the derivatives of the couplings in two different ways, using partial integrations. We obtain, after some calculation on the Vandermonde determinant,

in z

0 ,

n;::: -1 , (2)

270 These a.re the Virasoro constraints at finite N. In this derivation, it is evident that they arise from the reparametrization of the eigenvalue coordinates. As a natural extension of the above derivation and also as a warm-up to derive constraints of the two matrix model, consider the higher order differential operators [16] d

~ ~? ( d~i

)m

or

~(

1

-

( d

~i d~i

)m

(3)

Let us introduce the following notations 00

j

- -Etglct-1 l=l

d dj

,-t-t.!_ - -f: t=o 09t

(4)

In ref. [16], we succeeded in writing the constraints in a closed form:

(5)

where (6)

The existence of such constraints is peculiar in view of the fact that the Virasoro constraints in the double scaling limit fully characterize a space of correlators of topological gravity [7, 8]. We have checked [16] in the lowest few cases that these higher constraints are in fact reducible to the Virasoro constraints: Virasoro (7)

In the approach based on the Dyson-Schwinger equation, we study a relationship of all possible singlet operators and their correlation functions. An alternative ap-

271 proach is the one based on orthogonal polynomials, where one studies all the matrix ( both diagonal and off diagonal) elements of the eigenvalue operator ~ as well as its . te COnjuga

a

d)."

Let us enumerate basic properties of the orthogonal plynomials P,. (~) C..- 1 ~n-1

+... ;

hn+I

= = =

ZN

=

~P.. (~)

8n,m

Pn+I +e.., P,. (~) + R..Pn-1 (~)

= ~" +

,

j d~ e-V(>.;g,) P,..;r;(~) Pm.;r;(~) =< n I m > R,.+Ihn N-1

, (8)

N!Il h; i

The basic equations in this approach are the matrix elements of the Heisenberg algebra and a response of the system to a change of parameters, namely, a parametric derivative of the normalization constants hj: 8,J

=

< i I [ ~, ~] 1i >

(9)

d~,.lnhj = -~(a~,.ut)

(10)

where t, = t,. (g;) are a set of parameters which are functions of the original (bare) couplings g;. The first equation (9) is regarded as a discrete version of the string equation. The second one (10) may be called a discrete flow equation. The basic equations derived in the continuum theory are already visible when N is kept finite. In order to demonstrate the equivalence of the two approaches, let us rederive the Virasoro constraints from the equations (9),(10) above. Begin with (11) N-l

Take a trace of this relation and multiply by ZN. Namely,

L

j=O

< j I·· ·I j > ZN.

272 Using eq. (10) (t( = g(), we find

For the second term of the right hand side, use a formula which holds for any one-body operator N-1

~ < j I0

(A,\, d,\d )

N ( d ) I j >=>ave

J=O

t=1

1

(13)

'

where >ave denotes an averaging with respect to the partition function (eq. (1)). ( The derivative is in between the two determinants and does not act on the potential.) Eq. (13) readily follows from

A(,\t. · · · ,\N)

= det (Pi-1(,\j))

(14)

The calculation has reduced to the one mentioned in eq. (2). We obtain the Virasoro constraints at finite N again. The properties of the orthogonal polynomials have a curious connection [9] to the classical Toda lattice equation. Let

4>,. = lnh,.

(15)

From eqs. (9),(10), we find a hierarchy of classical differential equations whose first member is the classical Toda lattice equation: (16)

We leave the details to the references [9]. It is our hope that explicit results for the physical quantities come out from this correspondence at finite N. Note that we are still dealing with quantum theory. The feature that quantum correlators obey classical integrable differential equations is shared by other exactly solvable models of quantum field theory [17]. It is instructive that the Kazakov's original loop e> ,

(19)

which becomes, after using the factorization property of singlet operators in the planar limit,

.

Kw(l) =

rt w(l')w(f -l')

lo

.

(20)

This is Kazakov's loop equation [10]. For more complete discussion as well as its extension to the supersymmetric loop equation and the determination of its critical points, see the recent preprint [11]. Let us finally discuss how to take a double scaling limit of the Virasoro constraints at finite N [6] to obtain the Virasoro constraints indexed by half integers ( a single twisted boson) [8]. First, the potential must be fine-tuned in order to reach the kth multicritical point of Kazakov [10] describing 2d gravity coupled to (2, 2k - 1) nonunitary minimal conformal matter. Let a be the lattice spacing in the level space of the orthogonal polynomials. As the leading contribution to the matrix elements of ~ is 2, we rescale as

(21)

274

with N

--+

oo and a --+ 0, keeping

(22)

9at = 1/ ( a2+1/A: N) =finite

In order to drive the system slightly away from the critical point, we add to the original potential a source which has a nontrivial continuum limit:

(23) In the right hand side, we reexpanded the source in terms of the polynomial bases around the origin. This is added to the original couplings 9t· We then undo the procedure to find a dressed ( renormalized) source expressed in terms of the original bases {{(2 - ..\;)t+1 / 2 }}. Schematically, 9t

it + 9t

-

i~•c)

-

= it + •·• •

(24)

We find

i z~> [{{gt}} +{{it}}]= 0 ~(l

1/2) ·(•c)_8_

'

82 + !~ 2 L..., ·(•c) ·(•c) t=1UJt-1UJn-t

J!tw) n

-

L!,tw)

= f: (l + 1/2) i~•c) ~•c) + 1~

L...,

t=O

+

Jt

j:l

t=O

j{_tw) -1

·(•c)

j:l

uJt+n

t=1

j:l

j:l

(25) '

'

UJt

= ~ (l + 1/ 2) L...,

n ;:= 1 '

·(•c) _8_ 8 ·(•c)

lt

lt-1

+ !8Jo·(•c)2

'

This is the Virasoro constraints of a twisted boson derived in [8] in the double scaling limit. It is worthwhile to emphasize that the limiting process can be postponed until the very end [6).

B.

Two Matrix Model

Let us now turn to the two matrix model. One starts with Mehta's formula (18]: z~> ({{g}1l}}, {{g}2)}}] =

1

dM1dM2e-trV

(37)

rather than the partition function. Or consider a once iD.tegrated quantity

P (p)

= - 11'~1m tr ln (P - h+ iO)

(38)

The coordinate z of the one particle hamiltonian h originates from the eigenvalues of the matrix variable. Virasoro constraints are obtained from the reparametrization of eigenvalues but the procedure is not identical to that of the one matrix model [25]. Our basicformulas (eqs. (5), (6), (34), (35)) can be regarded as a kind ofbosonization of nonrelativistic fermions. A ramification to W geometry has been pursued recently [26]. What we have been witnessing is the transmutation of the eigenvalue coordinates into the target space degrees of freedom. This phenomenon is also present in one dimensional matrix model [27], but conceptual understanding seems to be still lacking. Constraint equations in general provide nontrivial information on target space physics

278

such as physics of black holes (28]. Clearly much more work must be dedicated on these issues together with the issue of the definition of nonperturbative 2d gravity. I thank Joe Lykken for helpful discussion.

References (1] E. Brezin and V. Kazakov, Phys. Lett. B236 (1990) 144; M. Douglas and S. Shenker, Nucl. Phys. B335 (1990)635 ; D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127; Nucl. Phys. 8340{1990)333. (2] M. Douglas, Phys. Lett. B238 (1990) 176. (3] T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B238 {1990)279. (4] F. David, preprint Mod. Phys. Lett. A5(1990) 1019; preprint SphT /90-127 ; See also S. Chaudhuri and J. Lykken, preprint Fermilab-Pub-90/267-T. [5] E. Marinari and G. Parisi Phys. Lett. 247B(1990) 537. (6] H. Itoyama andY. Matsuo, Phys. Lett. B255 (1991)202. (7] E. Verlinde and H. Verlinde, Nucl.Phys. B348 (1991) 457. [8] M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys.A6 {1991)1385;·R. Dijkgraaf, H. Verlinde and E. Verlinde, Nucl. Phys. B348 (1991) 435. (9] L. Alvarez-Gaume, C. Gomez and J. Lacki, Phys. Lett. 253B (1991)565; E. Martinec, Comm. Math. Pbys. 138 (1991)437; A. Gerasimov, A. Marshakov, A. Mironov and A. Orlov, Nucl. Phys. B357 {1991)565, M. J. Bowick, A. Morozov and D. Shevitz, Nucl. Phys. B354(1991) 496; SeeM. Bowick, this proceedings; also L. Alvarez-gaume, Helvetica Physica Acta 64 (1991 )360 lecture 8 for an excellent account. (10] V.A. Kazakov, Mod. Phys. Lett. A4 (1989) 2125. [11] L. Alvarez-Gaume, H. Itoyama and J. Maiies, preprint ITP-SB-91-39.

279

[12) T. Banks RU-90-52, preprint, to appear in the proceedings of the Cargese workshop on random surfaces; G. Moore, Commun. Math. Phys. 133 (1990)261; A.S. Fokas, A.R. Its and A.V. Kitaev Clarkson Universitypreprint, INS164, November 1990; W. Ogura, preprint, VPI-IHEP-90-6. [13) D. J. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 717; E. Brezin, M. Douglas, V. Kazakov and S. Shenker, Phys. Lett. 8237(1990)43; C. Cmkoviz, P. Ginsparg and G. Moore, Phys. Lett. 8237 (1990) 196; [14] T. Tada and M. Yamaguchi, Phys. Lett. 8250 (1990) 38. [15] M. Douglas, preprint, to appear in the proceedings of the 1990 Cargese workshop on two dimensional gravity and string theory. [16] H. Itoyama andY. Matsuo, Phys. Lett. 8262 (1991)233. [17] A.R. Its, A.G. Izergin, V.E. Korepin and N.A. Slavnov Int J. Mod. Phys. 84 (1990)1003. [18] C. Itzykson and J. B. Zuber, J. Math. Phys. 21 (1980) 411; M.L.Mehta, Commun. Math. Phys. 79 (1981) 327. [19] H. Kunitomoand S. Odake, Phys. Lett. 2478 (1990)57; M. Kreuzer, R. Schimmrigk, Phys. Lett. 2488 (1990)51; M. Kreuzer, Phys. Lett. 2548 (1990)81. (20] T. Tada, Phys. Lett. 2598 (1991)442. [21] J. deBoer, preprint THU91/08. [22) E. Gava and K.S. Narain, Trieste preprint IC/91/62. [23] M. Fukuma, H. Kawai and R. Nakayama, preprint UT-572, KEK-TH-272, November 1990. J. Goeree, Nucl. Phys. 8358 (1991) 737 [24) I.K. Kostov, Nucl. Phys. 8326 (1989) 583; Phys. Lett. 8266 (1991) 317. [25] U. Danielson and D. Gross, preprint PUPT-1258. See D. Gross, this proceedings. [26] Y. Matsuo, preprint LPTENS-91/19 [27] S. Das and A. Jevicki, Mod. Phys. Lett. AS (1990)509

[28] E. Witten, preprint IASSNS-HEP-91/12, March 1991.

280 TOPOLOGICAL GRAVITY WITH MINIMAL MODELS

Keke Li* California Imtitute of Technology, Pa1adena, CA 91 U5, USA

ABSTRACT It is briefly explained here that multi-matrix models of two-dimensional quantum gravity can be interpreted as toplogical gravity coupled with topological minimal models.

Soon after the exact solution of the hermitian matrix model of two-dimensional quantum gravity 1 , a remarkable connection with topological gravity was discovered2 • It was noted2 that the correlatio n functions of the one-matrix model at its lowest (Gaussian) critical point can be interpreted as given by the intersection theory on moduli space of Riemann surfaces, which defines two-dimensional topological gravity3 much like the intersection theory on the instanton moduli space defines topological Yang-Mills theory in four dimensions•. This connection, established by deriving and solving certain recursive relations that follow from the topological definition of correlation functions, can also be demonstrated from more explicit field theoretic formulations of topological gravity5 •6 • While the one-matrix model corresponds to topological gravity, the n-matrix model7 was suggested to correspond to topological gravity coupled with some topological field theory with n "primaries" 8 • It turns out that the topological field theories in question are the twisted N

=2

superconformal minimal models 1 •9 , as has been shown in [10) and will be briefly summarized in this lecture. Topological matter theories are given generically by topological 0'-models, which may be obtained in general by twisting N

= 2 su-

persymmetric O'-models 11 • Physical operators in topological 0'-model are invariant

*

Resean:h supported in part by the US Department of Energy under contract DE-AC0381ER40050 and by the Weingart Foundation. E-mail: kekeli0theory3.caltech.edu

281 under a BRST transformation. The BRST charge is given by one of the two (chiral) supercharges, thus the physical operators correspond to the chiral superfields. They correspond to the so-called chiral primary fields 12 in case the original N

=2

model is also conformally invariant 1 • The N = 2 superconformal minimal models, with central charge c = 3k/(k + 2) and k + 1 chiral primary fields, are the simplest N = 2 supersymmetric matter theory in two dimensions. Their twisted version will

give the simplest topological matter theories, which may natually be called topological minimal models . It is then not surprising that the kth model coupled with topological gravity will correspond to the ( k + 1)-matrix models. To describe topological conformal field theory in general, let the standard N = 2 superconformal alg ebra be generated by the Laurent coefficients of the holo-

morphic stress tensor T0 (z), supercurrents G(z) and G+(z), and the U(1) current

J(z), as well as their anti-holomorphic counterparts. The corresponding topological conformal field theory has a new stress tensor T(z) = T0 (z)+ ~o.. J(z). Under T(z), the supercurrents G and o+ have new conformal dimensions 2 and 1 respectively. The spin 1 current G+ is identified as a BRST current, its zero mode Q

= J G+ be-

ing the nilpotent BRST charge. The topological nature of the twisted model follows from the fact the stress tensor is a BRST commutator: T(z) = -{Q,G(z)}, which guarantees that the correlation functions of BRST invariant operators are independent of the deformations of the metric. General arguments show that BRST invariant but non-trivial operators can be represented by the chiral primary fields 12 , which are defined to be those annihilated by Gn =

J z-n- 1 G(z) and Gt = f z-no+(z)

for n 2: 0. In the N=2 model, the dimension of a chiral primary field is always half of its U(1) charge. Thus in the topological model, with T(z) = T0 (z)

+ ~o.. J(z),

chiral primary fields always have dimension zero, which is indeed required for their correlators to be topological. Starting from a dimension zero BRST invariant local operator repres ented by the chiral primary field t/>, one may introduce the standard BRST multiplet that consists of operators (t/>,G-tt/>JLtt/>,G-t(Ltt/>), corresponding to (0,0), (1,0), (0, 1) and (1, 1) forms on Riemann surfaces. By using

o,. =

L-1

= -{ Q, G-t }, one sees that although G- 1 t/>, (;_ 1 t/> and G_ 1 (L 1 t/> are not

BRST invariant, their integrals over the appropriate cycles (for G_ 1 ,p and (L 1 ,P) or the surface (for G-tG-tt/>) do give BRST invariant quantities. A chiral primary

282 field carries the U(1) charge under J(z) (and J(z)), whose conservation will give an important selection rule of the theory. Since the BRST operator Q has charge 1, the U(1) current J(z) is identified with the ghost number current as in age neral topological field theory. It is easy to see that J(z) has a background charge c/3 on the sphere (or (c/3)(1- g) on a genus g surface), where cis the central charge of the original N

= 2 superconformal algebra.

Consider now the topological minimal models obtained from the minimal N

=2

models with central charge c = 3k/(k + 2), where k is a non-negative integer. These N = 2 minimal models are given by an ADE classification 13 , which arises because of the intimate connection between N = 2 superconformal algebra and SU(2) Kac-Moody algebra14 • For a generic non-negative integer k, there is a model labeled by Ak+t· Fork even, there is an additional model labeled by Dt+2· For k = 10, 16 and 28, there are additional models labeled byE&, E1 and E 8 respectively. The spectrum of chiral primary fields in these models is particularly simple to describe in terms of the exponents of the simply-laced Lie algebras labeled by the ADE series. The set I of the exponents is given by I

= {1, 2, ... , n} for

An, I

=

{1,3, ... ,2n- 3,n -1} for Dn, I= {1,4,5, 7,8, 11} forE&, I= {1,5, 7,9, 11, 13, 17} for E1, and I= {1,7,11,13,17,19,23,29} for Es. Let h be the dual Coxeter number: h = n+1,2n-2, 12,18 and 30 for An,Dn,E&,~ andEs respectively. The chiral primary fields in the AD E series of N = 2 minimal models have dimensions (hL,hR) = (a2"h1 , a2"h 1 ) and U(1) charges (qL,qtt) = (aj;" 1 , ahl) with a E I. Thus in the corresponding topological model, the set of physical operators have dimensions (0, 0) and ghost charges ( a;t, a;t) with a E I. Let and 0 as their descendants. Since both topological gravity and topological minimal models are conformally invariant field theories, it is possible to solve the coupled theory exactly and then prove their equivalence with the multi-matrix models. However, essentially because of the topological symmetry and the minimal nature of the matter theory, some very simple observations and arguments are already sufficient to demonstrate this equivalence. First of all, the operator content (2) indeed agree with the multimatrix models (at the lowest critical point). Furthermore, one may derive a ghost number selection rule from (2). The ghost number current from the gravity sector, being the sum of the holomorphic and anti-holomorphic ones, has a background charge -3(2g- 2). The ghost number current J + J* from the matter sector has

284 a background charge (2g- 2)k/(k

+ 2) = (2g- 2)(h- 2)/h.

The conservation of

the total ghost charge leads to the selection rule for a non-vanishing correlation function (ll; Un;,a.}g:

L 2n;- 2+ 2(c:\-1) = (2g -2)2(h: 1).

(3)

i

This selection rule indeed agrees with the result from multi-matrix models. To i llustrate the selection rules (3), consider the example of topological gravit y coupled with the A2 minimal model. There are two chiral fields, the identity and Q(o,o). They become two (gravitational) primary fields, P and Q = Qo · P, when coupled to gravity. It follows from the selection rule (3) that their only non-vanishing correlation functions on the sphere are (PPQ) and (QQQQ), agreeing with the result of the two-matrix model8 • The reason for (PPQ) '/: 0 is simply that three punctures are created so that the sphere becomes rigid ( (PP P) = 1 in the gravity sector), while (Q(o,o)) '/: 0 in the matter sector. The reason for (QQQQ) '/: 0 is that the position of the fourth puncture corresponds to a (complex) moduli which must be integrated over the sphere, and (Q(o,o)Q(o,o)Q(o,o) J Q~1 • 1 >) '/: 0 in the matter sector. A complete demonstration of the equivalence with the multi-matrix models becomes possible after the discovery that the complete solution of the theory is determined and characterized Virasoro and W constraints15 • The structure of these constraints shows that the interactions among physical operators in these models (at their lowest critical points) consist of contact terms only. Such a contact term can indeed be computed by using the free-field formulation o f topological gravity 6•

The consistency of such contact terms can then be shown to lead to the Virasoro

constraint, thus demonstrating the equivalence of topological gravity with th e onematrix model6 • As was shown in Ref. 10, similar arguments can also be used to show the equivalence between topological gravity coupled with topological minimal models and the multi- matrix models. To describe this argument in more detail, note that the position of an operator Um,a

in a correlation function corresponds to a moduli (of punctured Riemann

surface) that must be integrated over the surface. There will be special contributions to this integral when

Um,a

approaches possible singularities on the surface, which

285

may be the positions of other operators or the nodes corresponding to the pinching of the surface. These contact interactions may in general involve more than two operators or nodes. Consider the two-point contact term fv, um,.,.lun,,s), where D. is an infinitesimal neighborhood of the position of the operator Un,.S· The situation of one puncture approaching another is described in the stable compactification of moduli space of Riemann surfaces by the formation of a sphere which contains the two punctures and is connected to the original surface by a node. Its physical implication is that fv, Um,tlun,) is represented by a local operator inserted on the original surface. By the ghost charge conservation, this local operator can only be Um+n-l,a· This is due to the non-degeneracy of the ghost spectrum, a property

which is not true for non-minimal models in general. Consider the descendant um,l of the puncture operator P

=

uo,t·

Since it contains no contribution from the matter

sector, its two-point contact terms is particularly simple to evaluate6 •9 , giving

(4) The contact coefficients of Um,l 's satisfy a commutator algebra6 which is isomorphic to Virasoro algebra and may be represented as

A local consistency condition for the contact coefficients is that the integrals of two operators Un,l and Um,l in a small neighborhood of a third operator u,.,.S must be independent of the order the two integrals:

(6) In principle there could be special contributions to (6) when the three operators coincide. However, collecting various two-point contact contributions to both sides of (6) using (5), one finds that (6) is already satisfied as the consequence of the commutator algebra {5) satisfied by the contact coefficients given in (4). This is a non-trivial property of the contact coefficients of puncture operator and its descendants. It means that the physical consistency condition {6) is satisfied by allowing

286

only two-point contact terms. The three-point contact terms among these operators do not contribute. This is an essential feature of topological gravity without matter6 , and it remains to be true for the puncture operator and its descendants in topological gravity coupled with minimal matter. One may also formulate a similar consistency condition on correlation functions, which then immediately implies a set of recursion relations. Consider the contact contributions to the integral of the position of

O'm,l

in t he correlation function

un,,a;). In addition to the contact terms of um,l with another operator, there are two kinds of factorization terms when O'm,l approach possible nodes on the surface. The first kinds of factorization terms, corresponding to O'm,l approaching a node associated with the pinching of a handle, are given by (um,t ll:=l

•

{um,l

II

m

O'n;,a;}A,g

=L L

•

B:,a{O'n-2,aO'm-n,la-a

n=2 aEI

i=l

II

O'n;,aJg-1•

(7)

i=l

The second kinds of factorization terms correspond to O'm,l approaching nodes associated with the cycles dividing the surface into two parts. They are given by •

{um,l

II

m

O'n;,a;)A•,g

i=l

=E E

B~':'a

n=2aEJ

·L S•XuY

{O'n-2,a

II iEX

O'n;,a;}flt {Um-n,A-a

II

(8) O'ni,ai}fl2>

jEY

·-·· +•2

where Es=XuY denotes the sum over all divisions of the set of labels S = {i1 ,i2 , ... ,i.} into two subsets X andY. Note that if a E I, then h- a E I. Again, the possible terms in (7) and {8) are determined uniquely by the ghost charge conservation. The factorization coefficients B~a and B~':'a may be obtained by a field theoretic analysis, but the consistency condition as outlined in following fixes their values completely. A natual consistency requirement is that the value of the correlation function (un,lO'm,l n~=l O'n;,a;}

must be independent of the orders of the integrals of the

positions of O'n,l and O'm,l· Since the surface integrals are independent of the order of the integrals, they do not contribute to this consistency condition. The contact terms (4) and the contact algebra {5) then imply that the only contributions to the

287

integral of position of O'n+m-1 in

(O'n+m-1 n:=1 O'n;,a;)

come from the contact terms

of O'n+m-1 with other operators and the factorization terms of O'n+m-1 with nodes. The fact that the integral of the position over the rest of the surface never contribute is a consequence of the non-commutativity of the contact algebra (5). The same consistency requirement also ~ves rise to a set of linear relations for the two-point factorization coefficients with the solution B:,a = ba and B~':'a = bafa. In addition, the consistency conditions imply that contact and factorization terms involving more than two punctures or nodes must vanish, which is a global manifestation of the absence of three-point local contact interactions involving O'm,1· The constant ba = b~a-a can be absorbed into the normalization of the operator O'n,a· It is convenient to choose ba = h/2(h + 1). Collecting the two-point contact terms and factorization terms, one obtains the following set of recursion relations

h +1

-h-(O'm,1

II.

O'n;,a;)g

~ +h l.ti II O'nj,Oj ), = L..J(ni )(O'm+n;-1,a;

i=1

i

j;li

1 ..

•

L {(O'n-2,aO'm-n,la-a II O'n;,a.}g-1 n=2aEI i=1 L {O'n-2,a II O'n;,a;)ft (O'm-n,la-a II O'ni,ai)g, }.

+ 2L

+

S•XUY

•••t+12

iEX

kEY

(9) The ratio a of the factorization coefficients is determined to be 1 by the consistency of (9) with {0'1,1) = l/24, where 0'1,1 is the dilaton operator1•6 and I is the number of chiral primacy fields which equals to the rank of the simple Lie algebra labeling the model. The set of recursion relations (9) may be rewritten as the set of constraints on the partition function (viewed as the generating function), giving rise to precisely the Virasoro constraints that have been derived from the multi-matrix models. The Virasoro recursion relations (9) in general reduce the number of the u,.., 1 operators in correlation functions. To obtain a complete solution of the theory, one needs also the recursion relations that can reduce in the correlation functions the number of

O'm,a

operators with a

-:f; 1. As

O'm,a

with a -:f; 1 involves con-

tributions from the matter sector, the evaluation of its contact terms with other operators appears to be delicate. However, the precise contact coefficients of O'm,a

288

are determined by consistency from those of O'n,l given in (5): (10) where c~.IJ is the OPE coefficient, and the contact algebra is

I O'm,aiO'n,t)- I O'n,tiO'm,a) = h ~ 1 (an- m)lum+n-t,a)• ln. ln.

(11)

The contact algebra (11) implies that the contact coefficients of O'm,a generate the extension of the Virasoro algebra (6) by the Laurent coefficients of a spin-( a+ 1) current. Consider the local consistency condition: (12) Using (10), one finds that the contributions from two-point contact terms alone do not satisfy (12) for a =F 1. Contrary to the case of a consistency condition (12) for a

= 1 discussed earlier, the

:f: 1 requires non-trivial three-point contact terms.

It also uniquely determines the three-point contact coefficients from the two-point

contact coefficients. In general one may formulate a local consistency condition, similar to (12), for s operators all approaching one another. This consistency condition will determine the s-point contact coefficients in terms of the contact coefficients of less then s operators. As the consequence of the non-commutativity of the twopoint contact algebra, all multi-point contact coefficients may be determined from the two-point contact terms by this inductive procedure. The precise coefficients of these multi-point contact and factorization interactions can also be determined from the two-point contact terms (10) by consistency of the recursion relations for

O'm,a·

Consider the integral of the position of

O'm,a

in a correlation function

(um,a ll:=t u,.;,a;). The recursion relation express this integral as the sum of the surface term corresponding to the integral over the surface excluding the singulari-

ties, and the contact and factorization terms involving possibly many operators and nodes. Consider the correlation function (um,aun,t ll:=t u,.;,a;), and demand it to be independent of the order of the integrals of the positions of the two operators

289

O'm,a-

and O'n,l· Because of the non-commutativity of the contact algebra (4.2), this

condition requires that the surface term must vanish. In addition, it also implies some relations among the multi-point contact and factorization coefficients, from which one may solve for these coefficients in terms of several unknown constants, much in the same way as was described earlier for the Virasoro recursion relations. In terms of these coefficients one may write down the recursion relation for

um,a-

explicitly, and the compatibility of these relations with different a and in particular with the Virasoro recursion relations (9) (for a= 1) will determines the remaining unknown constants completely. One finds, as a consequence of the contact algebra (5), that the recursion relation for

O'm,a-

contains up to (a+ 1)-point contact and

factorization terms. The result of such an analysis i s that the additional recursion relations, when formulated as consistent cons traints on the partition function, must correspond to precisely the W constra ints. The fact that all multi-point contact and factorization coefficients are completely determined (from the two-point contact terms) is the consequence of the uniqueness of this W algebra associated with the simply-laced Lie algebra labe lling the minimal model. Since the Virasoro and the W constraints completely determine all correlation functions, their agreement with the multi-matrix mo del thus demonstrate the equivalence to be shown. In summary, part of the arguments given in Ref. 10 showing the equivalence between topological gravity coupled with topological minimal models and the multimatrix models as described briefly here has been based on the demand of the consistency of the theory. It would be satisfying to develop a more rigorous and systematic procedure to exactly solve topological gravity coupled with miminal models. On the sphere there is no moduli, so one may consider only the matter sector, and compare solution of the pure matter system with the genus zero predictions of the W constraints. This study has been done with a complete agreement 16 • The nature of contact interactions, which is fundamental to the analysis described above, should also be examined more carefully, and this should be possible along the line of Ref. 17. Topological ( cohomological) field theory is always associated with a moduli problem. Such a topological interpretation of topological minimal models has also been given by using their formulation in terms of gauged WZW models 18 • There remain many open questions in the topological approach to two-dimensional

290

quantum gravity. For example, it would be interesting to better understand the connection of topological gravity with ordinary gravity formulated as a quantum Liouville theory. As one perturbs topological gravity and breaks the topological symmetry, many more physical operators will appear. A good understanding of such phenomenon will perhaps provide insight on the physical significance of topological field theory in general. There have been many current studies of the d = 1 non-critical string theory. It would be very interesting to discover a topological field theoretic description.

References 1. E. Brezin and V.A. Kazakov, Phys. Lett. 2368 (1990) 144; M. Douglas and

S. Shenker, Nucl. Phys. 8335 (1990) 635; D. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127. 2. E. Witten, Nucl. Phys. 8340 (1990) 281. 3. J.M.F. Labastida, M. Pernici and E. Witten, Nucl. Phys. B310 (1988) 611; D. Montano and J. Sonnenschein, Nucl. Phys. 8313 (1989) 258. 4. E. Witten, Comm. Math. Phys. 117 (1988) 353. 5. J. Distler, Nucl. Phys. B342 (1990) 523. 6. E. Verlinde and H. Verlinde, Nucl. Phys. 8348 (1991) 457. 7. M. Douglas, Phys. Lett. 8238 (1990) 176. 8. R. Dijkgraaf and E. Witten, Nucl. Phys. 8342 (1990) 486. 9. T. Eguchi and S.-K. Yang, Mod. Phys. Lett. A21 (1990) 1693. 10. K. Li, Nucl. Phys. 8354 (1991) 711; 725. 11. E. Witten, Comm. Math. Phys. 118 (1988) 411. 12. W. Lerche, C. Vafa and N.P. Warner, Nucl. Phys. B324 (1989) 427; E. Martinec, Chicago preprint Print-89-0373. 13. A. Cappelli, C. ltzykson and J.-B. Zuber, Nucl. Phys. 8280(FS18] (1987) 445. 14. A.B. Zamolodchikov and V.A. Fateev, JETP 63(5) (1986) 913; Z. Qiu, Phys. Lett. 8188 (1987) 207 and 8198 (1987) 497.

291

15. R. Dijkgraa!, H. Verlinde and E. Verlinde, Nucl. Phys. B348 (19 91) 435; M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A6 (19 91) 1385. 16. R. Dijkgraa!, H. Verlinde and E. Verlinde, Nucl. Phys. B352 (19 91) 59. 17. J. Distler and P. Nelson, Comm. Math. Phys. 138 (1988) 273. 18. E. Witten, preprint IASSNS-HEP-91-26, June 1991.

292

2D SUPERGRAVITY AND SUPERMATRIX MODELS

J. L. MANES • CERN, Theory Division CH-1211 Geneve 23, Switzerland ABSTRACT We consider a very general class of models based on supermatrices and show that they are equivalent to ordinary matrix models. This suggests that using supermatrices is not the right way to find a discretization of 2-D supergravity.

For d :5 1 the sum over triangulations performs very efficiently the integration over moduli space necessary to compute string amplitudes in the continuum limit for any order in perturbation theory [1],[2],[3]. This is reasonable because the sum over triangulations in terms of large N matrix models gives an explicit procedure to sum over metrics modulo diffeomorphisms on surfaces of arbitrary topology. One might expect that similar arguments for supermoduli spaces should be possible. So far there is no construction of the lattice analogue of a superconformal model coupled to 2d-supergravity. In the same way that ordinary matrices provide triangulations of two-dimensional geometries, one might imagine that the sum over supermatrices should generate a discretization of supergeometries. We have analyzed [4] a very general class of models based on (NIM) supermatrices [5] with the following block decomposition 4)

=( ~ ~ )

(1)

where A and !l are arbitrary hermitian bosonic matrices, whereas Ill is fermionic. The simplest case we consider is a one-supermatrix model

Z(NIM)

=j

D4>exp[-.8V(4>)]

(2)

where V(w) is a general SU(NIM) invariant potential. We use a representation in terms of moments [6] to compute (2) when 4> is a (Nil) supermatrix and the potential V is cubic, obtaining (3) Z3(Nil) "'Z3(N- IIO) This is a very simple result which suggests an exact cancellation between bosonic and fermionic degrees of freedom. Since the computations leading to (3) are not •Permanent address: Departamento de Fisica, Facultad de Ciencias, Universidad del Pais Vasco, 48080 Bilbao, Spain

293 easily generalized, we have to use some more indirect methods. First, we derive the Virasoro constraints satisfied by (2). These are of the form

L,.Z(NIM)

=0

, n ~ -1

(4)

where L,. is a differential operator which depends on (NIM) only through the difference N - M. This leads us to conclude

Z(NIM) ,... Z(N- MIO)

(5)

thus confirming the idea of a cancellation between fermionic and bosonic degrees of freedom. Finally, we consider the structure of the Feynman diagrams in detail for theories based on any number of supermatrices subject to totally arbitrary invariant couplings among themselves, i.e., more general than the so called chain models (see for instance ref. (7]. We obtain (5) again. It is rather surprising that the models based on supermatrices are completely equivalent to ordinary matrix models. The correlators of invariant operators Str M" are the same as those of trM" in the (M- NIO) pure matrix theory. These operators do not exhaust the possible operators one could write down in supermatrix theories, but it is nevertheless disappointing to find such a drastic trivialization of the theory. This is a strong indication that using supermatrices is not the right way to discretize supergravity. References [1] E. Brezin and V. A. Kazakov, Phys. Lett. B236 (1990) 144. [2] M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 635. (3] D. Gross and A. A. Migdal, Nucl. Phys. B340 (1990) 333. (4] L. Alvarez-Gaume and J. L. Maiies, Supermatriz Models, CERN preprint CERNTH.6067/91, April1991. [5] B. DeWitt, Supermanifolds (Cambridge University Press, 1985). [6] T.S. Chihara, Breach,1978).

Introduction

to

Orthogonal Polunomials,

(7] M. Douglas, Phys. Lett. B238 (1990) 176.

(Gordon

and

294

SOLVING (l,q) KDV GRAVITY

GIL RIVLis*t

Lauritsen Laboratory of High Energy Physics California Institute of Technology Pasadena, CA 91125, USA ABSTRACT

We explicitly compute correlation functions of the (1, q) series of KdV gravity. We also find, from purely algebraic considerations, a 'ghost' number collllerVation rule. This rule applies to (1, q) models in the limit where the C081I10logical constant vanishes. A slightly weaker selection rule still applies for a general cosmological constant. We find the interesting result that some correlation functions vanish even when the selection rule above is satisfied. This may be traced back to the choice of integration constants or to the extra Virasoro (or W-algebra) constraints. We stress that the computation is non-perturbative and the results apply to descendant fields as well as to primary fields.

1. Introduction Due to lack of space in the proceedings I will only· !!tate the results. For details and a list of references see the original papers£1 •21. The (p,q) models of KdV gravity are defined with the aid of the qth order differential operator (1)

and its qth root R. R is the (formal) pseudo-differential operator that satisfies R!' = Q. Thr:~ ..1..1 unknown functions u;(x) are determined (in part) by the 'string equation'

[P,Q]

= 1,

(2)

with the choice of P = (RJ')+. 0+ is the differential part of 0. The fields in the theory are identified with powers of R

(3) *This work supported in part by the U.S. Department of Energy under Contract No. DE-AC0381ER40050. tAddress after 9/1/91 : Theoretical Physics Group, Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA. Email After 9/1/91: gilGphyeice.berkeley.edu

295

One computes correlation functions as follows. First one identifies the two-point correlation function of two 'puncture' operators as the lowest dimension function (4)

where ResO is the coefficient of n-• in 0. Then other correlation functions (with more fields) can be computed with the aid of the generalized KdV hierarchy~,

aRi

=

CJt.

[ . .1 Rft,H .

(5)

J

That is, one can show that

= ResR ('P1'P1'P;) = Res [R+,Ri] (,:\'P;)

(6) (7)

and so on. 2. (l,q) Models In (1, q) models P

=D and one can show that Q = D 9 +x.

(8)

Due to that simple structure, one can explicitly compute Ri to many orders

If= D; + ixn•-q + i(i- q) n•-q-l + i(i- q) x 2n•-2q +... . q

Zq2

2q

(9)

Also one can derive formulae for correlation functions. For exampleS

{'P~cq+q-I'P~2 } and

1

lc+l

= T+1 IJ(nq-1) q

5 58060&f (q2 - 25)(q2 - 1)(2q + 1)(2q + 5)(3q + 5) (4q + 5)(6q + 5)(8q2

-

(10)

n=l

(11)

13q- 13).

Further, the structure of the operators Ri and also different combinations of them (such as [ (Ri+, Ri] + , etc.) is of the form

R!']

00

i

0 -_ "L..- "L..- cr1;xHnn-iq-; , i=O j=O

~The flow parameters t; deec:ribe the evolution of R (and R') in the space of operators. §These fonnulae are given in the topological limit z - 0.

(12)

296

where n is the order of 0 and

are constants.

a;j

3. Selection Rules One can show a 'ghost' number can be assigned to each field (operator) gh1';=q+1-i

(13)

and that a correlation function (1'11';l · · ·1';.. } will vanish unless n

L: gh1';j :s 2(q+ 1).

(14)

j=O

This selection rule is true even when x =f. 0. In the limit x -+ 0 one can further show that the correlation function will vanish unless n

L ii = Omod (q + 1).

(15)

j=O

Combining the two selection rules one sees that the correlation function will vanish unless n

L: gh 1';J = 2(1- g)(q + 1),

(16)

j=O

where g is a non-negative integer or half-integer. It is tempting to identify g with the contribution to the correlation function from different genera. Half-integer g could correspond to bordered Riemann surfaces**. One should notice that the selection rule (16) does not imply that if it is obeyed then the correlation function does not vanish. It does imply that if it is not obeyed, the correlation function vanishes. Indeed there are examples of correlation functions that vanish even when (16) is satisfied. For example, restricting oneself to fields with ij :::; q (primary fields), one can show that (17) (1'11'.· .. ·1'.· } = 0 '1

'"

when 3 :::; n < 6. It is tempting to conjecture that these correlation functions will vanish for any n 2: 3.

4. Conclusions We have explicitly computed correlation functions of fields in the (1,q) model of KdV gravity. Though these models have a very simple string equation, their solutions possess very interesting properties. We have shown how ghost number **However, a direct computation for many correlation functions shows that g is always an integer for the (1, q) models. This means that there are further selection rules we were not able to derive.

297

conservation, a property of a topological field theory, arises algebraically in the (1,q) series. It was also easy to see how the genus expansion follows from this ghost number conservation. Some of these properties are special to the ( 1, q) models. Ghost number is not conserved at the higher critical points, since these points are reached by perturbing with operators carrying ghost number (in the topological sense). It is natural to suppose that solutions of the first critical theory are sufficient to derive, by perturbation, the solutions to the higher critical point theories. We have been somewhat conservative in our discussion of the (1,q) models. It has been claimed, at least for the one matrix model, that the partition function may be a special T function of the KdV hierarchy. This T function would the Virasoro constraints and, thus, be determined uniquely. For the multi-matrix models one has to impose further constraints to fix T uniquely, i.e., W-algebra constraints. For the one matrix model, the constraint L_ 1 T = 0 can be derived by integrating the string equation. The other constraints are determined using the KdV flows. This procedure would then provide an unambiguous way to determine any correlation function with or without punctures, given the T function. From KdV gravity we only have information about the specific heat, u9 _ 2 = D 2 log T. Even the string equation is only a constraint on the specific heat. Integrating this equation in order to derive the first Virasoro constraint requires dropping a constant of integration, which is, in general, a function of the t;, i =F 0. Indeed, since the KdV and string equations only involve the specific heat, the partition function is only determined up to which differ by multiplication by a function independent of t 1 • If one wishes to maintain ghost number conservation, then it is further constrained. We have thus followed a more conservative attitude and basically only discussed correlation functions derived from the specific heat and not the partition function. On the other hand, it might well be that the definition of KdV gravity is not restricted enough. The natural extension is to impose the Virasoro constraints in the one-matrix model or the l-Vn constraints in the n -1 matrix models. In the one-matrix case this indeed amounts to integrating the string equation and fixing the integration constants to be zero.

Acknowledgements A major part of this work was done in collaboration with David Montano.

References 1. D. Montano and G. llivlis, Nucl. Phys., B360 (1991) 524. 2. G. llivlis, Two Topics in f2D Quantum Field Theory, Ph.D. Thesis, Caltech, 1991.

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Quantiza tion Methods

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301

FIRST QUANTIZATION AND SUPERSYMMETRIC FIELD THEORIES W. Siegel1

Institute for Theoretical Physics State University of New York, Stony Brook, NY 11794-3840

ABSTRACT In this talk we sununarize general features of first quantization conunon to particles and strings, and discuss applications to the superparticle and NSR superstring.

0. OUTLINE The talk will cover the following topics:

i. Particles and strings We will first consider the similarities between particles and strings in first and second quantization: the BRST operator, vertex operators, gauge fixing, and the BRST cohomology.

ii. Superparticle Next, we'll discuss some problems and solutions for various methods of quantizing the superparticle: first ilk, second ilk, and covariantized light cone.

iii. NSR superstring without picture changing Finally, as an example of the above problems, we will analyze quantization of interacting Neveu-Schwarz and Ra.mond strings in a way more similar to the Veneziano string, focusing on: particle analogs, conformal field theory, and string field theory. This may eventually help us understand covariant quantization of the Green-Schwarz string. 1

Work supported by National Science Foundation grant PHY 89-08495. Internet address: [email protected], bitnet: wsiegel@sunysbnp.

302

1. PARTICLES AND STRINGS It is important to study properties common to particles and strings since: (1) This helps us to understand strings better; and (2) when string theory gets old, these properties are the ones most likely to survive.

BV 2nd quantization BRST 1st quantization [1]: There is a direct correspondence between everything in a Batalin-Vilkovisky second-quantized theory and the Becchi-Rouet-Stora-Tyutin first-quantization of the same theory: Fields tP+ and antifields 4>- ~(Co)= tP+ + CotP-· The fermionic bracket/inner product: ({AI~), {~IB)) The (interacting) BRST operator:

s = !)

H = {Q,bo-!}

BRST cohomology In terms of the ghost number operator J (where [J, QJ = Q and Jt = -J), the states of the BRST cohomology can be classified as:

303

J

states

-~

(ghosts of) global symmetries physical states antifields to physical states antifields to global symmetries

-t +t +~

For Yang-Mills/open strings, J = -~ is just the p = 0 Yang-Mills ghost. To write physical states as 11/1) = VjO) (for vertex operators V as above), IO) must be this state (as follows from how the states and operators act under Q, J, and Poincare transformations). The normalization of the vacuum state can be defined by:

(OieoQeoiO) "' 1

Yang-Mills The simplest way to obtain the free BRST operator for first-quantized YangMills is to add 2+2 dimensions to the light cone: Then the SO(D-2) spin operators are generalized to 0Sp(D-1,1j2) spin operaors M'i: i =(a, a), a= (c,c). These can be defined by M'i =

l[i) (j) I

(ijj) = 11'i :

(alb) = flab , (cjc) = i

The BRST operator Q is obtained as one of the fermionic components of the 2+2covariantization of the light-cone Lorentz generators J_,. Interactions can then be introduced by adding external Yang-Mills fields to the BRST operator by the gauge covariantization p --+ p + A: J = t[c, bj + iMcc

Qint = c[!(p + A) 2 + !Mab Fab] + iMca(p + A)a- Mccb ::} V = fa(iMc a+ CPa+ ckb Mba)e-ik·:t As for strings, BRST nilpotence implies the field equations:

Q 2 =0 [(p+At,Fab]=O The states in the cohomology are classified as above; in particular, the physical vector states can be expressed in terms of the vertex operator acting on the vacuum, also as for strings: J = -~ : IO) = lc) J = -t: VjO) = fae-ik-:tla) Another analog to string theory is the expression for the 3-particle vertex in terms of vertex operators and the vacuum: (OjVVVIO).

304

2. SUPERPARTICLE Problems and solutions: The history of covariant quantization of the superparticle is basically a series of problems and their solutions: (1) The original Brink-Schwarz superparticle (2] has second-class constraints; the only viable solution so far has been to get rid of them. All methods to quantize second-class constraints have turned out to be useless: (a) Dirac quantization is singular (3]; (b) BV first-quantization fails (4]; (c) harmonic superspace is nonlocal [5). (2) Even after a way to eliminate the second-class constraints was found, by replacing the BS action with a physically equivalent action (see below), it was later realized that the new action couldn't be quantized Lorentz covariantly. The solution was to add more variables and constraints. The result was a BRST operator whose cohomology was the desired physical spectrum. (3) It can easily happen that a consistent solution to the free theory won't generalize to allow interaction~, as tested e.g., by introducing external fields. One reason for such a problem is that the constraints used have too many derivatives; there may exist other, redundant constraints which are lower-order in derivatives, and therefore preferable. (Although such constraints were known before, they were ignored because of their redundancy.) Such constraints will show up when trying to close the algebra of constraints/BRST in the presence of external fields. (4) Even though the BRST cohomology gives the right spectrum, the states may not have well defined norms. One way this problem can show up (as with e.g., the spinning particle or R.a.mond string) is that the cohomology might not be unique. Adding a nonminimal term to Q can easily solve this (see below). The alternative, picture changing, is an annoying complication for gauge fixing and nonperturbative analysis. This nonminimal term is needed for all known covariantly quantizable superparticles [6). (5) Another possible problem with the cohomology is that even though the spectrum is correct a covariant analysis shows that these states appear in the form of the field strengths instead of the gauge fields. This is clearly a problem for interactions, since it prevents minimal couplings. It is also a problem if one wants to write field theory lagrangians. If one tries the usual ~Q~ lagrangian, it doesn't work because the norms are not well defined, as in (4). Picture changing doesn't seem to be enough help in this case, but the nonminimal coordinates introduced in step (4) again solve the problem. The nonminimal bosonic coordinates are chosen so that all unphysical bosonic coordinates (except p2 ) are paired as harmonic oscillators, introducing the implied boundary conditions of the wave functions in terms of these coordinates.

305

Covariantly quantizable superparticles There are several systems with first-class constraints which propose to solve the problems of the BS superparticle. All these can be classified as three types:

(I) First ilk This approach to covariantly quantizing the superparticle is characterized by fixing the second-class constraint da of the original Brink-Schwarz superparticle by squaring it. In the original version [7] (first applied to the string) the resulting first-class constriants were: .A = p 2 ' 8 = = d'"fo~~cd This set is not covariantly quantizable [8]. Later a better version was found by adding extra constraints and variables [9]:

;a ' c,.,.,

.A= p 2

,

8 = pd , C..,., = d'"follcd + ptoM6c) , V,. = M,. 6p6 + kp,.

M,., = r1,.r,1 =>

v,. = r,.v , v = r. P

(A variation of these constraints was also suggested, which used an M represented by a spinor coordinate [10].) Even this set is incomplete, since it needs an additional dimension-! constraint for a local gauge transformation (6], such as

£

= '"f116 M,.,d + kd

The resulting set of constraints is just the complete set of field equations for the field strengths. Similar 3D models have explicitly given correct field theories: For 3D N=l, the Q which follows from C= ~

,

£ = f"'da

,

E = f'"'da

after projecting to irreducible representations of fermionic oscillators r' gives the scalar multiplet and various (free) Chern-Simons actions. When bosonic ghosts occur (D>3 or 3D N>l), Q needs a nonminimal term.

(2) Covariantized light cone [11] Instead of the usual BRST first-quantization, this method follows the following steps: (a) Use light-cone first-quantization, (b) add 4+4 dimensions, so that the light-cone J_;--. the covariant Q. Q again needs a nonminimal term; the simplest way is to instead add 8+8 dimensions (at least for the fermionic coordinates), which had been suggested by similar considerations for the string.

(3) Second ilk [12] The covariantized light cone has no obvious correspondence to classical mechanics, but a similar Q can be obtained from an action with an infinite number of constraints:

306 (Variation: add redundant constraints pq,. (13].) Q again needs nonminimal terms; one can use those suggested by the covariantized light cone. It is not yet known which (if any) of these methods generalizes to the superstring.

3. NSR SUPERSTRING W /0 PICTURE CHANGING Picture changing in usual NSR (1) String field theory: Ramond states (in any picture) have J =integer =f -! => S "'~YQ~, where Y has J = -1 (14]. Result: The analysis offermionic fields is difficult even in the free gauge invariant theory, since this operator mixes levels. Since the fermionic kinetic term doesn't resemble the bosonic one, supersymmetry also mixes levels. Gauge fixing is also more complicated. These features might also complicate the nonperturbative analysis. (2) Conformal field theory: Even in the Neveu-Schwarz sector, picture changing operators must be inserted, or vertex operators must appear in different pictures (15]. This is not a real problem, and can be avoided by a different choice of IO). However, picture changing is unavoidable again in the Ramond sector. The supersymmetry operator must be defined in different pictures: In one picture it looks like b/b8, in another like f8, rather than the expected sum of the two.

Simple examples (1) The simplest nontrivial BRST operator is Q=cp2 +de dis a commuting c-number, e anticommuting. The cp2 cohomology works as usual; for the rest: ~(d,e) =A( d)+ eB(d) de~ c~

=0

=>

dA

=0

=>

A ....... b(d)

= deA => cB = d>. => B = constant

This state is not normalizable and has the wrong ghost number, and wrong statistics. One solution would be picture changing. (2) Better solution (16]

(~IQ~)

has the

Direct products of previous solutions are still not normalizable. Use the boundary conditions implied by recombining d's and e's into creation and annihilation operators:

= [a2,a\] = {h1oht2} = {h2,htt} = 1 ~ = A(atm)IO) + httBt(atm)IO) + ht2B2(atm)IO) + httht2c(atm)IO) 6~ = (at 1 h1 + ht 1a 1 )A ~ 8 1 = C = 6(af 2a1 )a2AIO) = 0 (attht + httat)~ = 0 ~ 82 = atAIO) = 0 (at,at2]

~

~"'

IO}

The ~Q~ action reduces to just f/~Df/1, where A = f/1 +cr/J. Reinterpreting e.g., a 1 and a 1 t both as creation operators would again give non-normalizable states.

Spinning particles (1) N=l (Dirac spinor) Fix as in example (2) above:

The light cone analysis is identical to the above. Covariant analysis: Eliminating antifields from ~Qt> gives the usual f/J,f/1 action. (2) N=2 (vector)

j de !~Q~ = -l(q.. A6J?- !(B- o· A)

2

With the wrong boundary conditions for the ghost oscillators,

where 9 and i satisfy difFerent boundary conditions to get a nontrivial norm. This has the same cohomology as above (spin 1), but the wrong fields and action.

308

NSR superstring [17] (1) Conformal field theory Add a nonminimal term to Q, as for the Dirac spinor:

Q=

j c(ip + il.P' · t/1 + ic'b+ ih',8 + ih,8') + ltP · 2

p -1 2 b+ ::YI-'

The nonminimal term can be generated by fixing a gauge where the world-sheet gravitino propagates. Now physical states in both the NS and R sectors have J = -j. As for Yang-Mills and the Veneziano string, IO) =Yang-Mills ghost. Then all vertex operators have J = 1, and amplitudes are calculated as for the Veneziano string [18]:

J

{01

v v v bo v ... bo v IO)

with bo insertions for propagators. No picture changing operators appear. Equivalently, starting with the 0Sp(112) invariant vacuum, the extra picture changing operator insertions can be chosen to act on the OSp vacua, giving the Yang-Mills ghost as the picture-changed vacuum. (2) String field theory The action is simply

s=

t(4>IQ4>) + !g(4>14> * 4>)

where "*" is a generalization of the usual product of the Veneziano string, with a midpoint insertion of "picture changing operators." 4> contains both the NS and R strings, depending on boundary conditions, as in the light cone theory. This is similar to Witten's original superstring action in the NS sector.

REFERENCES [1) C. Becchi, A. Rouet, and R. Stora, Phys. Lett. 52B (1974) 344;

I.V. Tyutin, Gauge invariance in field theory and in statistical physics in the operator formulation, Lebedev preprint FIAN No. 39 (1975), in Russian, unpublished; J. Zinn-Justin, in Trends in elementary particle theory, eds. H. Rollnik and K. Dietz (Springer-Verlag, Berlin, 1975); I.A. Batalin and G.A. Vilkovisky, Phys. Lett. 102B (1983) 27, 120B (1983) 166; Phys. Rev. 028 (1983) 2567, 030 (1984) 508; Nucl. Phys. B234 (1984) 106; J. Math. Phys. 26 (1985) 172;

W. Siegel and B. Zwieba.ch, Nucl. Phys. B299 {1988) 206; W. Siegel, Int. J. Mod. Phys. A 4 {1989) 3705. [2) L. Brink and J.H. Schwarz, Phys. Lett. lOOB {1981) 310. [3) M.B. Green and J.H. Schwarz, Nucl. Phys. B243 (1984) 285; I. Bengtsson and M. Cederwall, Covariant superstrings do not admit covariant gauge fixing, Goteborg preprint 84-21 (June 1984); T. Hori and K. Ka.mimura, Prog. Thoor. Phys. 73 {1985) 476.

309 [4) F. Bastianelli, G.W. Delius, and E. La.enen, Phys. Lett. 229B (1989) 223; J.M.L. Fisch and M. Henneaux, A note on the covariant BRST quantization of the superparticle, Universite Libre de Bruxelles preprint ULB-TH2/89-04-Rev (June 1989). [5) E. Nissimov, S. Pacheva, and S. Solomon, Nucl. Phys. B296 (1988) 462, B297 (1988) 349; R.E. Kallosh and M.A. Rahmanov, Phys. Lett. 209B (1988) 233. [6) W. Siegel, Boundary conditions in first quantization, Stony Brook preprint ITP-SB-9098 (December 1990), to appear in Int. J. Mod. Phys. [7) W. Siegel, Nucl. Phys. B263 {1986) 93. [8) U. Lindstrom, M. Roeek, W. Siegel, P. van Nieuwenhuizen, and A.E. van de Yen, Nucl. Phys. B330 (1990) 19. [9) W. Siegel, Phys. Lett. 203 (1988) 79. [10) M.D. Green and C.M. Hull, Mod. Phys. Lett. A5 (1990) 1399. [11) W. Siegel, Nucl. Phys. B284 (1987) 632; Int. J. Mod. Phys. A4 (1989) 1827; Universal supersymmetry by adding 4+4 dimensions to the light cone, in Strings '88, eds. S.J. Gates, Jr., C.R. Preitschopf, and W. Siegel, College Park, MD, May 24-28, 1988 (World Scientific, Singapore, 1989), p. 110; Introduction to string field theory (World Scientific, Singapore, 1988), pp. 45, 46, 60, 75, 110, 168, 211. [12) A. Mikovic, M. Roeek, W. Siegel, P. van Nieuwenhuizen, J. Yamron, and A.E. van de Yen, Phys. Lett. 235B (1990) 106. [13) E.A. Bergshoeff, R. Kallosh, and A. Van Proeyen, The cohomology of the Brink-Schwarz superparticle, CERN preprint CERN-TH.5788/90 (August 1990). [14) J.P. Yamron, Phys. Lett. 174B (1986) 69; E. Witten, Nucl. Phys. B276 (1986) 291. [15) D. Friedan, E. Martinec, and S. Shenker, Phys. Lett. 160B (1985) 55; Nucl. Phys. B271 (1986) 93; V.G. Knizhnik, Phys. Lett. 160B (1985) 403. [16) W. Siegel and B. Zwiebach, Nucl. Phys. B288 (1987) 332. [17) N. Berkovits, M.T. Hatsuda, and W. Siegel, The big picture, Stony Brook preprint (August 1991). [18) S.D. Giddings, Nucl. Phys. B278 (1986) 242; S.D. Giddings and E. Martinec, Nucl. Phys. B278 (1986) 91.

310

TWISTORS AND THE GREEN-SCHWARZ SUPERSTRING Nathan Berkovits Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook, N.Y. 11794, U.S.A.

After replacing the spacetime variables of the d-dimensional Green-Schwarz superstring Lagrangian with twistor variables (d=3, 4, 6, or 10), the first-class constraints are separated out from the second-class constraints. These first-class constraints are shown to form an N=d-2 super-Virasoro algebra, out of which can be constructed a manifestly super-Poincare invariant BRST charge with only two levels of reducibility. When d=lO, all potential quantum anomalies in the algebra vanish and the BRST charge is nilpotent.

I. Introduction

Both twistor theory and string theory have evolved in unpredictable directions. Whereas twistor theory began as an attempt to formulate quantum gravity1, its most fruitful application has been in the solution of classical Yang-Mills equations. String theory, on the other hand, was originally developed to understand hadronic interactions, and was only later realized to describe quantum gravity. Another common feature of the two theories is that they both abandon the concept of second-quantized fields depending on points in spacetime. In twistor

311

theories, the second-quantized fields depend on points in the space of light-like trajectories (i.e. twistor space); in string theories, the fields depend on onedimensional "strings" in spacetime. An obvious generalization is to consider "twistor-string" theory in which the second-quantized fields would depend on strings in twistor space. 2 What would be the benefits of considering such a hybrid theory? It turns out that twistor theory and string theory have complementary sets of advantages and disadvantages. By combining the two theories into one, it might be possible to eliminate the disadvantages (hopefully, without eliminating the advantages as well). For example, two difficulties in twistor theory are the representation of massive particles and the construction of rules for calculating scattering amplitudes. In string theory, however, these two problems are easily solved. Massive particles are represented by higher resonances of the string, and scattering amplitudes are calculated by summing the exponential of the free action over all possible twodimensional surfaces that join the incoming and outgoing strings. Since neither of these solutions depend on the embedding space of the string, it is plausible that they will also be useful in a twistor-string theory. String theory, on the other hand, has problems with describing spacetime supersymmetric field theories in a Lorentz-covariant manner. Twistor theory does not suffer from such problems. For example, a covariant description of the massless superparticle (the first-quantized version of abelian super-Yang-Mills) is much simpler if one embeds the particle in twistor space, rather than in spacetime.3 In this talk, it will be shown that simplifications also occur in covariantly describing the Green-Schwarz superstring (the string analog of the superparticle) if one similarly embeds the string in twister space, rather than in spacetime. 4

312

II. The Green-Schwarz Superstring There are presently two different formalisms for studying ten-dimensional superstring theory, the Neveu-Schwarz-Ramond formalism formalism

6 •

5

and the Green-Schwarz

The N.S.R. formalism has the advantage of being based on a simple

geometrical principle -- worldsheet N=l superconformal invariance.

7

This princi-

ple completely determines the structure of interactions and allows the S-matrix scattering amplitudes to be expressed in terms of modular parameters of N=l super-Riemann surfaces.

8

The disadvantages of the N.S.R. formalism are that

scattering amplitudes involving fermions are much more difficult to evaluate than those involving bosons and that spin structures need to be summed over in order to project out unwanted states.

9

These complications have obscured the space-

time supersymmetry of the S-matrix. The Green-Schwarz formalism, on the other hand, is manifestly spacetime supersymmetric. The disadvantage with this formalism is the absence of any underlying geometrical principle. Although a light-cone formulation of the interacting Green-Schwarz superstring does exist, the lack of a geometrical interpretation for the interaction term in the Lagrangian (it is not a simple overlap integral as in the bosonic string or as in the supersheet formalism of the N.S.R. string) has prevented the scattering amplitudes from being expressed in terms of modular parameters of Riemann surfaces.

10

Perhaps by studying the

gauge invariances of the Lorentz-covariant Green-Schwarz Lagrangian, it will be possible to discover a geometrical structure analogous to the N=l superconformal invariance of the N.S.R. formalism. In its first-order "Hamiltonian" form, Type Iffi superstring Lagrangian is:

11

the d-dimensional Green-Schwarz

313 2tr

L ==

J dv ( P ,.X" -i p.i• -i ;.i" + an,.n" + &fl,.fl" + p• D. + »• tJ. )

(2.1)

0

where a, &, p•, and

»• are Lagrange multipliers,

and ' signify

:T

and : 11 , e• and

9• are real 2d-4 component anti-commuting spacetime chiral spinors with periodic boundary conditions,

p.

and ;. are their conjugates and are antichiral spinors,

D.

= ,. + n,.r:111' - f-b (assume P0>0), the usual d component spacetime vector variable can be replaced with a 2d-4 component bosonic spinor twistor variable. The extra d-3 gauge components of the twistor variable (because of the mass-shell constraint, pP has only d-1 independent components) are parameterized by a d-3 dimensional sphere. It is interesting to note that exactly when d=3, 4, 6, or 10, this sphere is parallelizable where the parallelizing transformations can be identified with multiplication by an element of unit norm in the division algebra of the real, complex, quaternionic, or octonionic numbers. In terms of these twistor variables, the massless particle Lagrangian takes the form:

In first-order form,

316

w.>.·

r:w.

=2 +c. ("'(>.•rt.,>. 1 ) ->.· (>.,w.)) where w. is a 2d-4 component anti-chiral spinor that is conjugate to x•, and 6" ,e. are Lagrange multipliers. Note that since x•rt. ("'(>.•rt.,>.")

r,:•w, ->.• (>.

6 w6 ))

is

identically zero, only d-3 of the 2d-4 constraints are independent (these constraints generate the d-3 gauge transformations of ). • that parallelize

s"~).

In a similar way, the Virasoro constraints for the Green-Schwarz superstring, TI"n,. = fi"fi,. =0, can be solved by using bosonic spinor twistor variables. 15 How-

ever, because [TI",

n"],o~o,

the equation TI"'"' x•rA>. 6 contains second-class con-

straints. These bosonic second-class constraints will turn out to be the worldsheet supersymmetric partners of the fermionic second-class constraints that are already present in the Green-Schwarz Lagrangian. After replacing TI" with 2w

Jdu(>."rA>. 6 -iB·r~B' o

x·r~x·

2tr

6)

and frP with

== Jdu(x·r~x• +ii·r~e' 6 ) 0

=Pt

x·r~x·

(this implies that

where P 0 ·Po=m 2 may be non-

zero due to higher resonances of the string), the Green-Schwarz Lagrangian in (2.1} becomes: 2w

L=

"

f0 du[ -"' (f0 dq ( >."(q)r~>. 6 (q)-i11"(q)r~ll' 6 (q) -pt

))

~>.'(u)rt1 >. 1 (u)-ill'(u)rf1 9' 1 (u)) {}T

. .• .. ,. +fJ•D.+P:>.."v. +z. ,.Pt +

-tp 0 0 -tp 0 11

0

(3.1}

2w

-ro,. f dq (>."(q)rA>.6 (q)-i11"(q)rAB' 6 (11)- x•(,)rAX 6 (q)-i9"(q)rM' 6 ( 17)) J . 0

Note that because >. • always appears quadratically in the Lagrangian, one can choose it to have either periodic or anti-periodic boundary conditions. In first-order form,

316 21r

L =

. ~· -tp. v;.•.. Id u [w•"'• +w•" -tp va• +ct •a +ct.• v;.. + p•D.+P 0

0

0

0

2>oA

v.+

(3.2}

0 21r

"/Opi dq(>."(q)r!t>. 6 (q)-i9"(q)r!tll' 6 ('7)- ~·(q)r!t~ 6 (q)-i9"(q)rf6 i' 6 (q)) J where !w., u..J are conjugate variables to [>.•, ~·J, 0

(/

a.(u) = w.(u)+

I dq[>.•(q)r!t>-'('7)-ifl• (q)r,r.,fl' 4 (qJ -Perl r .. ,. >.6 (ul + z&' r .. ,. >-' , 0 (/

a. = u..iPC r .. ,. 96 (u)- z&' r .. I' fl' 6 (u) , 0

(/

I dq[~•(q)r!t~ 6 (q)+i9•(q)rt.9' 4 ('7) -P&'l r .. ,. B' 6 (u) + '>iP&'r.. ,. 96 (u)- z&'r.. ,. B' 6 (u). 0

This Lagrangian is invariant under transformations generated by the first-class parts of

a.,

c., D.,

and D., as well as under the spacetime supersymmetry

transformations generated by

IV. First-Class Constraints Just like the fermionic constraints, the bosonic constraints are now half firstclass and half second-class. By using the twistor variables, however, it is possible to separate out the first-class constraints from the second-class constraints

without the disadvantages of the previous approaches.

317

It is easy to check that the following combinations of

c.

and D. are first-class

constraints (from now on, only the unbarred right-moving constraints will be discussed; after replacing ~ with --:-, the barred left-moving constraints can be OtT ut1' treated identically):

G(u) = >.• D. +B' • c.

r•(u)

= ll.(>.'f:'4>. 4)r,fc.

=

>.• p. +II' • w.+'>!.P 0 ~(>. •r!'b11 6 )

->.•(>.•c.)

= '>!.(>.'r:4>. 4)f,fw.

(4.1)

->.•(>. 6w6)

u• = ~,t(>.'r:'4 >. 4 )r,fD. ->.•(>. 6 D6 ) -(>.'rt.,ll' 4)r,fc6 +B' •(>. 6c.) +>.•(11' 6 C6)

= '>!.(>.'ff4>. 4)r;p6 ->.•(>.•p.) -(>.'ff4 11' 4 )r~uw6 +11'•(>. 6 w6 ) +>.•(II' 6w6 ) + P 0 p(1-.•r::,>. 4)rvabfGII' -11.>.•(>. 6 rt:,ll'))

with Poisson brackets: (4.2)

-il(>.{f:'4T~+•-l)r,f +l[ll,•>.t+lll>.: -(ll{ff4>.k~rpu)G,.+•-k-l

.

Note that the r• constraints generate the d-3 transformations of the >. 6 variabies that leave the combinations >. • ff4 >. 4 invariant, and the u• constraints are their worldsheet supersymmetric partners. Because the 2d-4 components of T• and

u• are therefore not all independent, one must either introduce ghosts for ghosts or sacrifice manifest Lorentz invariance. However unlike the wr,fD6 constraints

318

where an infinite tower of ghosts was necessary, the

r•

and

u•

constraints require

only two levels of ghosts for ghosts. The d first-level zero modes for the r• constraints are >."rf., since >. ~rf. r• vanishes identically. These first-level zero modes require a single second-level zero mode, "'r, 4P" 4 , since (>.'r, 4P>. 4 )>.~rf. also vanishes identically. The counting is correct, since the number of independent components of r• is d-3

= 2d-4 -d +1 .

Similarly for the u• constraints, the d first-level zero modes come from the vanishing of

>.~rf.u• -8'~rf.T•,

and the single second-level zero mode comes from

It is also possible to keep only the d-3 independent components of r• and u• by breaking manifest Lorentz covariance. By relating these irreducible constraints to parallelizations of the d-3 dimensional sphere, it is easy to show that they form an N=d-2 super-Virasoro algebra

16

(see reference 4 for details).

V. Second-Class Constraints After covariantly separating out the first-class constraints from (3.2), there still remain d-2 bosonic and d-2 fermionic second-class constraints. Although it is not known how to treat these second-class constraints covariantly, it is easy to show that in light-cone gauge, they imply the correct physical Hilbert space. Using the d-2 independent bosonic and fermionic first-class constraints to generate gauge transformations, the following light-cone gauge can be chosen:

>. 1=v'Pt, >.' =0 for i=2 to d-2, and

IJ' =0 for i=l to d-2.

(5.1)

In this gauge, the remaining second-class constraints in (3.2) take the form

wr =-~PeT A;"

for i=d-1 to 2d-4 and n;-!0

(5.2)

319 p;"=-P 0+e,.•cori=d-lto2d-4 andalln.

and

These imply the following Dirac brackets: 'A;"'

>-t

VPo

VPo

n

[ ~, ~ J = 20;;5,.+•

and

9;"'

Bt

VPo

VPo

{ ~, ~}

= %5;;5m+•

Since these are the same commutation relations as those of the usual GreenSchwarz light-cone fields, quantization of the twistor-space action in (3.1) gives the correct physical Hilbert space.

VI. Construction of a BRST Charge For the Green-Schwarz superstring, one can construct a manifestly superPoincare invariant BRST charge out of the first-class N=d-2 super-Virasoro constraints with only two levels of reducibility. However, because this BRST charge does not involve the Green-Schwarz second-class constraints, it would not be valid to identify the physical Hilbert space of the Green-Schwarz superstring with states of a fixed ghost number in the BRST charge's cohomology. Nevertheless, construction of a nilpotent BRST charge out of the Green-Schwarz first-class constraints when d=lO may be useful as an example of a free field representation of the N=8 superconformal algebra. For any reducible set of first-class constraints, Batalin and Fradkin have developed an iterative procedure that is guaranteed to produce a BRST charge that is nilpotent up to quantum normal-ordering effects. 17 For each zero-mode of the constraints, an additional ghost for ghost needs to be introduced. For secondlevel zero-modes, one needs to introduce ghosts for ghosts for ghosts. All of the terms in the Green-Schwarz BRST charge that involve three or fewer ghosts have

been explicitly evaluated, and the result is:

320 2~

Q,..d. =fdufd,P [-fll'H+.B.

V•+i(A•r!j,pb)'Yp+(A•r!j,Ab)~P6

(8.1)

0

+ [3i(,B.D.;A•)'rP -i (,B.r~(. D.;A')'{ +i (A•,B.)D.;1" -i (D.;.B. r:r(.A')'rv +i (A. D.;.B. )'rP -i ll'"/"

+~' 1# -%(D.;ll')(D.;1P)j ~#

+ [i(A • D.;.8.)6 -3i (.B.D.;A •)6- 1PD.;1p -i (A• .B.)D.;S +i ll'll -

~i ll'1 S +%(D.;ll')(D.;6)j'6] ,

where ll'=c+2i,Pd and .8.=/.+f/Je. are the ghosts for the first-class constraints, H=G-2i,PL and v•=T"+f/JU", ryP=hP+i.Pg" are the first-level ghosts for ghosts,

is

the

ghost

second-level

for

for

ghost

ghost,

- ·a + ·'· a p-· a +·'· a ~ · a +·'· a and 6= a +f/1 a are the conJ·ugate ll'=•8d 2'~'ac• =ae. '~'aJ.' ,,.=•ii;i' '~'ah"' -aj ak ghosts,

A • =).. • -,PO' •,

and D.;=

:,p -i ,P :u •

Because of Jacobi identities, only one type of quantum anomaly can consistently be introduced into the N=d-2 super-Virasoro algebra. 18 Because this anomaly affects the central charge of the Virasoro subalgebra, it is enough to calculate a zero central charge for the combined matter-ghost system to prove the absence of quantum anomalies in the algebra and the subsequent nilpotence of the BRST charge. Using conformal field theory methods, 19 the calculation is as follows: Fitld3

(o•,x•)

ll'

.B.

1"

6

Weight (-1,-12) {-12,-1) {0, 12) (-3/2,-1) {-2,-5/2) Mu/tip. 2d-4 1 2d-4 d 1 Contrih. (12-6d)/24 -15/24 +{30d -60)/24 -2ld /24 +33/24

(6.2) =

{d-10)/8

So the N=8 super-Virasoro algebra formed out of the Green-Schwarz first-class constraints in ten spacetime dimensions has no quantum anomalies and the BRST

321 charge is nilpotent.

VII. Concluding Remarks In this talk, it has been shown that by embedding the Green-Schwarz superstring in twistor space, rather than in spacetime, the gauge invariances of the covariant action are much easier to analyze. In d dimensions, these gauge invariances form an N=d-2 super-Virasoro algebra, and when d=lO, quantization of the first-class constraints does not contribute normal-ordering anomalies to the N 8 superconformal algebra. Covariant quantization of the Green-Schwarz second-class constraints remains an unsolved problem. For the case of the Neveu-Schwarz-Ramond superstring, the principle of N=l superconformal invariance was used in defining the S-matrix scattering amplitude as a functional integral over N=l super-Riemann surfaces of the exponential of the first-quantized action. If the second-class constraints can be better understood, it is possible that the Green-Schwarz S-matrix scattering amplitude (in twistor space) will be similarly definable as a functional integral over N=8 super-Riemann surfaces of the exponential of the first-quantized twistor-space action. Recently, several authors have proposed that the on-shell constraints for the second-quantized fields in ten-dimensional supergravity and super-Yang-Mills can be understood as integrability conditions in twistor loop-space. 20 It would be interesting if these integrability conditions (and the resulting on-shell constraints) are related to N=8 superconformal invariance of the twistor-space Green-Schwarz superstring action in the same way that the on-shell constraints of ordinary gravity are related to the conformal invariance of the bosonic string action.

322 Acknowledgements I would like to thank Tom Banks, Michael Douglas, Machika Hatsuda, Yutaka Matsuo, Martin Rocek, Natan Seiberg, Alexander Sevrin, Sang-Jin Sin, Peter Van Nieuwenhuizen, John Yamron, and especially Warren Siegel for useful discussions. This work was supported by the National Science Foundation under grant# PHY89-08495.

References (1)

Penrose,R. and MacCallum,M.A.H., Phys.Rep. 6C, 241{1972).

(2)

Shaw, W.T., "Twistor Quantization of Open Strings in Three Dimensions", February 1987, Twistor Newsletter 23. Cederwall,M., "An Extension of the Twistor Concept to String Theory", Goteborg preprint 89-15, April 1989.

(3)

Ferber,A., Nucl.Phys. B132, 55(1978). Bengtsson,I. and Cederwall,M., Nucl.Phys. B302, 81 (1988). Eisenberg,Y. and Solomon,S., Nucl.Phys. B309, 709(1988}. Sorokin,D.P., Tkach,V.I., Volkov,D.V., Zheltukhin,A.A., Phys.Lett. B216, 302{1989}. Berkovits, N., Nucl.Phys. B350, 193{1991).

(4)

Berkovits, N., Phys.Lett. B241, 497(1990}. Berkovits, N., Nucl.Phys. B358, 169{1991).

(5)

Ramond,P., Phys.Rev. D3, 2415(1971).

323 Neveu,A. and Schwarz,J.H., Nucl.Phys. B31, 86(1971). (6)

Schwarz,J.H., Phys.Rep. 89, 223(1982).

(7)

Gervais,J.L. and Sakita,B., Nucl.Phys. B34, 632(1971).

(8)

Friedan,D., Martinec,E., and Shenker,S., Nucl.Phys. B271, 93(1986). Berkovits,N., Nucl.Phys. B304, 537(1988).

(9)

Kostelecky,V., Lechtenfeld,O., Lerche,W., Samuel,S., and Watamura,S., Nucl.Phys. B288, 173(Hl87). Berkovits, N., Nucl.Phys. B331, 659(1990).

(10) Mandelstam,S., ''Interacting-String Picture of the Fermionic String", in 1!185 Santa Barbara Worhhap an llnjfjcd String Theone«,

eds. Green and Gross

(World Scientific, 1985), p.577. Restuccia,A. and Taylor,J.G., Phys.Rev. D36, 489(1987). (11) Siegel,

w.,

"Covariant

Approach

.Wmpo«ium on Anoma/je• Geometqr and TopologJi ,

to

Superstrings",

in

eds. Bardeen and White (World

Scientific, 1985), p.348. (12) Mikovic,A., Rocek,M., Siegel,W., van de Ven,A., van Nieuwenhuizen,P.,

Yamron,J., Phys.Lett. B235, 106(1990). (13) Gates,S.J.Jr.,

Grisaru,M.T .. , Lindstrom,U., Rocek,M.,

Siegel, W., van

Nieuwenhuizen,P., and van de Ven, A.E., Phys.Lett.B225, 44(1989). (14) Nissimov,E., Pacheva,S., and Solomon,S., Nucl.Phys. B297, 349(1988). (15) Fairlie,D. and Manogue,C., Phys.Rev. D36, 475(1987). (16) Ademollo,M., Brink,L., D'Adda,A., D'Auria,R., Napolitano,E., Sciuto,S., Del

Giudice,E., DiVecchia,P., Ferrara,S., Gliozzi,F., Musto,R., and Pettorino,R.,

324

Nucl.Phys. Bl14, 297(1976). Englert,F.,

Sevrin,A.

,Spindei,Ph.

,Troost,W.

and

van

Proeyen,A.,

J.Math.Phys. 29, 281(1988). (17) Batalin,I.A., and Fradkin,E.S., Phys. Lett. B122, 157(1983). (18) Osipov,E.P., Lett. in Math.Phys. 18, 35(1989). (19) Polyakov, A.M., Phys. Lett. B103, 211(1981). (20) Witten,E., Nuci.Phys. B266, 245(1986).

Howe,P .S., 'Pure Spinors, Function Superspaces, and Supergravity Theories in Ten and Eleven Dimensions", Stony Brook preprint ITP-SB-91-18, April 1991. Bergshoeff,E., Delduc,F., and Sokatchev,E., Phys. Lett. B262, 444(1991).

325

Q-QUANTUM MECHANICS D.G.CAIDI

Dept.of Physics and Astronomy, State University ofNew York at Buffalo Buffalo, NY 14260 ABSTRACT Two formulations of q-quantum mechanics based on quantum deformations of the Heisenberg equations of motion, are discussed. In one the commutaiDr is replaced by the "quommutator": [A,B]q = qAB • (1/q)BA. The odler involves using the quantum bracket of the time derivative of an operator in the equation of motion. Both have advantages and difficulties, but the conclusion is that these q-quantum mechanics appear to be sensible.

This is a report of work done in collaboration with A. Chodos. [1] Quantum groups [2] have been found to play important roles in integrable systems, including exactly solvable lattice models, as well as conformal field theories (arising in statistical mechanics systems and in string theories), and in tOpological field theories and related knot theory. In all of these the Yang-Baxter equation plays a central role. [2,3] In connection with the study of quantum groups, there have been various investigations of deformed harmonic oscillator algebras and thereby deformed Heisenberg algebras [4]. For example, one can use the "quommutator" instead of the commutator to get the deformed Heisenberg algebra for a single oscillator: qbbt- (1/q)btb [b,bt]q = 1. These have been studied for their own sake, and also for constructing representations of quantum groups. In the light of all this, a natural question arisies, whether it may be possible to use the idea of a quantum deformation in a yet more central way, directly in the dynamics and structure of quantum mechanics itself, so that one considers a quantum deformed quantum mechanics, or a q-quantum mechanics. This is guaranteed to have immediate physical consequences, and, it appears so far, that they can be made sensible. Following the approaches for writing the commutation relations of a quantum group, but now using them in the Heisenberg equations of motion, with no suggestion of a quantum group structure, we have two ways (at least) to deform. Letting O(t) be an observable in the theory, we have:

=

method 1)

method 2)

•

i [H(t) , 0 (t) ]q = i [qHO - (1/q)O H] = 0 (t)

i [ H(t), 0 (t)] =

[0• (t)]q =

(1)

q0/2 -q-012

(2) q1/2 - q-1/2

Each method has its pros and cons. Method 1) appears to have a problem with unitarity, but one can formally integrate the equations and one can make sense of expectation values of observables. Method 2) is highly non-linear and so, difficult to integrate; however, it appears to be unitary. With the method 1), H is not a constant of the motion: H(t) Ho [1 - i r H 0 t]·1 , with r = q - 1/q , H0 = H(t=O) . Then in general, formally we find for 0 (t):

=

326

0 (t) = [ 1 - irHot J-q/r 0 (0) [ 1 - irHot ]l!qr .

(3)

Despite problems with unitarity, we do have quite sensible and finite expectation values of operators, taken in eigenstates of Ho. We considered examples from classical mechanics, and in quantum mechanics those of the free particle and the hannonic oscillator. In method 2), the Hamiltonian itself is a constant of the motion and energy in conserved. Furthermore, hermiticity appears to be preserved in the cases when q is treated as a formal variable, or when q = eie. With q = eh and h = sinh(b/2), we find for 0 (t): 2t

roCt)

h

= J0(0)

dO (4)

sinh- 1(ih [H,O 1)

Due to the non-linearities, it is difficult to proceed further in the general case. But for the free particle: X(t) - X(O) = t (2/h) sinh- 1(h p), which is a well-behaved, single-valued function. This result is different from the usual quantum mechanical result, and that obtained from the quommutator deformation. Time evolution is obviously unitary here in this free case, and, we believe, in general. These are indications that the two methods are not equivalent Besides its intrinsic interest and its value as a tool for probing the essentials of mechanics, one of the possible practical applications of a q-quantum mechanics comes from the fact that the quantum deformation parameter acts as a cutoff which discretizes the time-evolution of a system. One of the many open questions is whether and how either or both of our methods of deforming the Heisenberg equations, can be generalized to many degrees of freedom and finally field theory. This will be of interest, since it is clear much can be learned from these deformations. We saw this, for example, in studying Eq. 1, where there is a lack of unit&Jity. However, what the lack ofunitarity seems to imply, is that just as in the passage from:.classical to usual quantum mechanics one gives up determinism in favour of probafri:lity amplitudes which are defined as matrix elements in a Hilbert space, here one is abandoning the physical significance of the Hilbert space inner product but, it appears, retaining the meaning of an expectation value, the appropriate ratio of such inner products. Alb in all, q-quantum mechanics is an excellent laboratory for exploring what is truly essential for a sensible mechanics.

1. A. Chodos and D. G. Caldi, Yale preprint YCIP-P9-91, Feb. 1991. 2. P. P. Kulish and N. Yu. Reshetikin, J. Sov. Math. 23 (1983) 2435; E. K. Sklyanin, Func. Anal. Appl. 16 (1982) 27; V. Drinfeld, Sov. Math. Dokl. 32 (1985) 254; M. Jimbo, Lett. Math. Phys. 10 (1985) 63; 11 (1986) 247; Commun. Math. Phys. 102 (1986) 537. 3. M. Jimbo,lnt. J. Mod. Phys. A4 (1989) 3759 and refs. therein; L. Alvarez-Gaum6, C. Gomez and G. Sierra, Phys. Lett. 220B (1989) 142; Nucl. Phys. B319 (1989) 155; E. Witten, Nucl. Phys. B330 (1990) 285. 4. L. C. Biedenharn, J. Phys. A. 22 (1989) L873; A. J. Macfarlane, J. Phys. A. 22 (1989) 4581; J. A. Minahan, Mod. Phys. Lett. A 5 (1990) 2625.

327

DIFFERENTIAL REGULARIZATION AND RENORMALIZATION: RECENT PROGRESS* Daniel Z. Freedman t Department of Mathematics and Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 U.S.A. ABSTRACT We discuss briefly recent progress in the development of the dift'erential regularization method: the 1-loop effective potential; llimpliflcation of integrals in ,P" theory; unitarity in ,P" theory; and the fermion triangle graph.

1.

Introduction

Differential regularization 1 is a new real-space computational method in perturbative quantum field theory which regularizes ultraviolet divergences and simultaneously defines renormalized amplitudes. The method was applied in Ref. [1] to massless ,P" theory throu~ 3-loop order. Virtually all required calculations were easy, and the amplitudes were shown to satisfy the renormalization group equations. This provided a stringent test of both the practical usefulness and the consistency of the procedure. Many 1-loop calculations in gauge theories were also performed in Ref. [1] with little difficulty, but more higher loop computations and further systematic development is needed to learn whether the method can really be useful for chiral gauge theories such as the standard model. I believe that the ideas and techniques of the method were presented clearly in Ref. [1] and need not be repeated here. Instead, I summarize briefty below several topics where there has been progress during the 3-month period after the Stony Brook Conference. The advantage is that readers will learn the current state of affairs well before the new material appears in standard papers. But there may be a slight risk of error because some of the topics disc:ussed are not yet as certain as a standard paper requires. I hope that this risk does not prove to be material. • This work is supported in part by the National Science Foundation grant #8708447 and by the Department of Energy under contract #DE-AC02-76ER03069. t

e-mail: [email protected].

328 The springboard for the new method is the observation that essentially all primitively divergent graphs are well defined in real space for non-coincident points, but too singular at short distance to allow a Fourier transform. A regularization procedure must supply a prescription for the real space amplitude which makes the short distance singularity harmless. In massive ¢> 4 theory with Lagrangian (1)

the propagator is t:.(x,m)

= 4 ~ 2 ~K1(mx) 1 = -471'2

(2)

[..!_ x2 + R(x)]

where the short distance properties of the Bessel function have been used in the second line, so R(x) can be viewed as a known function with a logarithmic singularity. To see how differential regularization works, let us calculate the 4-point function up to 1-loop, including the effect of mass which was not done in Ref. [1]. The bare amplitude is, using Xij = x;- Xj,

r< 4>(xl,x2,X3,X4) =

A2 -Ac5(x21)6(xal)c5(x41) + 32 71' 4

x { c5(x21)(x;) to confirm that -y(.A) and c5(.A) are of order ..\2 and do not contribute to (6) in 1-loop order.] The function lnx 2 M 2 fx 2 has Fourier transform -4?r 2 ln(p'J / M 2 )/p2 where M = 2Mh and 'Y = 1. 781 is the Euler constant. We define the Fourier transform of 1/z4 in differential regularization, using (4) and then formal integration by parts, viz. eipz 1 lnz 2 M 2 d'z- = - - d'ze'''"D--:::--

J

z4

4

J ·

1 2j = 4p

"

a-ze

z2

;,.-ln z2 !-!2 z2

(8)

p2

= -11"2ln~

M The momentum space 4-point function is then (9) where q;j

= q; + qj and (10)

330

Since R( x) has Fourier transform also (8}} to write 2 2 p2 F(p ) = -11' In M2

+

411' 2 [ ,.2.lom2 -

jd' k [(p - k

1 )2

tr], we use convolutions (and

+ m2 + (p -1 k )2 ]

[

k2

+1 m2

1] - k2

(11)

The ultraviolet finite momentum integral can be easily done using Feynman parameters leading to

The amplitude (9), (12) differs only in the renormalization point from that obtained after renormalization using any regularization procedure. Where has the usual ultraviolet divergence gone? It is hidden in the contribution of a short distance surface term which was ignored in (8). Later we will come back to discuss these surface terms in more detail. 2.

The One-Loop Effective Potential in ;• Theory

The effective potential plays a central role in the determination of the ground state and its symmetry properties in most treatments of quantum field theory. It is defined in terms of the momentum space 1PI vertex functions r(q;) at zero momentum, (13} Thus momentum space methods are most natural for V(~), but it is not hard to use differential regularization. One needs to observe that _-t denote the functional integration with the weight given in eq. (6). Since the gauge-fixing function is arbitrary we may conclude that the nilpotency of the gauge covariant BRST operator

(12) is the necessary and sufficient condition for the gauge independence of the background functional (6). Equation (12) is also known to be the fundamental equation of the string field theory (5), following from the string field theory Lagrangian.

(13) In the second quantized theory the canonical momenta are realized as derivatives over the coordinate, P = We denote by d the differential operator, related to the free BRST operator as

/o·

{)

{)

d({)Q'Q)

= !l(P = fJQ,Q),

(14)

and the covariant differential operator, related to gauge covariant BRST operator introduced above, as

D=d+A.

(15)

The consistency condition of the first quantized theory, or the classical equation of the second quantized theory take the geometrical form of the Chern-Simons type equation for the curvature of the target space:

(16)

345

3

D=lO superspace and chiral superfields

According to the superparticle theory (3], the flat D=10 superspace can be characterized by are D=10 anticonunuting the classical coordinates X",B;, where X" is a D=10 vector and Majorana-Weyl spinors of positive chirality, Ct = 1, ... ,16; p = 0,1, .... Our notations and conventions are those of [4). The important ones are the following. The spinors of positive chirality have spinorial indices Ct up, the ones of negative chirality have them down. The '"(matrices have their two indices both up or both down, and they are symmetric.

e;

The supersymmetry charges and covariant derivatives are defined as follows.

8

+ "Bp)o

q~

l}8o

~

~-

p

8

p

1

(17)

(,Bp)o.

These charges and covariant derivatives form the twisted N = 2(p + 1) supersymmetry algebra, {~,q~}

+2

,,pli"1

1

{~,tip}

-2 ,,pfJp/

1

{d",q'} = 0.

(18)

!'tom the charges of the twisted N = 2(p + 1) supersymmetry algebra given in eqs. (18) we build two combinations,

(19) They form the following algebra:

{f!,FJ} = 0 I

{fl:,PJJ {J'!,i'J}

(20)

This algebra has Abelian subalgebras which remind the Abelian subalgebras in D=4 where chiral and antichiral subalgebras exist,

{qA,qB} {q..t,,f} {qA,Il

(21)

A, A being 2-component spinor indices. The treatment of D=10 flat superspace (20) and gauge theories in this superspace will have some analogies and some differences with 4D flat superspace (21) and gauge theories in it.

346 The irreducible representation of the algebra (20) is realized by putting the constraint which is analogous to the chiral constraint in D=4. We can put the constraint (we will call it chiral constraint by analogy)

fl: ~(X",8,) = (q"+l- d")or ~(X",8,) = 0.

(22)

The corresponding superfield can be real (as different from the chiral 4D superfield), since the constraint is real. The solution can be written in the formal way as (23) where 111(X",89 ) is an unconstrained superfield and the product includes all fl:. We will call the superfield satisfying the constraint (22) the chiral superfield. From all the charges of the twisted N = 2(p + 1) supersymmetry algebra given in eq. (18) only

a

o f.i/J q =a8o+,..vo

(24)

commutes with the constraint fl:. Therefore the chiral superfield has the trivial dependence on 8p+t, even though it depends on all Bp, p = 0, 1, ... and not only on 80 as the the normal N = 1 superfield in z = (X, 80 ) superspace does. The solution to the chirality constraint can be presented by expanding the superfield in powers of 8t, 82 , ••• , where the coefficients of the expansion are normal N = 1 superfields,

(25) The chiral constraint allows to express all components of this superfield through the first one and its covariant derivatives. For example, the first in the set of constraints (22) is

(26) When applied at 81 = 0, it gives ~~(X",80 ) = (~(X",8o).

(27)

Note, that ~ are the covariant derivatives for N = 1 superfield ~(X", 80 ). Alternatively, one may choose the constraint

(28) We will call this constraint and the corresponding superfield antichiral, by analogy with D=4 superspace.

347 To describe the gauge theory, one proceeds in a standard way of introducing connections for each direction in flat superspace,

..,., =

.r;;.

~(QP+I = ~(q"+l + dP..a +A") 2 a + D") a 2 a + AP+I a a

'

:f! = ~(Q~+ 1 D~) = ~(~+I + A~+l - ~ -A~) . -

(29)

One can rewrite this equations as follows:

D"a

a

aoa - ;JO, + A~ ' " llll aoaap+l + pvp+J + AP+I a

(30)

0

The gauge symmetry is introduced into the space by choosing the parameter of the gauge transformation to be a covariantly chiral superfield,

:f! A(X",8t) = 0. The integrability of this constraint requires the

{;P,.~}

vani~hing

(31) of the curvature,

= F:3 = 0

(32)

0

One can choose the basis in which covariant chiral derivatives are free and

:f! = fl: ,

A~- A~+ 1 = 0 ,

_1! = f!

+ A~(z) ,

(33)

where A~(z) is an unconstrained superfield. It is possible to impose the set of constraints on torsion and curvature in the classical superspace z = (X", 8,) and to solve them to describe the on shell geometry. In addition to eq. (32), we impose the conventional constraint by expressing the vector connection through the spinor one: 1 • • (34) 2({.1!,.1",;} + {.F,;.-1!}) = 11 v,.c5"·'.

-r:

The on shell constraint which is added to the geometry is

{_1!,F,;}=O.

~5)

The solution of this last constraint requires the connection on shell to be of the form A~(z) = ev (f!e-v) ,

(36)

where V(z) is an unconstrained superfield. It is necessary to solve .Bianchi identities in the presence of all above mentioned constraints. We have found that the solution requires the particular properties of spinor-vector curvature,

(_1!, V....,) =Ill.., .

(37)

This spinorial superfield must be covariantly antichiral and satisfy gauge covariant Dirac equa· tion. Thus the geometry of the superspace described above is an alternative description of the on shell D = 10 supersymmetric Yang-Mills theory. The difference with the usual (X, 80 ) superspace is the fact that it is related to the first quantized superparticle action and that the geometry will be shown to be a solution to the nilpotency condition of the gauge covariant BRST operator.

348

4

First quantized superparticle

The classical action of the superparticle [3) is

sc1 =Pi+ f:r~Ptip + T4t/>4},

(38)

p:O

where f/> 4 = {g, (P, qp} are the Lagrange multipliers to the constraints2. The first-class constraints are 1 2 T. = { 2P , Kp, P} . (39) Here we are using the following notations: Kp

+ q") .f ' ~(JI' + qP+l) .

(d"

(40)

These constraints generate local symmetries of the action: reparametrization, tt- and e-symmetries, respectively. Note, that the commutator of the fermionic tt- and {-symmetries has reparametrization in the right-hand side,

(41) In this sense the fermionic symmetries of the world-line are quite fundamental and can be considered as the "square root" of the Virasoro constraint. The phase space of the first quantized superparticle theory [2)-[4) in addition to classical coordinates (X" ,8;) and their canonical momenta contains a set of ghost fields and their canonical momenta. They are related to the local symmetries of the action. The gauge-fixed action in arbitrary gauge, including the light-cone one and the free covariant one, has the following form [3):

(42) where flaRST is a nilpotent ghost number +I BRST operator, whiclt will be presented below, and Ill is an arbitrary gauge fixing function of ghost number -1.

In the gauge fixed-action (42) b,c is a couple of reparametrization ghosts. The spinors (classical and ghost together) form a spinorial multiplet of 0Sp(9.1/4) supergroup, consisting of spinors of alternating Grassmann parity and chirality (9)- [11). We are using the notations of (4), where the spinors of 0Sp(9.1/4) are presented using labels of SU(2) representations, e.g. 8(j,m), and/or by using their original labels as they come from 2The aecond ilk action in ref. ['1] can be obtained from the action (38) at (P = O,p = 1,2, .... This leads alao to a different aet of ghosts and a different B RST operator.

349 superstring quantization, where 8.,..

= 8(j = p + q/2,

m

=p-

q/2) and

jlt+t,q+t

= 8(j =

p + q + 1/2, m = q- p- 1/2), p ~ q ~ 0. The SU(2) label m is related both to ghost number

and to conformal charge in string theory. The nilpotent ghost number +1 BRST operator, which was found in (2], is given, according to [3)- (4), by the following expression:

=

=

We &SSume a sum over p 0,1, ... and ao a. In (43)

+I,p+l p=O

(-_\P+I "'+I -

, 8,, - 28p+l,pb) f8p+t,p+t + 28pH,p+l b + _\Pol'+

(44)

The BRST operator commutes only with N = 1 supersymmetry and does not commute with all of those extra supersymmetries which are not the symmetries of the physical spectrum. It was shown in (2) that the cohomology of the nilpotent operator (44) is given by the 8+8 supersymmetric Yang-Mills mtiltiplet.

In what follows we will use a version of the BRST operator related to the one in eq. (44) by canonical transformation 3 n' = e•ne-• 1 (45) where I

(46)

q"+l) .

(47)

• = bc,+IF: and

F: = ~(dl' -

3 1 am grateful to W. Siegel for euggeetion to represent the BRST operator, which we have WJed for the eecond quantization, in thia particular form.

350 Consider now the functional given in eq. (6). The set of classical and quantum coordinates of the superparticle is defined above, the free BRST operator is given in eq. (45). The gauge covariant BR.ST operator will be introduced in a way that the connection field A(Q) can be absorbed in the change of variables in the path integral if the connection is a pure gauge only. Together with our choice of the BRST operator (45) we suggest a particular choice of coordinates Q: Q = {Z,Y},

(48)

where

{X",o;,b,c;+ 1 = .Xp+t,p+d, {.V+1.q,op+2,q+IJ, P~ q

z y

~

o.

(49)

With this choice of coordinates the free BRST operator d is

d=

~P 2 :b + c;+tf! + bc;+t in the conformal field theory is related to .\ through the relation: (35)

We can now construct the solution to any arbitrary order in .\ following the procedure outlined above (for details see ref.(13]). In particular we can study the order .\3 contribution to the solution, and ask if we can see the change in the background charge from Q to Q'. (Note that Q 2

(ccFT- dcFT)/3

= (8y3 /C~,,) + O(y4 ).)

of the background charge in the

X0

-

Q'2 is given by to

For this, we note that the presence

system is equivalent to giving the dilaton a

background value proportional to QX 11 [24]. Thus a change in Q will correspond to a change in the background value of the dilaton field. Examining the solution to order .\3 we find that such a background is indeed present in the solution. In particular, we can calculate Q - Q' from the solution and find that [13]: I

Q- Q

8y3

= 6Q(C,,,) 2

(36)

in agreement with eq.(31) to this order. Note that if we did not know the central charge of the perturbed conformal field theory, the above procedure would give us a way to calculate this.

t

This state is not actually a physical state, since tp does not have dimension (1,1). It turns out, however, that in obtaining the solution in power series in A, we need to treat this state on the same footing as a physical state, otherwise in the process of obtaining the solution we need to invert a matrix, one of whOile eigenvalues is proportional toy. As a result, the inverse of this matrix, acting on an order y"+l term, will give an order y" term, thus upsetting the counting of powers of y [13].

370 This finishes our discussion of explicit construction of the solutions of string field theory. We can now ask·: given these solutions, what evidence do we have that they indeed represent string theory formulated in the background of perturbed conformal field theory? In order to answer this question, we need to compare string field theory action 5'(1)) formulated around these new backgrounds with string field theory action 5'(1)) formulated directly in the background of the perturbed conformal field theory. First we can compare the coefficient term in

S with the

Q8

of the quadratic

BRST charge Q8 of the perturbed conformal field theory. It

turns out that to order A these two charges are related to each other by an inner product preserving similarity transformation (25), provided we identify A~o) with

gA;j../2 as in eq.(35). Hence the spectrum of Q8 and Q8 are identical, and the kinetic term of S is identical to the kinetic term of S after a linear field redefinition. The next question is whether the interaction terms of Sand Scan also be shown to be identical. In this case it turns out that to order A, if we calculate the S-matrix elements from the actionS, they are identical to the S-matrix elements calculated in string theory in the background of perturbed conformal field theory [26)! This, in turn, gives a strong indication that there is, in fact, a field redefinition which takes us from s(l)) to S(.)). Generalization of these results to higher orders in A has not been carried out. Also, so far, the explicit field redefinition which takes us from

S to S has

not been constructed.

To summarise, we have shown how we can use string field theory to demonstrate that conformal field theories which are related by marginal and nearly marginal operators can be regarded as different classical solutions of the same underlying string field theory. We have also discussed how to construct explicit classical solutions of string field theory representing these perturbed theories. In this process, we have obtained string field theoretic expression for the ,8-function and the central charge of the perturbed conformal field theory. Finally, for c = 1 conformal field theory coupled to gravity, we have explicitly constructed an asymptotically flat, static (possibly singular) solution labelled by infinite number of paf This has been shown for S-matrix elements with arbitrary number of tachyonic external legs, and also for S-matrix clements with three arbitrary external legs.

371

rameters. For a specific set of values of the parameters the solution reduces to the recently discovered black hole solution. Acknowledgements: I would like to thank the organisers of the conference for their hospitality during my stay at Stony Brook.

REFERENCES [1] M. Saadi and B. Zwiebach, Ann. Phys. 192 (1989) 213; T. Kugo, H. Kunitomo, and K. Suehiro, Phys. Lett. 226B (1989) 48; [2] T. Kugo and K. Suehiro, Nucl. Phys. B337 (1990) 434. [3] M. Kaku and J. Lykken, Phys. Rev. D38 (1988) 3067; M. Kaku, preprints CCNY-HEP-89-6, Osaka-OU-HET 121. [4] H. Sonoda and B. Zwiebach, Nucl. Phys. B331 (1990) 592. [5] H. Rata, Phys. Lett. 217B (1989) 438, 445; Nucl. Phys. B329 (1990) 698; B339 (1990) 663.

[6] M. Saadi, Mod. Phys. Lett. A5 (551) 1990; Int. J. Mod. Phys. A6 (1991) 1003. [7] B. Zwiebach, Mod. Phys. Lett. A5 (1990) 2753; Mod. Phys. Lett. A5 (1990) 2753; Phys. Lett. B241 (1990) 343; Comm. Math. Phys. 136 (1991) 83. [8] L. Hua and M. Kaku, Phys. Lett. B250 (1990) 56; M. Kaku, Phys. Lett. B250 (1990) 64.

[9] B. Zwiebach, Phys. Lett. B256 (1991) 22; preprint MIT-CTP-1908. [10] C. Schubert, MIT preprint CTP 1977. [11] A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B241 (1984) 333. [12] A. Sen, Phys. Lett. B241 (1990) 350. [13] S. Mukherji and A. Sen, preprint TIFR/TH/91-12 (to appear in Nucl. Phys. B)

372 (14) S. Mukherji, S. Mukhi and A. Sen, preprint TIFR/TH/91-28. [15) R. Dijkgraaf, E. Verlinde and H. Verlinde, Comm. Math. Phys. 115 (1988) 649. (16) S. Mukherji, S. Mukhi and A. Sen, preprint TIFR/TH/91-25 (to appear in Phys. Lett. B). (17) S. Elitzur, A. Forge and E. Rabinovici, Preprint RI-143-90. (18] G. Mandal, A.M. Sengupta and S.R. Wadia, preprint IASSNS-HEP-91/10. (19) E. Witten, preprint IASSNS-HEP-91/12. (20) M. Rocek, K. Schoutens and A. Sevrin, preprint IASSNS-HEP-91/14. [21] K. Bardacki, M. Crescimannu, and E. Rabinovici, Nucl. Phys. B344 (1990) 344. [22] R. Dijkgraaf, E. Verlinde and H. Verlinde, preprint PUPT-1252, IASSNSHEP-91/22. [23] I. Bars and D. Nemeschansky, Nucl. Phys. B348 (1991) 89. [24] R.C. Myers, Phys. Lett. B199 (1988) 371; I. Antoniadis, C. Bachas, J. Ellis and D. Nanopoulos, Phys. Lett. B211 (1988} 393, Nucl. Phys. B328 (1989} 117; S.P. de Alwis, J. Polchinski and R. Schimmrigk, Phys. Lett. B218 (1989} 449. [25J A. Sen, Nucl. Phys. B345 (1990) 551. [26) A. Sen, Nucl. Phys. B347 (1990} 270.

373

LORENTZ-HARMONIC (SUPER)FIELDS AND (SUPER)PARTICLES

E. SOKATCHEV LPTHE, T.t,/, Univer$ite de Pam VII !, pl. Jtu$ieu, 75!51 - Pam Cedez 05, .France

Abstract

We apply the recently developed concept of a compact Lorentz-harmonic space, which is isomorphic to the sphere sD- 2 , to the problem of covariant quantization of the superparticle in D = 3, 10. We study the structure of the representations of the Lorentz group realized on harmonic functions on sD- 2 • The crucial difference between compact and non-compact harmonic analysis is explained. The massless harmonic fields depend on one space-time coordinate z++ only, and on D-2 harmonic coordinates. It is shown how ordinary massless fields can be obtained from the harmonic ones by means of covariant integration on sD- 2 . We construct a Lorentz-harmonic superspace, which involves only one quarter of the usual number of Grassmann coordinates. It resembles very much the light-cone superspace, however it is Lorentz covariant. This framework is used to formulate a superparticle action, in which all the constraints are first class, Lorentz-covariant and allow a straightforward canonical quantization.

1

Introduction

The problem of the quantization of the Brink-Schwartz superparticle [1] (and the related problem of the Green-Schwartz superstring [2]) has a long history. The simplest and historically the first approach made use of the light-cone frame. It allowed to separate the Grassmann constraint 1 Da = 0 of the BS superparticle into first and second class parts, and thus to perform canonical Dirac quantization. The high price to pay was the loss of Lorentz invariance. Siegel (3) proposed a modification of the BS superparticle, in which the Grassmann constraint was (fD)a = 0 instead of Da = 0. The singular matrix f (on shell P 2 = 0) played the role of a Lorentz-covariant projector which singled out 1 Da is the conjugate momentum of the Grassmann coordinate 9°, after quantization it becomes the spinor covariant derivative.

374 the first-class part of the constraints D 01 = 0. However, a problem remained, this time it appeared when trying to fix the gauge invariance generated by the constraint (fD) 01 = 0 (the so-called "kappa symmetry"). This invariance affects only half of the Grassmann coordinates, but that half could not be separated out in a Lorentz-covariant way. In 1986 it was suggested [4] to combine the advantages of the light-cone approach with manifest Lorentz-invariance, using vector harmonic variables for the Lorentz group. They were supposed to play the role of bridges bringing (but not breaking!) the Lorentz symmetry SO(D- 1, 1) down to the light-cone subgroup 50(1, 1) x SO(D- 2). This idea was further developed in [5],[6],[7], where the necessity to use spinor harmonic variables was underlined. However, all those attempts suffer from a principal problem: the coset SO(D -1, 1)/[50(1, 1) x SO(D- 2)], which the harmonic variables describe, is a non-compact space. This means that the functions on it, which contain finite-dimensional representations of the Lorentz group, cannot be integrated invariantly and do not have a harmonic expansion [8]. Moreover, a number of basic features of the compact-space harmonic analysis [9] have been assumed to apply to the non-compact case case as well. Unfortunately, this is not the case. For example, in the compact case of SU(2)/U(1), the harmonic condition n++. = up. L "LT>. P. .,

d e t L = 1. ........ -r

(35)

It turns out [10] that this condition removes only one degree of freedom, after 45 degrees of freedom left. This is precisely which II u II has 16 X 16- 1- 210 the dimension of 50(9, 1), so II u II parametrizes the group space of 50(9, 1). The

=

382 harmonic functions f!, ...Am.A, ... AJu±) are homogeneous under the right action of S0(1, 1) x S0(8), so they depend on 45- 1 - 28 = 16 variables, which is the dimension of the coset S0(9, 1)/[S0(1, 1) x S0(8)]. Finally, the restriction to the sphere S0(9, 1)/[S0(1, 1) X S0(8) X K] is achieved by imposing the conformal condition

sa""

r'l D ++i )A • • ( U ±) - 0 A, ... AmA, ... A,.

=}

!9A, ... AmA,• ... A,. -- !9A, ...AmA,• ...A,. • • (U +)

1

(36)

where n++i

= ('""')AA [u+orA ___!!__.. - ut"'-f}-1 au-A A au-ooA

(37)

I

"'

is the conformal boost harmonic derivative (-y1 is an S0(8) 1 matrix). The vector harmonics are given by (38)

Their properties can be established with the help of (34), (35) and their corollaries. The covariant measure on S 8 is (39)

The on-shell fields are, e.g.,

j du+1 6u1"'cJ~ 17 (x++,u+), fJ,..(fl')opt/J13 = 0 Ft(x) = j du+l 6 u~Au!PcJ-tSAA(x++,u+), Ft = (r,...,)~F,...,,

'ljJ 0 (x) =

fJ"F,._ 11

5

= E,._•••.,).PfJ.,F>.p = 0.

(40)

Lorentz-harmonic supersymmetry

We begin by recalling how one constructs the representations of the D-dimensional supersymmetry algebra (41) in the massless case. One goes to the light-oone frame in which P,.. and decomposes (41) into

= (P-, 0, ... , 0),

Here Q'! = t(r±r=F)~Qp are the two light-cone projections of the D-dimensional spinor Qa. Clearly, half of the SUSY gener.atOI"S (Q!) anticommute with everything, so they do not take part in the construction of the Fock space. At· this stage the original symmetry SO(D -1, 1) is broken down to S0(1, 1) x.SO(D-2). If D > 3,

383 one can go one step further by dividing the remaining half ( Q;;) into pairs of creation and annihilation operators which satisfy the algebra

(43) Then one chooses a vacuum state 10 >, iimiO >= 0, and uses the generators qm to build up the Fock space 10 >, qmiO >, qmq,.jO >, · · ·, (qt!O >. Note that the number N of the creation operators is 1/4 of the original number of components of Q,. At the Fock-space stage the SO(D-2) symmetry is broken down to SO(D-4)x S0(2). We shall follow the spirit of the above procedure in order to construct a massless on-shell Lorentz-harmonic superspace. The resulting superspace will look very similar to the so-called light-cone superspace [12] (see the comments at the end of this section). The principal difference will be that we shall use Lorentz-covariant light-cone-like projections of the supersymmetry generators, obtained with the help of the spinor Lorentz harmonics. Take, for example, the case D = 3. The generator Q, is decomposed into Q± = u±Q,, after which (41) becomes

(44) (recall that for Lorentz-harmonic on-shell fields p++ = po = 0). The essential part of this algebra consists of the generators Q- and p--, has the form of N = 1 SUSY in one-dimensional space, and can be realized in the Lorentz-harmonic superspace: {x++,o+,u~},

.5x++=iu+ 0 t,e+, .58+=u+crta, .5u~=O.

(45)

A superfield 4>q ( x++, o+, ut) in this superspace has the expansion:

(>q(x++' e+' u~)

=

P~]18

= EijlctP'k/8.

(55)

Note that the superfield ~ is anti-analytic, i.e. it depends on 8['- only. The reader may have noticed the resemblance between the D = 10 Maxwell superfield (52) and the D = 4 N = 4light-cone superfield of [12]. The 9 structure is exactly the same. The principal difference is that~ (52) depends on one space-time variable ( x++) only. The rest of them have been replaced by the 8 harmonic coordinates u+ of 8 8 • This allows us to recover manifestly Lorentz-covariant fields from the components of~ (52) by means of the integral transform (20) (see (40)). Also, in the light-cone superspace of [12] only half of the SUSY algebra is realized linearly. In our case, after removing the harmonic dependence, we find supermultiplets of the full (80(D- 1, 1) covariant) SUSY.

385

6

Particles and superparticles

The on-shell Lorentz-harmonic (super)fields can be regarded as wave functions for quantized (super )particles. We shall show this in the simplest case D = 3. The particle action consists of two parts. The kinetic part

introduces conjugate momenta for XoiJ = (T")apx" and for the Lorentz harmonic variables u~. The latter satisfy the defining condition u+"u~ = 1, therefore there are 3 independent momenta for them (after quantization they become the harmonic covariant derivatives [4]). The second part of the action introduces the constraints which the wave function ~(P, u) is supposed to satisfy. Those are: p++~ := P0 pu+"u+ 13 ~ = 0,

pD~ := Papu+"'u- 13 ~

= 0,

D0 ~

= q~,

n++~ = 0.

(57) It is easy to check that these are first class constraints. The first two of them can be replaced by an equivalent pair of constraints: p++~ =

0, P2 ~ = 0.

(58)

Indeed, P 2 = _p++ p-- +(p0) 2 , so (58) implies (pO? = 0-+ p0 = 0 (the same will be true in higher dimensions, where p0 will be replaced by the transverse momentum P;, (.P;)l = 0 -+ P; = 0). The third constraint means that the wave function carries an S0(1, 1) weight, so it is defined on the coset S0(2, 1)/S0(1, 1). The last constraint restricts~ to the compact harmonic space S 1 "'S0(2, 1)/[S0(1, 1) x K]. All of those constraints are introduced into the action by Lagrange multipliers:

The quantization is straightforward. The momenta Pap, n++, n--, D 0 become space-time and Lorentz-harmonic derivatives. After that the constraints (57) imply that the wave-function is a Lorentz-harmonic field ~q' (x++, u+ ), which describes a massless field, if the weight q' is an integer :::; -2 (see (30)). Note that the classical value of q in (59) does not necessarily coincide with the quantum weight q' of the wave function, due to ordering ambiguities in the operator D 0 = u+"'a;au+a- u-aa;au-a. The supersymmetrization of the action (56), (59) is very easy. One replaces ±"'13 in (56) by the supersymmetric one-form ±"'13- ifJ(ajJI3) and introduces a kinetic term -iDaiJa for 8"' ( after quantization Da becomes the spinor covariant derivative Da = ofotr- PQfJfJP). Finally, one adds the Grassmann constraint w-u+Q Da (meaning

386

that .Pis a function of e+ = u+"'8a only) to (59). The resulting superparticle action is

S

=

j dr[Pap(x"'P- i8, 4>*). It should satisfy

•

Sc~(l/>)

= S(4>,0): the classical action is recovered when all antifields are set to

zero. • The extended action should satisfy the master equation, which at the classical level reads ( S, S) = 0.

::a.,

BpS is a matrix of rank N • S is a proper solution, which means that S"'IJ on the stationary surface where N is the number of fields 4>A. The last condition gives the possibility of a gauge fixed action. In fact, the gauge fixed action will just consist of the same extended action, but where we choose other coordinates as fields, such that theN non-trivial directions of S"p(z) = TJ"'"'S-,p(z) are fields rather than antifields. The second requirement implies that on the stationary surface S"' p( z) satisfies (8)

where ~ means modulo field equations of the extended action. A nilpotent matrix of size 2N x 2N has rank $ N, and only rank N if all its zero modes are contained in the matrix itself. This implies that we have for arbitrary smooth local functions v" ( z) the implication (9)

for some local smooth functions w"'(z). In our example we have thus to extend the action by terms depending on anti fields, because the gauge invariance implies that there is a zero mode. We have to extend the action such that this zero mode appears as S'c where cis a new field to be introduced, which is the ghost of the symmetry. So the extended action is

S(4>, 4>*)

=j

dt P"'.ic,..-

~gP"' P,.. + x;cP" + g*c.

(10)

In this case we are already finished. It is clear that the terms linear in the antifields are the BRST transformations of the corresponding fields. This extended action satisfies the master equation, and it is proper. The master equation includes in this case just the invariance of the action under the symmetry. But in more complicated theories the same principles include also the closure of the algebra and all the relations found in the previous years about open algebras, and other similar complications. The properness implies that all zero modes, ... have to be included. The master equation

392 can always be solved perturbatively in antifield number (that is the same as the ghost number for the antifields and zero for fields.) When we have obtained the extended action, the physical variables are represented by an antibracket (AB) cohomology at ghost number zero. phys. variables => local AB coho at ghost nr. 0.

(11)

This means the following. The operation which consists of taking an antibracket with S is nilpotent. Indeed from the Jacobi identity (S, (S, F)) = !((S, S), F) = 0. The antibracket cohomology are the local functions F which have SF= (F, S} = 0 and where two solutions are equivalent which differ by (S, G), where G is any local function. Cohomology is now defined with= instead of~ in eqs.(3-4). This will allow to do field redefinitions which change the field equations. For general gauge theories the fact that the physical variables are equivalent to this antibracket cohomology is proven using the language of homological perturbation theory in [6, 5].

9.2. Canonical transformations In the example it is easy to define the fields such that the action is gauge fixed. We have at this point the bosons XP., PP., g and c* and the fermions x;, P;, g* and c. It is clear from Eq. 10 that we have a non-trivial kinetic term between g* and c. Thus we will define g* as a field, which we will call b : g*

= b;

g = 1- b*.

(12)

This is a canonical transformation which means that the transformation preserves the brackets : calculating brackets in the old or new variables is the same, or in otht'r words the new variables also satisfy Eq. 6. Therefore they also conserve the master equation (S, S) = 0. Canonical transformations from {~~·} to {~~~ 1*} for which the matrix

J:~

I.,.

is invertible, can be obtained from a fermionic generating function F( ~. ~'*). This useful formulation [2] is explained in detail in the appendix of [3]. The transformations are defined by ~~A= oF(~,~~·) ~· - oF(~,~~·) (13) o~A

A-

o~A

.

Canonical transformations are an important part of the formalism. In this one concept we find several steps which people do in quantisation procedures. • Point transformations are the easiest ones. This are just redefinitions between the fields ~~A = JA(~). They are obtained by F = ~AJA(~) which thus determines the corresponding transformations of the antifields. The latter replace the calculations of the variations of the new variables. • Adding the BRST transformation of a function sill(~) to the action is obtained by a canonical transformation with F = ~A~A +Ill(~). The latter gives ~lA =~A ;

~~

= ~~ + OAIII(~).

(14)

393

• Redefine the symmetries by adding equation of motions ('trivial symmetries'). This is obtained by F = 4>~4>A + 4>~4>~hAB(4>) (15) (the first term is the identity transformation). • Elimination of auxiliary fields can be done by canonical transformations (see appendix B of [4]). This procedure will then also give the 'compensating transformations'. The canonical transformations keep by definition the master equation invariant, and because they are non-singular, they also keep the properness requirement on the extended action. Of course in the new variables, we do not see the classical limit anymore. But the most important property is that the antibracket cohomology is not changed. This should be obvious from the definition. 3.3. The gauge fixed theory After our transformation Eq. 12 we end up with the following extended action:

S

= jdt P"X,. -

2 +be+ X*cP" !p 2 ,.

2• + 1b*P 2

(16)

Now the action obtained by putting all antifields to zero is 'gauge fixed'. For a gauge fixed action one can prove [5, 6) local AB coho =*" local BRST coho.

(17)

The antibracket cohomology is now represented by the BRST cohomology for some operator

n4>A

as I

= o4>·

(18) +•=O The combination of Eq. 11 and Eq. 17 imply that the physical variables arc represented by the local BRST cohomology at ghost number zero. In contradistinction to the antibracket cohomology we have to use the field equations (of this gauge-fixed action) in the analysis of the BRST cohomology. A

3 ...f. Trivial variables We have done here the gauge fixing by just one canonical transformation : Eq. 12. This is usually possible for actions which are linear in derivatives. But in general it is not always possible to find suitable covariant variables. Therefore we add new trivial variables. Trivial variables are a set of fields with corresponding antifields, for which terms are added or subtracted from the extended action. They are separately solutions of the master equation and carry no antibracket cohomology. The simplest examples are adding a bosonic A and fermionic b and corresponding antifields by an extra term

394

in S of the form b* -\. One checks that this trivially satisfies the master equation, that one has changed the number of fields + antifields by 4 and increased the rank of S01 p by 2, and that there is no change in the anti bracket cohomology. Other examples are just adding a bosonic-\ with S = -\ 2 or a fermionic b with S = (b*) 2 • The addition of such trivial b, -\ sectors is part of the scheme which Batalin and Vilkovisky suggested [1] to obtain the gauge fixed action. Then they propose to do a canonical transformation of the type Eq. 14. If a 'gauge fermion' can be found which satisfies certain conditions then the action is gauge fixed. However, for a givf'n gauge theory it is not clear that such a gauge fermion exist. In other words, choosing the right variables for the gauge fixed theory has an arbitrariness and there is no guarantee that such a covariant basis exist. 3.5. The superparticle As an example we mention that such a procedure has not yet been found for the Brink-Schwarz (BS) superparticle action (forD= 10) [11]

Sc1

•

= P" X,. -

•

1

0 f/0 - 2gP" P,..

(19)

The classical variables are the coordinates X,., their conjugate momenta P 11 , the einbein g and the fermionic variable 0 which is a ten-dimensional Majorana-Weyl spinor. The action is infinitely reducible, which means that one needs an infinite number of ghosts, called O,o with p = 1, ... , oo to be added to the classical variables in order to obtain the minimal solution for the extended action [12, 13]. Then one adds fields as suggested by Batalin and Vilkovisky. That implies two infinite pyramids of fields. Steps which people took to get a 'gauge fixed action' are now recognised as being no canonical transformations or adding variables which are not 'trivial' [4]. Therefore the BRST operator which was found for the final gauge fixed action had extra solutions for its cohomology which do not correspond to physical variables of the classical action. The spinor variables in the gauge fixed action are given in table 1 with their ghost number. All the fields with 2 non-zero indices have been introduced as trivial sectors. Nevertheless, there were counting arguments in favor of this set of fields based on considerations about orthosymplectic symmetry [14]. And indeed for this gauge fixed action another BRST operator has been found [15], which has the 'right cohomology ', that is the fields occurring in the d = 10 super-Maxwell theory. However, this BRST operator does not follow from a quantisation procedure on the BS action. The gauge fixed extended action at this point is of the form

(20) where CAB is a constant non-singular matrix, with inverse CAB and n is the BRST current. We can then perform again a canonical transformation to a classical action. It is in fact the generalisation of the inverse of Eq. 12. All the fields of negative ghost number in table 1 are replaced by their antifields. E.g. the antifields of the fields of ghost number -1 become in this way fields of ghost number 0, and will be called

395

Table 1: Spinor fields in the superparticle gauge-fixed action.

-3

-2

0 8

-1

on

2

1

8to

j21

8u

931

922

820

830

821

..\

.vo

_xn

).20

).30

3

,X21

An ..\21 ..\22

..\31

gauge fields (as there was g in our example of the particle). So after this canonical transformation which is the inverse of a gauge fixing, all the fields of ghost number 0 are now classical fields. The classical action is then [16]

sc1 =

P

x - ~9P

2

00

+ 2: ..XPOp p=O

00

00

p=O

p=O

- 2: ,\P f(P- 2: {,\~'+1 -

8p+l ~ + ,\P + 8p f}7]p

(21)

=

where ,\P ).P.P, 8p = 8p,p, (P = -X~1,p and 7]p = 9~1,p+l' The full extended action follows from straightforward application of the quantisation principles. All the remaining fields in the table (or their antifields for the negative ghost numbers) now occur as minimal ghosts or ghosts for ghosts. There are a double infinite set of fields, of symmetries, of zero modes of this symmetries, ... , which is the reason why we have called this formulation of the superparticle the DISP (Doubly Infinite Symmetric superParticle). It has the same physical variables as the BS action, but allows a straightforward quantisation because in the Hamiltonian language there are only first class constraints. Other similar approaches, where the BS action has also been replaced by another classical action, allowing a consistent quantisation, have been given in [17].

4. Anomalies at the formal level In the full quantum theory, the master equation gets replaced by

(W, lV)

= 2i1i.AW,

(22)

396 where

~=nts of the metric, for which we take h ++-

9++ 9+- + V9

(31)

and its + +-+ - interchanged, and their antifields. The only solutions which can not be absorbed in a counterterm and which have the correct dimensionality can be written as (32) and its partner AR. If the right-left symmetry is not broken, then we can only find an anomaly proportional to AL + AR. Going back to the original variables 9a/3• ~ and c, and by adding local counterterms M, this can be written as (33)

This is the usual expression of the dilatation anomaly. In the conformal gauge = T/a/3• this expression vanishes, but this does not mean that there is no anomaly. This can be avoided by introducing a gauge choice 9a/3 = Pa/3 where p is some background metric. The anomaly is then a functional of this p. In the present scheme, this procedure does not make sense : the anomaly is a functional of the fields, not of background fields (and can thus be shifted by local counterterms, which are also field-dependent). The conformal gauge is obtained in a way analogous to Eq. 12 for the particle : g*"' 13 = -bOt/3 j 9a/3 = T/a/3 + b:/3. (34)

9a/3

The field g occurring in Eq. 33 is then a £unction of the antifields. This reinterpretation allows one to determine the form of the anomaly in any gauge, using canonical transformations. So t.he next question is : how do these affect the expmsion for the anomaly ?

398 Canonical transformations do not leave the !:l. operation invariant. But we have (35)

5. Regularisation 5.1. Introducing and eliminating Pauli- Villars fields We will use a Pauli-Villars (PV) regularisation. It will allow us to make contact with the work of Fujikawa [18] on obtaining anomalies from the non-invarianc~ of t.he measure. The consistency of the PV scheme will then imply that the obtained expressions satisfy the consistency equations, a point which is very unclear in Fujikawa's method. To start, we introduce PV partners for all fields and antifields. So we have z

={

4>A , 4>'A } ;

(3G)

where the latter are the PV fields. For the calculations at one loop it is sufficient to give the extra recipe that loops of PV fields produce an extra minus sign which reflects itself in a modified definition of !:l. : f:l.:: (-)A aA aA- (-)A_i!__i!_,

ax A ax A

(37)

To obtain this sign one can modify the definition of the path integral over these fiE-lds [7], or else introduce extra sets of fermions and bosons. The first method is certainly the simplest when we are only looking to one loop. The PV fields then have the same statistics as the ordinary fields, but one can say that the integration over these fields in the path integral is defined differently, such as to produce a minus sign. This is consistent for fields which occur only in loops, i.e. quadratically in the action. Th~ second method produces the minus in the loop by having opposite statistics for PV fields. E.g. for regularising a real boson field, we have to introduce a fermion. But as the kinetic operator of a boson is a symmetric operator, this would vanish for a fermion. Therefore one has to introduce a complex fermion (or 2 reals). Then one has over-compensated the loop of the original boson in the regularisation procedure. One thus introduces an extra PV boson. So in summary, one has introduced e.g. for a boson field t/J, 3 PV fields : the fermions XI and X2 and the boson Xo· Each of them have also their antifields. But again, this complication can be forgotten when considering 1 loop anomalies : one may just treat the PV fields as having the samE' statistics, and insert the sign by hand.

399 The PV fields are introduced as trivial systems in the limit M -+ oo. We add to the action a term M 2xATABXB where Tis an invertible matrix. This implies that the x• fields are not invariant under S while the x fields are in the image of this operator. In the language of Feynman graphs, these fields are thus very massive. Also, these fields should provide a regularisation of the original action. Therefore, neglecting the mass terms, they should produce the same vertices as the original action. The regularisation now consists in postponing to take the limit M -+ oo to the end of the calculation. We will define

{38) Let us first look at the massless part. It is the w 2 part of the action S~"9

=! (S(z + w) + S(z- w)) = S(z) + stv + O(w4 )

{39)

which automatically satisfies then the classical master equation. When looking only at one loop (the only case which we consider here), one can forget all terms of order w 4 • In the formulation with 3 PV fields for each ordinary field, of which and have opposite statistics from ¢A and x0 has the same statistics, we would write

xt

SoPV - 8t 82S + 2~-8o 8oS.

xt

{40)

where we defined the operators

The notation B2 implies also that the corresponding w fields appear at the right of S. This satisfies the master equation, and also f1SPV = 0, where we do not have to modify 6. this time. We can introduce this action from the start, even before gauge fixing. The reason is that introducing a PV regulator in this way 'commutes' with a canoniral transformation. By this we mean the following. Suppose that one has introdurro the PV fields using the above prescription and then performs a canonical transformation. Any canonical transformation on the z fields can be generalised to a canonical transformation on the z and w fields such that after this canonical transformation the same result is obtained as when introducing the PV sector only after the canonical transformation. So we have the following scheme S(z) ! PV S(z) + !w"SapwP

can.--+ trans£.

S(z) PV

! can. trans£. --+

S(z')

+ !w'"Sapw'P

So this part is fixed immediately when giving the theory.

(42)

400

On the other hand, for the mass term there is a lot of arbitrariness. First of all we have separated here fields from antifields. This should be done suitably for gauge fixing, i.e. SAs should have rank N. But once this has been specified Tis an arbitrary non-singular matrix, which may depend on fields or antifields. We will claim that the results for the anomalies for different choices ofT ('different regularisations') differ only by the variations of local counterterms. In other words, the anomalies do not change in the cohomological sense. Let us now look again at the full master equation for the regularised action

(43) First of all, due to the definition Eq. 37 one has l:l.S = O(w 2 ). But the anomalies now come from the violation of the 'classical master equation' due to the mass terms. This will be proportional to w 2 • Removing the PV fields by integrating them out will replace w 2 by a term of order 1i. The violation of the master equation (to order w 2 ) is given by

-i1iA = = =

l(sr•g, sreg)- i1il:l.sr•g (Sfj,S~e9 ) + O(w 4 )

+ 1iO(w2 )

-M 2 xATAsS 8 orW 01 - !M 2 xA (TAs. S(z)) x 8 ( - ) 8 +0(w 4 ) + 1iO(w2 ),

(44)

where S 8 "' is the left derivative of S w.r.t. cf>8 and right w.r.t. z 01 • Now we remove the PV fields. In a path integral this would mean that the XX terms are replaced by their propagator (which gives a 1i ). The x• terms are dropped in this step. We follow this idea and defining

(45) we obtain (46) Note that for gauge theories with a closed algebra K is the matrix of the derivatives of the transformation of fields w.r.t. the fields. In the limit M -+ oo we take the trace of this expression (for a field independent matrix T). Now we will regularise this using the regulator 0. The PV fields were a way to obtain a regularised definition of l:l.S. Without the PV fields (S, S) = 0 and ll.S =F 0, and we define ll.S as the expression in Eq. 46. Notf' that if we first take the limit M-+ oo (and forT a constant matrix), it corresponds indeed to the unregularised definition of l:l.S. 5.2. The integrals and the Fujikawa regularisation

401 We then replace the propagator by an exponential function. This will allow us to make contact with the Fujikawa calculation of anomalies (18]. 1 1-0fM2

= lo{

00

2

d>. exp(>.O/M )exp(->.).

For a local action, we split A= (a,x), B

(47)

= (b,y) and

where J is some differential operator. Also for the propagator we have (49) The anomalies are at this point A

=

fo"" d>. e-~ j dx j dy str Jt(y)c(x- y) · exp ( >.";i~)) c(x- y) fo"" d>. e-~ Jdx Jdy str c(x- y)J(x) exp ( ).";;:)) c(x- y)

(50)

where J and 'R are now considered as matrices in the a, b, and str denotes a supertract' over these indices. The · indicates that derivatives in the operators do not act further. Several lemma's have been obtained to calculate expressions of the form Eq. 50. I believe that the one in [9] contains most of the others. However, the formulas in [19] do in fact contain all this, and could be used to generalise them even furtht>r. One obtains an expansion in M 2 • The divergent terms when M 2 -+ oo can usually be eliminated by a local counterterm Ml! but the PV procedure introduces several copies of PV fields with masses adapted such that these terms disappear anyway. So we can forget about them. One discards also the terms which vanish in this limit. and the result is thus independent of M 2 • From the last expression, one can :;ec that the M 2 independent terms have only >.occurring in the first factor, and the integral over >. gives thus 1. As already mentioned, the above regularisation scheme is very similar to Jo'ujikawa regularisation [18] and to the heat kernel approach (20]. These approaches have to use special variables in order to avoid anomalies in 'preferred symmetries'. The Jacobian which is regularised in the Fujikawa approach corresponds to the first tt-rm in Eq. 48. By including the second term in that equation we can avoid this restriction to the 'Fujikawa variables'. 5.3. The Green-Schwarz superstring and the light-cone gauge A contradiction with the above statements on gauge-independence of anomalies seems to exist in the Green-Schwarz superstring. The classical action can bt> gaugt'fixed in the light-cone gauge [21] or in the so-called semi-light-cone gauge [22], both of which destroy the manifest rigid space-time super-Poincare invariance. In the former

402 there are no space-time anomalies. In the latter the local fermionic ~~:-symmetry is fixed by the unitary gauge r+o = 0, while the other local worldsheet symmetries are covariantly quantized. It was claimed by Kraemmer and Reb han that anomalies in the semi-light cone approach do m;t cancel [23). On the other hand, M. Chu [24) claimed that in a Hamiltonian formulation, the definition of the Lorentz generators can be changed in order 1i such that anomalies disappear. Using our methods (although we did not yet completely formulate it in the way described above) we redid the calculation [9) and obtained .A= -12 2!"AL. Adding local counterterms this anomaly could still be moved to an anomaly in global Lorentz symmetries, but it could not. be canceled. There seems therefore a contradiction between gauge independence and the different results in light-cone and semi-light-cone gauge. The difference between these two gauges is in the bosonic sector. The arguments why they should be the same should go by using canonical transformation as in the general theory written above. One could even ask the question just for the bosonic string. The calculation of anomalies in the conformal gauge, which was reviewed above gives as a final result that anomalies cancel for D = 26. On the other hand in the light cone gauge this requirement comes from the analysis of the quantum rigid space-time Lorentz algebra. Jt only closes when D = 26. Are these two calculations connected ? Is there a canonical transformation between both ? We do not have a final answer to this question, bnt it seems that one can not avoid non-local canonical transformations to eliminate t.he ghosts in going to the usual light-cone gauge fixed action [10). Therefore, we can see no contradiction with the statements above. If one does not perform local canonical transformations, it is not clear that anomalies should be conserved when going to ihf' light-cone gauge. 6. Conclusions

The Batalin- Vilkovisky formalism is a convenient framework for the quanti:sation of gauge theories. One first builds an extended action, function of fields and antifieldl:l which satisfies 3 requirements : a classical limit, the master equation and a properness condition. The latter implies that there are enough non-trivial directions in the Hessian of this extended action, such that when choosing (by a canonical transformation) the 'fields' as those directions in the extended space and antifields as the ot.her directions, one obtains a gauge fixed action. Using antibracket cohomology onE' can prove that this action describes the same physical states. For the quantum theory one needs another master equation. A theory has no anomalies if a local action can be found which satisfies this equation. Possible consistent anomalies are solutions of antibracket cohomology equations at ghost number 1. For local actions, the definition of the operation 8 which occurs in the master equation needs regularisation. By introducing Pauli-Villars fields one can motivate

403 the regularised definition

D..S

= J~! dx j

dy str 8(x- y)J(x)exp

c~:)) 8(x- y)

(51)

where infinite terms are removed. J and 'R. are given by Eq.45,48, 49 using an al·bitrary matrix T which 'determines the regularisation'. The anomaly is then a function of fields and antifields, and does not change under local canonical transformations (in the cohomological sense), in particular they are gauge independent. They are also independent of the regularisation, in particular of the choice of T. For the superparticle and for the superstring we found difficulties with the BrinkSchwarz and the Green-Schwarz actions. For the BS superparticle, which is a good formulation in the light-cone gauge, we could not find canonical transformations or the right trivial variables to obtain a gauge fixed action. We found however another action, the DISP superparticle, which describes at the classical level the same physical states, and allows a straightforward quantisation [4]. Other modified suitable actions have been found by other groups [17]. For the superstring also the Green-Schwarz action, which is a good formulation in the light-cone gauge, has anomalies in a semicovariant gauge (covariant in the bosonic sector). This is not in contradiction with the general results, as going from the covariant to the light-cone gauge involves nonlocal canonical transformations. So far also no gauge-fixing procedure has been found for the Green-Schwarz action. Probably we also need in this case a modified action similar to those for the superparticle.

7. References 1. I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D28 (1983) 2567 (E:D30 (1984) 508). 2. I.A. Batalin and G.A. Vilkovisky, Nucl. Phys. B234 (1984) 106. 3. W. Troost, P. van Nieuwenhuizen and A. Van Proeyen, Nucl. Phys. B333 (1990) 727. 4. E.A. Bergshoeff, R. Kallosh and A. Van Proeyen, preprint CERN-TH.6020/91, SU-ITP-888, KUL-TF-91/5. 5. J.M.L. Fisch, On the Batalin- Vilkovisky antibracket-antifield BRST formalism and its applications, Ph.D. thesis, ULB TH2/90-01. 6. J. Fisch, M. Henneaux, J. Stasheff and C. Teitelboim, Commun. Math. Phys. 120 (1989) 379; J. Fisch and M. Henneaux, Commun. Math. Phys. 128 (1990) 627. 7. A. Diaz, W. Troost, P. van Nieuwenhuizen and A. Van Proeyen, Int. J. Mod. Phys. A4 (1989) 3959. 8. M. Hatsuda, W. Troost, P. van Nieuwenhuizen and A. Van Proeyen, Nucl. Phys. B335 (1990) 166.

404

9. F. Bastianelli, P. van Nieuwenhuizen and A. Van Proeyen, Phys. Lett. B253 (1991) 67. 10. T. Kugo, S. Schrans, W. Troost and A. Van Proeyen, work in progress. 11. R. Casalbuoni, Nuovo Cimento 33A (1976) 389; L. Brink and J.H. Schwarz, Phys. Lett. B100 (1981) 310. 12. R. Kallosh, Phys. Lett. B195 (1987) 369. 13. U. Lindstrom, M. Rocek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B224 (1989) 285. 14. I. Bars and R. Kallosh, Phys. Lett. B233 (1989) 117. 15. E.A. Bergshoeff, R. Kallosh and A. Van Proeyen, Phys. Lett. B251 (1990) 128. 16. R. Kallosh, Phys. Lett. B251 (1990) 134. 17. A. Mikovic, M. Rocek, W. Siegel, P. van Nieuwenhuizen, J. Yamron and A.E. van de Ven, Phys. Lett. B235 (1990) 106; F. EJ3ler, E. Laenen, W. Siegel and J.P. Yamron, Phys. Lett. B254 (1991) 411; F. EJ3ler, M. Hatsuda, E. Laenen, W. Siegel, J.P. Yamron, T. Kimura and A. Mikovic, preprint ITP-SB-90-77; M.B. Green and C.M. Hull, Mod. Phys. Lett. AS (1990) 1399, and in Strings '90, eds. R. Arnowitt et al. (World Scientific, Singapore, 1991) p.133; N. Berkovits, Phys. Lett. B247 (1990) 45. 18. K. Fujikawa, Phys. Rev. Lett. 42{1979)1195; 44(1980) 1733; Phys. Rev. 021(1980)2848 (E : D22 (1980) 1499). 19. P.B. Gilkey, J. Diff. Geo. 10 (1975) 601; lnvariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem (Publish or Perish, Inc.) Math. Lect. series no. 11, 1984. 20. M. Liicher, Ann. of Physics 142 (1982) 359; J.W. van Holten, J. KowalskiGlikman and D.N. Petcher, Nucl. Phys. B309 (1988) 680; K. Fujikawa., U. Lindstrom, N.K. Nielsen, M. Rocek and P. van Nieuwenhuizen, Phys. Rev. D37 (1988) 391. 21. M. Green and J. Schwarz, Nucl. Phys. B243 (1984) 285. 22. S. Carlip, Nucl. Phys. B284 (1987) 365; R. Kallosh and A. Morozov, Int. J. Mod. Phys. A3 (1988) 1943. 23. U. Kraemmer and A. Rebhan, Phys. Lett. B236 (1990) 255. 24. M. Chu, Nucl. Phys. B353 (1991) 538.

CFT and Related Topics

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407 SOME ASPECTS OF FREE FIELD RESOLUTIONS IN 2D CFT WITH APPLICATION TO THE QUANTUM DRINFELD-SOKOLOV REDUCTION PETER BOUWKNEGT

CERN-TH, CH-1P.11 Geneva ~3, Switzerland JIM McCARTHY

Department of Phy$iC$ 1 Brandeu Univer$ity Waltham, MA 0~~54, USA KRZYSZTOF PILCH

Department of Phy$iC$1 Uni1Jer8ity of Southern California Los Angele$, CA 90089-0484, USA

ABSTRACT We review some aspects of the free field approach to two-dimensional conformal field theories. Specifically, we discuss the construction of free field resolutions for the integrable highest weight modules of untwisted affine Kac-Moody algebras, and extend the construction to a certain class of admissible highest weight modules. Using these, we construct resolutions of the completely degenerate highest weight modules of W-algebras by means of the quantum Drinfeld-Sokolov reduction. As a corollary we derive character formulae for these degenerate highest weight modules.

1. Introduction Free field techniques have been widely used in the study of two-dimensional conformal field theories. Realizations of the chiral algebra by free fields, resolutions which project onto its irreducible representations from the Fock space modules, and chiral vertex operators intertwining between resolutions, form the tools of this approach (see the review [1], and references therein). In this paper we intend to highlight several points which have not been emphasized in the literature, and refer the reader to [1] for a more complete discussion of the basics. The main theme will be the application to quantum Drinfeld-Sokolov reduction, which allows one to investigate virtually all properties of W-algebras using corresponding properties of Kac-Moody algebras. Our presentation of this

408

application is very much inspired by, in particular, references [2-8] in which most of the results can be found. The paper is organized as follows. In Section 2 we review the free field realizations of affine Kac-Moody algebras, and their associated screening operators. The construction of intertwining operators between free field Fock spaces, and how these can be used to build a complex of free field Fock spaces yielding a resolution of an irreducible highest weight module, is explained in Section 3. Here we will not restrict ourselves to integrable highest weights - relevant for the corresponding WZW-model -but also treat a class of admissible weights introduced by Kac and Wakimoto [9-11). This is the class of weights relevant for the quantum DrinfeldSokolov reduction since they lead to completely degenerate highest weight modules for the corresponding W-algebra. This will be discussed in Section 4. Finally, in Section 5, we apply the results of Section 4 to derive character formulae for the completely degenerate W-algebra modules, and compare them to the characters obtained in the coset model approach to W-algebras. For basic notations used throughout this paper the reader is invited to consult Section 1.1 in [1).

2. Free field realizations of affine Kac-Moody algebras In this section we briefly outline the construction of realizations of the ( untwisted) affine Kac-Moody algebra g on free field Fock spaces. The first example of such a realization was constructed by Wakimoto for s1{2) [12]. In general one proceeds as follows [13,14). Let G be the (complex) group corresponding tog and let B_ be a Borel subgroup. A character XA : B_ -+ (C* defines a holomorphic line bundle over the flag manifold B_ \G. The group G, and thus also g, act on (local) sections of this line bundle. Upon introducing complex coordinates z 0 , a E ~+• on a maximal cell of B_ \G this becomes a realization of g in terms of linear differential operators. Restricting this realization to polynomials in Z 0 provides us with the "analogue" free field realization for finite dimensional Lie algebras. 1 This realization can be lifted to a realization of the affine Kac-Moody algebra gat level k [13]. Specifically, introduce a set of bosonic first order fields (,8°(z ), -y 0 (z)), a E ~+• of conformal dimension (1, 0) (corresponding to a~.. and z 0 , respectively), and a set 1

See [1] for an algebraic equivalent of this construction.

409 of rankg =f. scalar fields 4>i(z), with operator product expansions -y'(z),B'(w)"' haa' /(z- w) and ;(z)i(w) "' -6ii ln(z- w). Introduce furthermore the Fock spaces :FA = :Ft ® :FfJ-, that are freely generated from the vacuum lA) by the oscillators .B~ , a~ for n < 0, 'Y~ for n $ 0, and labeled by the momentum zero mode of the scalar field p;IA) = a+A;IA). Here a.+ 1 = Jk +hv, and we will also use the notation a_, where a+a- = -1. The following expressions for the Chevalley generators define a realization of gr. on :FA of highest weight A e;(z)

= pa;(z) + ... ,

h;(z)

= -a-(a~ · io(z)) +

L

(a,

aEA+

f;(z) = a_-ya;(z)(a~ · io(z))

an: -ya(z).B.. (z):,

(2.1)

+ ....

Suppose we introduce a degree by deg(,B') = -1 ,deg('Y') = 1 ,deg(i) = 0. Then in e;( z) the dots stand for terms of deg 2;:: 0 in terms of ,B-y-fields only. The term of deg = -1 is basis independent while the higher degree terms depend on the particular choice of basis. The dots in /;(z) stand for terms of deg $ 1 in terms of ,B-y-fields only, and are basis dependent (unfortunately even the terms of highest degree). They are however completely fixed once a particular choice for e;(z) has been made. The Virasoro algebra acts on FA by means of the Sugawara construction, which in the realization (2.1) takes the form

T(z)

L :,B'(z)O-y'(z):

= -t : 81/>(z). o(z): -O+P. ilf4>(z)-

(2.2)

aEA+ The Lo-eigenvalue of the Fock space vacuum lA), i.e. its conformal dimension, is given by hA = !a~(A,A + 2p). Finally, we need operators acting between Fock spaces labeled by different weights A - the so-called screening operators - which will be used to build operators that intertwine with the action of gr.. In the flnite-dimensional analogue problem they arise from the (left) action of n+ on B J, giving differential operators which clearly commute with n+ in the realization. Lifting these to gr. gives operators of the form

s;(z)

= _ (.Ba;(z) + ... ) e-ia+ad>(z),

i

= 1, ... ,f.,

(2.3)

410

where the dots stand for terms of deg ;;:: 0 in terms of ,B-y-fields only. The operator products of s;(z) with e;(z) and h;(z) are regular, while

.,., 2(k + h v) (-1- -i+;·f>(w)) f,1·( Z )81·(W ) ""'-oiJ ( ) Uw e • a;,a; z-w

(2.4)

For g~: = sl(,;h, (2.4) can be verified directly, using the explicit realization of the currents f;(z) and s;(z) [15,14). In the general case this result, conjectured in [1), can be proved by observing that an algebraic field redefinition reduces the computation to an sl{2) subalgebra [7).

3. Intertwiners and resolutions The operator product expansions of s;(z) with e;(z), h;(z) and /;(z) discussed in the previous section provide the starting point for constructing intertwiners, which are mappings between Fock spaces that commute with the action of the current algebra. In particular (2.4) suggests we consider the (formal) operators

QA,A·(C): FA-+ FA•, QA,A•(C) = (s; 1

...

s;") =

fc dzt ... dzn

Si 1 (zt)

... s;"(zn),

(3.1)

where A' = A- .8 (note that J3 = Ei O'j is a sum of po&itive roots), and C is a suitably closed multi-contour as discussed below. More precisely, when acting on a vector v E FA,

Q A,A' (c)V =

"L_;

lc{ dZt • • • dZn

lt, ... ,lnEZ C

II< II

ZJ: - Z/

1: ... > lzn I (for z; =/= 1) where z; is integrated counterclockwise from 1 to 1 around the origin z = 0. The ambiguity in the phase of the integrand is fixed by analytic continuation from the region on the positive real half axis where 0 < Zn < ... < z 1 • With these conventions the operators [ s;, ... s;,] and [ s,.(id ... s,.(;,j) related by a permutation 1r are not necessarily proportional. We will show however that within [ ... ] the s; 's satisfy the Serre relations of U9 ( n_ ), thus clarifying, in this "dual" context, that the space of potential intertwiners is isomorphic to a quantum group Verma module. Observe that upon interchanging two screening operators s; and s i one picks up a phase factor q(;,;l. This motivates the following definition of the adjoint operator (3.12) More generally (ad s;)x

= s;x- q(n, xvs, LA) ~ L'{:', where L'{:' is an irreducible highest weight module of the algebra W~:[g]. 6 We will provide some evidence in support of this conjecture in Section 5. Assuming then that H~n)(n, xvs, LA) ~ L'{:' sn,O' there exists a natural candidate for a resolution of L '{:', obtained simply by applying the functor Hq(n, x vs, · ) to a resolution of the i~:-module LA. So suppose, following Section 3, that we are given a resolution of the i~:-module LA where the differential dis constructed out of the screening operators s;(z) and the terms are free field Fock spaces .rln) :::; J1n) ® Ffl'Y. Then, using the fact that [ Q, d] = 0, as well as the result (see e.g.

[4-6]) (4.9) 5 In the case of W-algebras it is conventional to parametrize highest weight modules through Aa+ = Aa+ + Aa_. 6 For weights A not of the form (3.23) this is not necessarily true. One can show, for instance, that for dominant integer weights H~n)(n, xvs, LA) ~ 0 for all

n E ~ [15,6]. Also, the fact that H~)(n, xvs, LA) is an irreducible W-algebra highest weight module is not a priori obvious but nevertheless appears to be true for weights of the form (3.23).

419 we have

Lr/ ~ Hq(n,xvs,LA) ~ Hq(n,xvs,Hd(J1.*>• ®J="fl'Y))

~ Hd(Hq(n, xvs,

J1•>• ® :Ffl'Y)) ~ Hti(J{*>•).

(4.10)

That is, we have constructed a resolution of theW-algebra module Lr/ in terms of free field Fock spaces J{n>•. In fact, one can show that in Q-cohomology (4,5] (4.11)

as well as (4.12) Equation (4.11) implies that the differential d of the resolution (4.10) can be replaced by the differential J obtained from d by replacing the screening operators s;(z) by their Q-cohomologous counterparts s;(z) in terms of scalar fields tjJ(z) only. Furthermore, since the W-algebra is exactly the commutant of the screening charges j s;( z) as remarked before, this confirms that the W-algebra commutes with J and thus acts on all the terms in the resolution. For g = sl{2) the differential J can be recognized as the differential of Felder's complex that provides a resolution of the Virasoro degenerate highest weight modules (33]. This observation constitutes a proof of (4.10) for sl{2), as has been discussed in (4]. Equation (4.12) gives an explicit expression for the Virasoro part of this commutant and confirms equations (4. 7) and (4.8). For g = An one can find a generating function for the other generators Wk(z) of theW-algebra by means of the so-called quantum Miura transformation (2,3] n+J

L Wk(z)(ao 3. It is hoped that this will provide a new playground in the search for a realistic string theory as well as new models for critical phenomena. Acknowledgements: It is my pleasure to thank the organizers of this conference for their hospitality. Discussions with M.B. Halpern and R. Cohen were very helpful.

446

REFERENCES 1. G.V. Dunne, I.G. Halliday and P. Suranyi, Phys. Lett. B213 (1988) 139.

E. Kiritsis, MDd. Phys. Lett. A4 (1989) 437. 2. M.B. Halpern and E.

Kiritsis~ Mod.

Phys. Lett. A4 (1981) 1373; Erratum

ibid. A4 (1989) 1797. A. Yu. Morozov, A.M. Perelomov, A.A Rosly, M.A. Shifman and A.V. Turbiner, Int. J. Mod. Phyw. AS (1990) 803. 3. Y. Kazama and H. Suzuki, Mod. Phys. Lett. A4 (1989) 235 4. M.B. Halpern, talk in this proceedings and re£. therein 5. R. Cohen and D. Gepner, Mod. Phys. Lett. A6 (1991) 2249 6. P. Di Vecchia, J.L. Petersen and H.B. Zheng, Phys. Lett. B162 (1985) 327; W. Boucher, D. Friedan and A. Kent, Phys.. Lett. B172 (1986) 316; S. Nam, Phys. Lett. B172 (1986) 323; P. Di Vecchia, J.L. Petersen, M. Yu and H.B. Zheng, Phys. Lett. Bl74 (1986) 280.

447

RECENT DEVELOPMENTS IN THE VIRASORO MASTER EQUATION M.B. Halpern Department of Physics, University of California Berkeley, California 947!20, U.S.A. ABSTRACT The Virasoro master equation collects all possible Virasoro constructions which are quadratic in the currents of affine Lie g. The solution space of this system is immense, with generically irrational central charge, and the solutions which have so far been observed are generically unitary. Other developments reviewed include the exact C-function, the superconformal master equation and partial classification of solutions by graph theory and generalized graph theories. 1. The Virasoro Master Equation

Affine Lie algebra or current algebra on S 1 [Ja(m) ' J(n)] b

=

'J.

~ ab

cJ(m+n) c

a,b = 1, ... ,dimg

+m

Dynkin :::>Maximal, diagonal etc.) of the Virasoro master equation and its companion, the superconformal master equation [25]. In particular, the last two sets of level-families are at least N = 1 superconformal, and I will discuss the metric (m) and superconformal metric (m(N = 1)) ansii.tze below. As an example, the value at level 5 of SU(3) (20] c(

(SU(3)s)~ h' :::> h are

L(h),

Lh'- L(h),

L9

-

Lh'

+ L(h)

(2.5)

where L(h), called the bottom of the nest, is any construction on h. Restriction of the bottom to the subgroup construction Lh gives the affine-Sugawara nests [17], which include the affine-Sugawara constructions, the coset constructions and the coset nests. 4. Automorphisms [14, 20, 21]. Aut g is a covariance group of the master equation on affine g, and (2.6) wE Autg is an automorphically-equivalent solution when L"b is a solution. 5. Self K-conjugate constructions [21, 22, 30]. These constructions are automorphically equivalent to their K -conjugate partner (2.7a) (2.7b)

and live with half affine-Sugawara central charge on Lie group manifolds of even dimension. In particular, each self-complementary graph of graph theory (see Section 6) is a level-family of self K-conjugate constructions [21] on the orthogonal groups, and self K-conjugate constructions have also been observed on SU(2n + 1) [22]. 6. Spectrum. The L"b-broken conformal weights of the affine primary state with matrix representation T are the eigenvalues of the matrix [4, 17] (2.8) are similar results are easily obtained for the broken conformal weights of affine secondaries. A relativistic, conformal and diffeomorphism-invariant world sheet action [28] has been obtained for the generic high-level smooth solution of the master equation. A cornerstone of these theories is K -conjugation covariance, which dictates that the theory of level-family Lis gauged by its commuting K-conjugate theory L. As an introduction, the classical Hamiltonian of theory L is

(2.9)

452

L:

where L: and are the high-level or semi-classical limits [20] of Lab and £ab, vm is a ~agrange multiplier analogous to Ao in Yang-Mills theory, and the "Gauss law" Km = (K,K) are the commuting Virasoro operators of the K-conjugate theory L. Further discussion of the corresponding world sheet action, including a preliminary study of the classical primary fields, is found in Ref. [28]. K-conjugation also plays a central role in Ref. (29], which proposes a scheme to compute the fusion rules of the generic theory. The construction of full irrational conformal field theories is one of the most important open problems in the program.

3. The C-Function and a C-Theorem Consider the function on affine-Virasoro space [30] (3.1a) (3.1b) where Tab= :JaJb: = Tba,Pab,cd is the natural metric on the space and {3ab(L) is defined in (1.4). The function A(L) is an exact C-function on affine-Virasoro space because 8A(L) = -12.P. acd(L) = 0 (3.2) 8£ab ab,cd!J is the Virasoro master equation and

A(L)i.a=o = c

(3.3)

is the central charge in ( 1.4b). All the conformal field theories of the master equation are fixed points of the associated flow (3.4) which satisfies the C-theorem (31]

A(L) ~

o

(3.5)

on positive integer level of affine compact g. The flow equation (3.4) bears a strong resemblance to the expected form of an exact renormalization group equation. Such a connection will not be straightforward, however, because the generic off-critical quantum field theory (3.6)

453

is scale-invariant and non-relativistic when Lab does not satisfy the Virasoro master equation. See Ref. (30] for further details and applications of the C-function, including a study of the flow on the space of graphs and related connections between graph theory and Morse theory.

4. The Superconformal Master Equation The superconformal master equation (25] collects all the solutions of the Virasoro master equation on 9:r x SO(Fh which are at least N = 1 superconformal. In this construction, the general supercurrent is (4.1a)

A=l, ... ,dimg, l=l, ... ,F

(4.lb)

where JA are the currents of the "bosonic" affine g, with Killing metric GAB, and S1 are the world sheet fermions of level 1 of SO(F), with carrier space metric 1'/IJ· The coefficients eAI and tiJK are called the vielbein and the 3-form respectively. The system is at least N = 1 superconformal when the vielbein and 3-form satisfy the superconformal master equation (25] eAI

= e 8 JeCKeD 1 (6~Gcv

+ fEBA fcvE)TJJK

1 + t~JK(2tMNLeAR1JLR + eBMeCN fBcA)1JJM1'/KN tlJK

=

eALeBIKtiJ]MGAB1'/LM

+ 2eAleBJeCK fABDGvc

+ (~tP[IJtK]MNtRLQ + 2tMPltNQJtLRK}TJPQ1'/MR1'/NL 3 2 where the cyclic product

C = -eAI e8 JGAB1'/IJ

(4.2a)

1 + -tfJK tLMN1JIL1'/JM1'/KN 4

(4.2b)

(4.2c)

(4.3) is totally antisymmetric when A 1J is antisymmetric. Special cases of the superconformal master equation were obtained in (32, 33]. The superconformal master equation is a consistent ansatz of the Virasoro master equation on 9:r X SO(F)t, in that the superconformal system contains the same number of (cubic) equations and unknowns. This allows the generic counting (25]

454

of superconformal constructions and computation of the superconformal fraction of conformal field theories by comparison with the Virasoro master equation. Further details are found in Ref. [25], including the corresponding stress tensor, super afunction, super C-theorem and theN= 2 superconformal master equation. See also Sections 5 and 7.

5. Graphs and Generalized Graph Theories In order to classify their conformal and superconformal field theories, the master equations are generating graph theory (21] and generalized graph theories [22-26], including the 2-dimensional simplicial complexes [27]. So far, we have found sevent "graph-theory units" SO(n)diag :graphs ofSO(n)

SU(n)memc :sine-area graphs of SU(n) SU(II~nt)metric

: sine($area) graphs of SU(Ilin,)

SO(n)diag[":~J :signed graphs of SO(n)

X

SO(n(n -1))/2)

SU(n)metric[":~J :signed sine-area graphs of SU(n)

X

SO(n2 - 1)

SU(Ili'ni)metric[":~J :signed sine($area) graphs of SU(II;n,)

SO(dimSO(n)) [~; 1 ]

:

X

SO(IIin~- 1)

2-complexes of SO(dimSO(n)).

(5.1)

whose conformal or superconformal field theories are classified, as shown, by the graphs and generalized graphs. I will not have space to review the sine-area graphs [22-24} or the sine($ area) graphs [24], but signed graphs are simply ordinary graphs with extra ± signs on the graph edges . . tA.mong these, the first three units are examples of the metric ansatz [21,22,24) 9metric

(5.2) on affine g, while the second three units, which involve signed generalized graphs, are : examples of the superconformal metric ansatz [26, 23, 24] 9metric [ 1 ]

"::0

(5.3) on affine g X SO(dimg). The (purely fermionic) constructions on 2-complexes are also superconformal. tAn eighth graph-theory unit is the N

=2 bosonic ansatz of Kazama and Suzuki (34].

The metric and superconformal metric ansiitze are part of a prescription [21-24] which generates a graph-theory unit of conformal field theories and a graph-theory unit of superconformal field theories in each so-called "magic basis" [22] of Lie g. The total number of magic bases, and hence the total number of graph-theory units of conformal field theories, is not known. In this development, the graphs and generalized graphs not only classify the conformal level-families of the ansatze, but an unsuspected Lie group structure, called generalized graph theory on Lie g [21-24], is seen in each of the graph theories. In particular, the conventional graphs live on the orthogonal groups, and the defining relations of each graph theory (edge-adjacency, graph isomorphisms, etc.) are expressed in terms of the structure constants of the magic basis. The interested reader should consult Ref. [24], which axiomatizes the subject. In what follows, I review only the classification aspect of conventional graph theory.

6. The Graphs of SO(n).nog The standard Cartesian basis of SO(n) is a magic basis [22] with adjoint index a = ij ,

1 :5 i

< j :5 n

(6.1)

where i and j are vector indices of SO(n), and the diagonal ansatz SO(n)diog on affine SO( n) has the form Lob= L;;,w = L;;1/J;;, 2 6;~c6;t (6.2a)

T( L) = 1/J;2 E L;; :J,~

:

(6.2b)

i N 2 >> N 3. I focus first on N 2 , which is the number of automorphically-inequivalent level-families in SO(n)aiag· Note that removal of graph labels is only a factorial reduction, so the asymptotic order of N 2 is the same as the total number 2< ~) of level-families (labelled graphs) in SO( n )aiag· We conjecture that, similarly, the number (2.1) of level-families on g is a good estimate of the number of automorphically-inequivalent level-families, at least for large manifolds. Also important is the number N3 > N 2 , where N 1 is the number of level-families in the full Virasoro master equation on SO(n). This inequality shows that affine-Virasoro space is a structure which is much larger than any particular graph-theory unit of conformal field theories.

7. Superconformal Constructions on Triangle-Free Graphs The level-families of the superconformal metric ansatz [26, 23, 24] 9metric [ ~~1 ] satisfy linear equations on each "triplet-free" signed generalized graph. The signs of the generalized graphs, like the graph labels, are further automorphic copies, so automorphically-inequivalent superconformal level-families are obtained by gaugefixing to the generalized graphs. I will discuss only the example [26) of SO(n)a;a9 [~~1 ] on affine SO(n) x SO(n(n1)/2), whose automorphically-inequivalent level-families live on each triangle-free graph Gn of order n,

(7.1)

458

The variables Aij(Gn,x) of the level-families live on the edges (ij) of each triangle-free graph, and determine the supercurrent according to

G=

Aij(Gn,x) 7 ··S·· k J,, .,.

2:

(7.2)

(ij)EE(G)

The linear equations of the ansatz are

2:(1 + Tn A(Gn))ij,kiAkl(Gn,x) = k-.1/lt • To determine the ground states of this theory, one has to solve ~ = 0. This is most easily accomplished by reverting to the group theoretic characterization of the ring, 'R. Indeed, in terms of the casimirs, XA, of G, one has to solve: Xr = const. ).. ,

XA = 0 ,

A= 1, ... , r - 1 ,

(3.3)

where Xr is the ca.'limir of maximal degree, g = mr + 1. It is now very convenient to consider XA restricted to the Cartan subalgebra (CSA) of G, and think of XA as a function acting on some basis elements of the CSA. Indeed, XA is a homogeneous polynomial of degree mA + 1 on the CSA, and this polynomial is W(G)-invariant, where W(G) is the Weyl group of G*. Let denote a vector in the CSA. Consider a Coxeter element t ,s, of W(G). This element has order g, and acting on the CSA, s has eigenvalues exp(27rim;/g), j = 1, ... 'r. Let e(l) be an eigenvector of s with eigenvalue exp(21rijg). Now observe that because XA is W(G) invariant, we have:

e

where the last identity follows from the homogeneity of XA· However the foregoing implies that XA(e(t)) = 0, A = 1, ... , r - 1. Moreover Xr(e(t)) =F 0 since the vanishing of all the casimirs would imply e(t) :: 015 • Observe that all the W(G) images of e also satisfy Eq.(3.3). Finally, note that because the Landau-Ginzburg fields are all H'-casimirs, all the W(H) images ·of e(t) yield the same ground

* t

Convenely, a W(G)-invariant polynomial on the CSA can be mended to a CMimiron.G. A Coxeter element can be wriUen u a product r 1 r 2 ••• r., wheft r; ia the Weyl reflection in the airnple root a;. A Coxeter element dt!pellda Gn the dtoice of a ayatem of eimple root&, and the upon the ordering ol the r; in the roJ"eeiin& pi"'duct, but aU COKeter elemeota are conju&ate, and any auch element will auffice here.

468

state. This gives a one-to-one* mapping between the ground states of the perturbed model and the cosets of W(G)/W(H). It is interesting to note that in the conformal theory there is also a natural one-to-one association of Ramond ground states and the cosets of W(G)/W(H) 2 •8 • Presumably these associations of Weyl cosets with ground states in the conformal and perturbed conformal theories are related, but this is not obvious from the two constructions. One should also note that the perturbed superpotential in Eq.(3.2) is completely resolved (morsified), and only has massive perturbations.

9.!. Soliton Structure and Soliton Maue6 Suppose that the two-dimensional space time is lR x R with coordinates (u, t). We want to find the minimum energy configurations, q,i(u, t), subject to the boundary conditions: (3.4) q,i(O' = ±oo) = q,'f' , where q,'f' are two of the solutions to: ~:; = 0. In particular, we wish to determine the fundamental chiral solitons; that is, those single soliton states that are annihilated by half of the supercharges in the pertubed theory* . At is this point it is helpful to use the following physical picture as a guide: The conformal field theory has JJ = degenerate Ramond ground states that can be mapped into each other via operator product with the chiral primary fields. The perturbed theory also has JJ distinct degenerate ground states, and we are now seeking the chiral solitons that link these ground states. Such chiral solitons presumably have the chiral primary fields as their conformal progenitors- . The commutation relations of the perturbed superalgebra can be computed8 and one finds that the mass, M, of a solitonic state satisfies a Bogomolny bound:

I;fm I

M 2::: I~Wj

= I const. ~q,tl,

(3.5)

where ~W = W(q,t)- W(q,i) is the topological charge of the soliton, and this charge is proportional to ~q,l = q,f - q,}. The bound in Eq.(3.5) can also be established by semi-classical arguments16 •9 • The chiral solitons are precisely those solitons that saturate this bound, and because of this one can argue that the chiral solitons are fundamental. That is, the chiral solitons are generally not multi-soliton states. Therefore, to determine the mass spectrum of the (fundamental) chiral • To completely establish this one needs the theorem that the values of the casimirs on ~ uniquely specify~ up to Weyl images11• The perturbed theory haa four supercltargea: Q:!:and Q:!: and two appropriately chosen linear combinationa •· 8 of these charges annihilate chiral aolitona. b It would be nice to establiah this interpretation rigorously, and in particular aee how operatora and states behave aa one continuously deforms the theory away from the confonnal point.

*

469

solitons, one needs to solve the problem of which pairs of ground states are connected by such solitons. It is simplest to first state the solution to this problem and then justify it. To do this we need to introduce the 1oliton polytope, 'P. For a hermitian symmetric space G / H', there is a canonical representation, V, (called::: in our earlier works) of G such that (i) the representation is miniscule, i.e. all the weights of V have the same length, and (ii) the highest weight space I..X> is fixed by H'. (In fact .X= ~(pa- PH), where PG and PH are the Weyl vectors of G and H respectively.) For example, for the grassmanians:

_

.Gm,n

=

SU(n+m) SU(n) X SU(m) X U(l) '

(3.6)

the representation, V, is them index anti-symmetric tensor of SU(m + n). The important point is that, literally by definition, the weights of V are in one-to-one correspondence with the cosets of W(G)/W(H). Thus we may associate ground states of the integrable model with the vertices of a regular geometric figure whose vertices are the weights of V. This is the soliton polytope, 'P. If A1 and A2 are two weights of V whose corresponding ground states can be linked by a chiral soliton, then, from the results above, it is elementary to see that the mass of this soliton is given by: (3.7) where M 0 is some overall constant and eel) is the eigenvector of the Coxeter element introduced earlier. The characterization of chiral solitons is now elementary: two ground states are connected by a chiral soliton if and only if the corresponding weights, A1 and A2, on 'P differ by a root of G. While there is not a completely rigorous proof of this statement, there is very strong evidence that it is trues. There are some physical arguments, of which perhaps the best is based upon the relationship between the conformal theory and the perturbed theory8 • However, perhaps the most compelling evidence is the fact that the foregoing characterization of fundamental chiralsolitons satisfies a vast number of consistency conditions provided by resonances. Since this analysis leads to the soliton charges for the higher spin integrals of the motion, we will describe it here in some detail. 3.3. Higher Spin

Con~erved

Charge1

Consider any three vacua on 'P that are connected by a triangle of chiral solitons. Label these solitons by a,b and c. Project this soliton triangle into the the complex plane defined by e(I)· Then from Eq.(3.7) one sees that the sides of the triangles have lengths equal to the masses M., M• and Me of the three solitons. Let the angles be labelled by 8., 86 and 8e as shown in figure 1. Then some trivial trigonometry shows that:

(3.8)

470

Figure 1. The mass projection of the soliton triangle.

If one now imagines scattering soliton a and soliton b against each other, then there is a resonance to create soliton c at rest when soliton a and soliton b have rapidities i(9,. -11') and -i(9, -11') respectively. Now recall that there are conserved charges, q being the mass). Let qi•>, q!•> and qi•> be the spins s charges of the· solitons a, b and c. The resonance implied by the soliton triangle of a, b and c imposes the following constraint upon the q charges: (3.9) Considering every soliton triangle in the polytope provides a highly overdetermined system for all the spin s charges of the solitons (see figure 2, for example). The solution to this system of equations is also provided by the soliton polytope. Consider, once again, the Coxeter element of the Weyl group of G acting on the CSA. Let e(mA) be an eigenvector with eigenvalue exp(211'imA/g). Project the soliton triangles onto the complex plane defined by {(•)· The remarkable fact about the soliton polytope is that this new triangle has interior angles s9,., s9, and s9c mod 1r8 •17 •18 • Thus, modulo signs and an overall scale, one can identify qi•>, q1•> and qi•> with the side lengths of this projection of the soliton triangle. Moreover, one can orient the sides and thereby give these lengths a sign so that these signed lengths exactly satisfy Eq.(3.9). Thus the soliton polytope encodes all the information about all of the charges of the chiro.l solitons. The fact that there is a solution to such a highly over determined system of resonance constraints also provides good evidence that the original characterization of chiral solitons is correct.

471

9..4. Further Comment, on the Integrable SLOHSS Modeu The actual set of numerical values of the higher spin charges of the chiral solitons is not altogether surprising: Depending upon the soliton, the spin s charge, q~•l, is always some component of the eigenvector of the Cartan matrix of G with eigenvalue 2 - 2cos( ' 2" ). (The details of how these eigenvectors emerge in the projections of the polytopes may be found elsewhere11 •17 .) Thus one sees further evidence of the relationship to affine Toda theory. It is also important to remind oneself that the Landau-Ginzburg model is supersymmetric. This means that each soliton described above is, in fact, a (shortened) supermultiplet of four solitons, two "bosonic" and two "fermionic". In the 001"responding affine Toda theory, this supersymmetry appears only at the quantum level, and requires a special choice of background charge a particular value for the coupling constant. This coupling constant is also purely imaginary, just as in the non-supersymmetric affine Toda theories that describe integrable perturbations of non-supersymmetric conformal field theories.

Figure 2. The soliton polytope for the E1/ Es X U(l} model is obta.ined by taking V =: 56 of E7. This diagram shows the ~(l) (or mass) projedion of the polytope; the dob are the images of the vacuum staies. The central dot in the diagram represenb two vertices. The lengths of the soliton lines ue proportio11al to the 110liton masses. There are 756 solitons and 4032 soliton triugles. The system of equa~ions (3·.9) is thus

overdetermined by more

~han

a. fador of 5.

472 It is known that the classical Toda theory, with a purely imaginary coupling constant, has soli tonic solutions with physically sensible quantum numbers 19 • These interpolate between the vacuum states, which lie on the weight lattice of the corresponding group. At the quantum level, it will be necessary to truncate the soliton spectrum to provide a unitary Hilbert space. For the simplest affine Toda model, the sine-Gordon model, this quantum group truncation20 can be viewed as effectively reducing the infinite-well potential to a Landau-Ginzburg potential with a finite· number of wells, or as reducing the weight space of SU(2) to a finite weight diagram of some representation of SU(2). We conjecture that the G/H' Landau-Ginzburg solitons correspond precisely to such classical Toda solitons, after some form of (perhaps affine) quantum group truncation. This truncation should effectively reduce the whole weight space of G to the finite weight diagram of the representation V. The work described above characterizes all the chiral solitons. A consistent scattering matrix for these (supermultiplets of) solitons has only been computed for the simplest (minimal) models9 • It would be interesting to find S-matrices for some of the more complicated Landau-Ginzburg models. There is, however, a problem to be solved before one can do this. Simple kinematic arguments 9show that the chiral solitons cannot (except in the type A minimal models) form a closed scattering theory. One needs to add new states. One can make educated guesses as to what these states should be, and kinematic consistency can often be restored by adding some new "breather" states. It would be nice to have some method of determining the complete spectrum and then finding the S-matrix. Another interesting issue is raised by the circuitous manner in which the higher spin charges were determined. The masses of the solitons were obtained from the superalgebra and a Bogomolny bound involving a topological charge. The higher spin charges were then deduced from resonance consistency. Since the higher integrals of motion come from the super- W algebra generators, one might hope to obtain new Bogomolny bounds, and perhaps some new topological charges, and thereby determine the higher spin conserved charges of the solitons directly from the perturbed super-W algebra. The fact that the answers·are so beautifully encoded in the soliton polytope suggests that there may also be some simple underlying geometry to the perturbed super- W algebra. Finally, it should be remembered that the most relevant chiral primary perturbation is not the only perturbation of the N = 2 super-coset models that leads to a quantum integrable N = 2 supersymmetric field theory. For the minimal Aseries there are two other such perturbations9 •4 •21 • (One of these perturbations will be discussed in the next section.) There are also indications22 that there may be other perturbations of the SLOHSS models that give rise to integrable theories. It is certainly of interest to determine the soliton spectrum and S-matrices for these models.

4. Real Resolutions of A-D-E Singularities, Solitons and Fusion Rules Our purpose here is to make some, hopefully, amusing observations concerning perturbed A-D-E minimal models* . These observations will be discussed in more

*

Note that only the A and D series minimal models are SLOHSS models.

473 detail elsewhere. In contrast to perturbing by the most relevant cbiral field, as above, we now perturb the minimal models with the (F-component of the) unique, lea,.,t relevant operator, tPtop, of dimension c/6. As we will see, this type of perturbation also leads to interesting soliton structure t . For the A-series, it is known that this perturbation leads to an integrable theory9 •4 •24 •21 , and non-trivial integrals of motion have been constructed. Similar results can probably be established also for the D and E series. An important point to realize is that perturbing N = 2 theories by insertions of the forme- 2: ).t(J t+h.c.) in the correlation functions amounts to deforming the operator algebra in a way that can be characterized by an effective superpotential25 , W( t/>, .X) = Wo + L:.E'R g~:(.Xt) tPk· Here, the fields t/>1: denote generic elements of the chiral ring, and the coupling constants 91: are particular, non-trivial* functions of the perturbation parameters (or "flat coordinates"), At, and can be determined by using the techniques described in25 •26 ·2 7. For the minimal A series the effective superpotential for the perturbation with tPtop is

W _ _ z_,x!!:f!T, (_x-1/2.1./2) _ _1_J.n+1 _ n+1 'I' 'I' n+1 - n+1

_x-~.n-1 'I'

+0(.\2)

'

(4.1)

where Tn are Chebyshev polynomials. The corresponding potential I~~ 12 is a multiwell potential with n zeros along the real-4> axis. If one calculates the values of W at all the critical points, one finds that it takes only two values. Thus all the chiral solitons have the same mass ( M = 16WI). This is perfectly consistent with the conjecture that this model is equivalent to a quantum truncated, N = 2 supersymmetric sine-Gordon model 22 • (This conjecture is also supported by the structure of the quantum integrals of motion.) The potential in Eq.(4.1) received recently attention 28 •29 •22 •30 because of the remarkable fact that for A= 1, the structure constants Ctm" of the deformed cbiral ring, in a basis consisting of fields C)t( t/>, A) = - 8 ~~,;!.), coincide with the fusion coefficients Ntm n of SU(2)A:=n-1 WZW models. This is because one finds that:

C)t((t/>} .. , 1) = SSta '

(4.2)

liJ

where Sta is the modular transformation matrix of the SU(2) characters, and a labels then vacuum states, ~~((t/>) 4 ) =: 0.

t There is an analogous, relevant perturbation of general SLOHSS models and one can

*

construct S-matrices for the solitons28 • As remarked earlier, the most relevant chiral primary perturbation leads to a simple effective superpotential W(,P,>.) W 0 + >.,P 1 •

=

474

Apparently, similar properties hold also for other minimal models, where the dependence on the Hat coordinate). is 25 ·31

*:

Dn:

w

= ( -1)R- 1t.-transformations

64>i

i

.

;

1:

=k-11'++L-d';~:11'+11'++(++-t-)

6h±± =o±k±

+ k±~h±± -

h±±~k±

+ 2tcTH(II) ( >.±±~B±±± - B±±±~>.±±) DB±±±=o±>.H + 2>.±±~h±±- h±±o=F>.±±

(5.16)

+ k± ~B±±± ; 1:) 1c'f1!'± + >.'f'fd'.;~:11'±11'±

- 2B±±±~k± i

61r± = 8±

(

i

where T±±(II) = !11'~11'~. The field equation for the auxiliary fields is algebraic (5.17) but difficult to solve for 1r in closed form, so that again there is not a closed form for the action without 1r's. Nevertheless, one can solve for 1r to any given order in the gauge fields and the result agrees with that obtained by using the Noether method to find the corrections to (3.1),(4.6) to that order in the gauge fields.

511

To obtain a better understanding of these actions, it may be useful to consider setting the spin-three gauge-fields to zero in the actions considered above to obtain the coupling to pure (spin-two) gravity, which can then be compared with the conventional minimal coupling to gravity. The Noether coupling approach gives the action (5.18) Although one could calculate some of the higher order corrections to this and attempt to guess the general form, this is clearly not the best way of finding the coupling of a scalar field to gravity. The approach of [3] gives

(5.19)

and the

1r

field equation is (5.20)

which can be solved explicitly to give {5.21) Substituting (5.21) into (5.19) gives the complete non-polynomial form ef the action (5.22) This gives the full non-linear corrections to (5.18). Similarly, the Hamiltonian approach gives {5.23)

and the momenta can again be eliminated explicitly to give a non-polynomial action similar to (5.22).

512

Of course, most people would prefer to use a little geometry and write down the standard minimal coupling to a metric gp 11 (5.24)

If one chooses to parameterise the metric as

(5.25)

then 0 drops out of the action (5.24) as a consequence of classical Weyl invariance and (5.24) becomes precisely (5.22). Note that, contrary to claims sometimes made, (5.25) does not correspond to partial gauge-fixing; (5.25) is simply a convenient parameterisation of a general metric 9pv· However, for non-zero B, (5.20) gives an equation for 1f which is difficult to solve in closed form, although it is straightforward to solve order by order in B; substituting the perturbative solution to (5.20) back into the action recovers the results of the Noether method. However, just as the non-linearity in gp11 is best understood in terms of Riemannian geometry, it seems likely that the nonlinearity in B can also be best understood in terms of some higher spin geometry, which would also allow the coupling of W-gravity to more general matter systems. The two canonical approaches described here for W3-gravity also work for other W-algebras and other matter systems. The Hamiltonian approach clearly works quite generally; one writes the matter action in first order form and replaces time derivatives of fields in the W-currents by the corresponding momenta. The full action is given by adding the Noether coupling of these currents to Lagrange multiplier gauge fields. The resulting action is of the form (5.7) and so invariant under the transformations (5.9), (5.10). The covariant canonical approach of (3] has been generalised to w 00 (8], WN (4] and indeed free boson realisations of any W-algebra (4]. It has also been applied to non-linear sigma-models, free-fermion models (4] and supersymmetric models (30,10] and probably again applies quite generally.

513

6. The Geometry of W -Gravity The non-polynomial structure of gravity is best understood in terms of Riemannian geometry and this suggests that the key to the non-linear structure of W-gravity might be found in some higher spin generalisation of Riemannian geometry. The approaches of [3] and [11] describe W-gravity in an implicit form, but an understanding of the non-linear structure of the theory without auxiliary fields seems desirable. A 'covariantisation' of the approach of [3] is given in [6,9], but the resulting theories still have auxiliary fields. Other approaches to the geometry of W-gravity are presented in [31,32,33]. In [12], a geometric formulation of W 00 gravity was derived and this will now be briefly reviewed. Riemannian geometry is based on a line element ds = (gi;di are taken to satisfy harmonic oscillator commutation relations

(8.3) It is necessary to regularise (8.1), and it is convenient to define T =: !8+4>i8+4>i: so that L., = :Em: a~+ma~m :, where colons denote the normal ordering with

!

523

respect to the modes a~: i i . ; i ·- { a,.. am . a,..am ...

a;,..a~

if m > n

(8.4)

if n ~ m

Then the commutator algebra generated by the quantum operator T is given by the Virasoro algebra (2.1) with central charge c =D. It will be convenient to suppress delta-functions and numerical factors and present such algebras schematically as (8.5) The term linear in 1i is what one would expect from applying the Dirac prescription to (8.2), but there is in addition a central charge term of order h 2 , which corresponds to an anomaly, as we shall see. The stress-tensor (8.1) can be modified by adding a non-minimal 'background charge' term, T ~ T' = T + a;8!4>; where a; is some constant vector. The classical Poisson bracket algebra then has a central charge, [T', T'] ,...., T' + co where co = ha;a;, and the quantum algebra again takes the form (8.5), but with c = D + c0 fn. This factor of ;,_-I can be absorbed into a rescaling of the background charge a~ Vha, so that the stress-tensor becomes

(8.6) and the quantum central charge becomes c = D

8.2

LINEAR REALISATION OF

+ a 2 /24.

Woo

Let 4>i be D complex free bosons. Then the currents [39] n-1

W,..(z+) =

L f3n,r8~4>;8~-r ;p,

n

= 2,3, ...

(8.7)

r=1

for suitably chosen constants f3n,r generate a Poisson bracket algebra which is a certain classical limit of the W 00 algebra of [22], (note that this algebra is not the w 00 algerba of [21]) which has the generic form [22, 39]

[W,.., Wm] '""Wn+m-2

+ Wn+m-4 + Wn+m-6 + Wn+m-8 +...

(8.8)

This is a linear realisation of a classical limit of W 00 , as these currents generate variations of 4>' which are linear in 4>'. The quantum currents given by normalordering (8.7) generate an algebra of the general structure (for some constants

524 Cn)

which consists of the Dirac quantisation of the algebra (8.8), plus central extension terms of order 1i2 • This quantum algebra contains the Virasoro algebra with central extension 2D and is the W 00 algebra of [22] with c = 2D [42], generalising the c = 2 construction of [39].

8.3

NON-LINEAR REALISATION OF W3

ForD free real bosons, as was seen in chapter 3, the currents given by (8.1) and w = idijka+¢>ia+if,ia+¢>1c generate a classical w3 algebra provided the constants dijk satisfy (3.5). The corresponding transformations of 4>i are given by (3.6) and are non-linear for spin 3. The classical algebra (2.1),(2.2),(2.3) can be written schematicaly as (8.2) together with

[T, W]"' W,

[W, W]"' A,

A=:TT

(8.10)

In the quantum theory, the currents can be defined using the normal ordering (8.4). The spin-two currents again generate the Virasoro algebra (8.5) with c = D. The [T, WJ commutator now takes the form

[T, W] "'i1iW + 1i2 J,

(8.11)

consisting of the Dirac prescription term, plus a term of order 1i 2 which involves a new spin one current, J [2]. Thus although the classical algebra closes, the quantum one does not unless the d-tensor is traceless, diij = 0. Even if it is traceless, the [W, W] commutator gives

[W, WJ "'i1i: TT: +(D + 2)1i 2 (T2 •0

+ T) + 1i3 c',

which consists of the Dirac term, a central charge term of order h 3 proportional to c1 dijk~jk and a new spin-four current T 2•0 [2,43]. If ~ij =/= 0 there are extra terms in (8.12) involving J [43]. Since the coefficient of the spin-four current T 2•0 is non-zero for any D > 0, the quantum algebra never closes when the normal ordering prescription (8.4) is used, as the right hand side of (8.12) cannot be written entirely in terms of T, W and composites constructed using (8.4), such as : TT :, : TW : etc. To close this algebra, it is necessary to introduce J, T 2•0 as generators, and then to introduce further generators, such as rn,O = A.k .. amA.ianA.Ja+'I' n+lA.i8 A.i wm,n,O _d a+ '... [43] . +'I' +'fT - l]lc 'I' +'I''

=

525

However, instead of defining the composite operator : TT : using the prescrition (8.4), it could instead be defined as : TT: using (7.1). The two definitions are related by [2] (8.13)

so that they differ by a finite term of order 1i. Using this, the algebra (8.12) can be rewritten as

so that the coefficient of the spin-four current T 2•0 is now D- 2 instead of D + 2, with the result that the algebra closes non-linearly on T, W, : TT = if and only if d'ii = 0 and D = 2 [2]. For D = 2, the solution of (3.5) given by dm = - I t and d222 = 1t gives a traceless d-tensor and hence a closed algebra which becomes preciesly the W3 algebra (7.3),(7.4),(7.5) after rescaling the currents. This is the two boson realisation of the c = 2 w3 algebra given in [14]. This model is closely related to the Casimir construction of W3 [25,24]. The classical W3 algebra is realised by the Sugawara currents T = !tr(J+J+) and W = itr(J+J+J+), where J+ is an SU(3) Kac-Moody current [5]. In the quantum theory, these currents (after normal ordering) no longer generate a closed algebra, as the commutator [W, W] gives rise to the spin-four current T 2•0 = tr(J+o~J+) [25,24], which is similar to the current T 2•0 that arose in the free boson model. In the case in which the Kac-Moody algebra is of level one, it is possible to perform a truncation to a realisation of the quantum w3 algebra, and this is related to the fact that the level one Kac-Moody algebra can be constructed from the two boson model discussed above [24]. 8.4

W3 REALISATIONS WITH BACKGROUND CHARGES

As in the Virasoro case, one can consider adding higher derivative terms to the currents to give (8.15)

for some constants 0-i, ei;, k The current algebra will close on T, W, : TT: to give the w3 algebra (7.3),(7.4),(7.5) provided the tensors ai,ei;.li satisfy certain constraints [2]. For the D = 2 model these constraints are satisfied by choosing the only non-vanishing components to be given in terms of a free parameter a by a1 = a, en = a, e21 = 3a, h = 6a2, giving a model with central charge c = 2 + 24a2 [14]. For real a one obtains models with any value of c ~ 2, while

526

for imaginary a, a unitary theory can only be defined if the background charge a is chosen to take the discrete values a 2 = -[p(p + 1)t 1 for p = 4,5,6, ... giving the minimal series of representations of the W-algebra with central charge ep = 2-24[p(p+1)t 1 (14]. ForD f::. 2, the constraints on the coefficients in (8.15) were solved in [27], giving a realisation of the W3 algebra in terms of any number D of bosons, with arbitrary central charge c = D + a 2 /24. In particular, it is possible to construct in this way realisations of W3 with central charge c = 100 for which there is a nilpotent BRST operator. One can instead ask whether modifying the currents as in (8.15) can give an algebra that closes using the normal ordering (8.4), i.e. whether an algebra can be found in which [W, W] can be written entirely in terms of T, W and : TT : instead ofT, W and : TT:. It was shown in [53] that this is only possible if there are precisely D = 2 bosons, in which case ai = fi = 0 and €ij is proportional to Eij, so that the Virasoro subalgebra has c = 2.

8.5

ONE BosoN REALISATION OF

W00

=

For one boson 4>, the currents Wn = ~( 8+4> )n for n 2, 3, ... generate the w 00 algebra classically [8]. The w 00 algebra takes the schematic form

(8.16) All the currents except the Virasoro current W2 generate non-linear transformations and this non-linearity leads to new currents on the right-hand-sides of the quantum commutation relations e.g. [W2, W3] gives rise to the current J = 8+4> while [W3, W3] gives rise to T 2 •0 =: 8+4>8!4> :. However, one can again consider adding higher derivative terms to the currents Wn (as in (8.15)) to modify the algebra. It was shown in [56] that the coefficients can be chosen in such a way as to close the algebra, giving rise to the W 00 algebra of [22], which is of the schematic form (8.8), with central charge c = -2. This is precisely what is needed to cancel the contribution of +2 coming from using a (-function to sum the infinite number of ghost contributions [56].

527

9. W -Gravity Anomalies from Matter Integration There has been a great deal of recent work on the quantisation of W-graVity and the anomalies that arise [42-59]. The classical coupling of a free boson system toW-gravity is described by an action of the form (4.8), where So is the free boson action {3.1) and the h*A are gauge fields. In this chapter we will consider the integration over the matter fields q,i only, regarding the gauge fields as external sources, and study the anomalies that arise in the Ward identities corresponding to the classical W-gravity symmetries. No gauge fucing is needed as the gauge fields are not being integrated over, and the scalar fields have a well-defined propagator. In the next chapter, we will gauge fix a.ll the local symmetries, introduce the appropriate ghost fields and discuss the integration over all fields. For simplicity, consider a chiral W-gravity, so that the complete action is of the form (4.7), consisting of the free boson action(3.1) plus the linear Lagrange multiplier terms. Since this is just a free system plus constraints, the normal ordering prescription (8.4) is sufficient to subtract a.ll divergences. It is convenient to use the background field method, writing 4>1 as the sum 4>' +c,o' of a background field (iJi and a quantum field i)

St/1:~:; = iL(-It8:~:" (E'f.. !~0.~:>) n~O

(8)

v~:!:J

Given (8) it is not difficult to show that S

=I d z(F:~:i8'f4»i+lt/1:~:;8'ft/I±;_B'faa:~: -w'fc:~:-l8-4»i8+4»i-F+ip_i) 2

..

(9)

is invariant provided the B's transform as

and (11)

As an example, we construct a gauge invariant action for the non-universal super W 3 algebra discovered by Hull [4]. The generators are the energy momentum tensor T and its supersymmetry partner G of spin 1, W 3 and its supersymmetry partner H of spin ~· They form a closed algebra (4] if (12)

542

The non-chiral gauge invariant action follows then from the Eq. (9)

s=

j tfz( F*•a"f~' + i..P±•a"f,p*' - hnTu

- x"fG*

- Bn"f W±±± - /JnH H -}8-~'8+~' -

F +1F -•)

,

{13)

where

(14)

The gauge transformations can be determined from the Eq. (8) and (10). Obtaining the superfield formulation of (13) is straightforward [7]. It would be interesting to see whether our formalism can be applied to the case of Toda type theories where the potential term pefP., is present in the gauge fixed action {1 ). We expect that the free case p = 0, should follow straightforwardly from our formalism. Our results indicate that coupling of the W-gravities to the Wess-Zumino models may be realised through the non-abelian generalization of our formalism. Also it would be interesting to see whether the formal bracket structure can be explained through some generalized Hamiltonian formalism.

REFERENCES (1) A. Mikovic, Phys. Lett. 260B {1991) 75 (2) A. B. ZamolodchikoY, Teor. Mat. Fiz. 65 (1985) 1205 V.A. Fateev and S. Luk'yanov, Intern. J. Mod. Phys. A3 (1988) 507 [3) A. Bllal, Phys. Lett. 227B (1989) 406 I. Babs, Phys. Lett. 228B {1989) 57 C. N. Pope, L.J. Romans and X. Shen, Phys. Lett. 236B (1990) 173; Nud. Phys. B339 (1990) 191 (4) C.M. Hull, Nucl. Phys. B353 (1991) 707 [5) K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Phys. Lett. 243B {1990) 245 (6) E. Bergshoeft", C.N. Pope, L.J. Romans, E. Sezgin, X. Shen and K.S. Stelle, Phys. Lett. 243B ( 1990) 350 (7) F. Bastianelli, Mod. Phys. Lett. A6 (1991) 425

543

Anomaly-free W -gravity Theories

C.N. Pope*

Center for Theoretical Physics, Texas AI!!JM University, College Station, TX 778,/9-,jf!,jf!, USA.

ABSTRACT We give a review of some recent developments in the quantisation of W-gravity theories. In particular, we discuss the construction of anomaly-free W 00 and W3 gravities.

*

Supported in part by the U.S. Department of Energy, under grant DE-FG05-91ER40633.

544 1. Introduction

Two-dimensional gravity has provided a rich and fascinating field of study in the last few years. One can view two-dimensional gravity as being the gauge theory of the Virasoro algebra in two dimensions. Since higher-spin extensions of the Virasoro algebra exist, it is natural to investigate the corresponding two-dimensional gauge theories. Many of these higher-spin algebras, known as W algebras, have been discovered. The first, called W3, contains a current of spin 3 in addition to the usual spin-2 current, the holomorphic energymomentum tensor, of the Virasoro algebra [1). Subsequently, generalisations toWN algebras were obtained, which contain currents of each spin s in the interval 2 :5 s :5 N [2). A general feature of these extended algebras is that they are non-linear, in the sense that the operator product of two currents may produce terms that must be viewed as composite operators, built from products of the fundamental currents in the algebra. The reason for this is that to leading order, the 0 PE of currents with spins s and s' gives a quantity of spin s 11 = s +s' - 2, which may exceed N if N > 2. This argument for the occurrence of non-linearities breaks down if N = oo, and indeed a linear algebra of this kind, known as W00 , exists [3). In most other respects, W00 is qualitatively similar to the finite- N algebras. For example, they all have non-trivial central terms in the OPE of any pair of equal-spin currents. There is another related algebra, known as Wl+oo• which has currents of each spin 1 :5 s :5 oo [3,4). All of the W N algebras, with N finite or infinite, admit contractions to algebras that may, in a sense to be clarified below, be thought of as classical limits. In the case of W 00 , the corresponding contracted algebra is the w 00 algebra found in [5). For Wl+oo• its contraction, WJ+oo• is isomorphic to the algebra of area-preserving diffeomorphisms on a cylinder [3,5]. In this article, we shall review the construction and quantisation of two W-gravity theories. The first of these is based upon the classical two-dimensional gauge theory of the w 00 algebra [6). We shall see that in order to quantise the theory, it is necessary to renormalise the classical currents that generate the classical w 00 symmetry. After the dust has settled, one finds that the renormalised currents generate the full W00 algebra [7]. Thus the classical Woo gravity that one starts with ends up as quantum W 00 gravity. The second example that we shall consider here is that of W3 gravity. One might think that the non-linearities of the W3 algebra would make the analysis of this case much more difficult. Up to a point, this is true. However, the essential qualitative features are in fact remarkably similar to those of the W00 case. The starting point in this second example is the two-dimensional gauge theory of a contraction of the W3 algebra [8,9). It would therefore be more appropriate to call this classical W3 gravity, rather than w3 gravity. Again, we shall see that the quantisation process requires that the classical W3 currents must be renormalised. In the end, the renormalised currents generate the full W3 algebra (10,11).

545 2. Classical and Quantum W 00 Gravity The classical theory of w 00 gravity was constructed in [6]. In its simplest form, one can consider a chiral gauging of w 00 ; i.e. one gauges just one copy of the algebra, in, say, the leftmoving sector of the two-dimensional theory. We shall discuss non-chiral gaugings in more detail later. For now we just remark that, thanks to an ingenious trick introduced in [9], involving the use of auxiliary fields, the treatment of the non-chiral case can be essentially reduced to two independent copies of the chiral case. As our starting point, let us consider the free action S field in two dimensions, where L is given by

= 1/7r J d2 zL for a single scalar (2.1)

Here, we use coordinates z = x- and z = x+ on the (Euclidean-signature) worldsheet. This action is invariant under the semi-rigid spin-s transformations b.(8.Jil•at{la¢k

(4.2)

provided (4.3)

Note that we introduced d symbols cJii", which are totally symmetric. The commutator algebra of these transformations closes if ( 4.4)

It is given by

[8( ft), 8( f2)] ¢'

=

8( f3

= f20ft -

ft0t.2)¢'

[8(ft),8(>.2)] ¢; = 8(>.3 = 2A20ft- fto>.2); [8(>.1),8(>.2)W = 8(t.3 = -2(>.18>.2- >.2a>.l)o¢)otf).(2ab+ba)uu. In this limit we could reduce the).. anomaly to the minimal one by adding an extra term to the h transformation rule in eq. (4.18):

!

oh

4

= 15 (>.ab- ba>.)u.

(4.22)

However, it turns out to be more natural to make a different choice for Oextrah: (4.23) For this choice we have that u and v transform according to the operator product expansion in the limit c-+ ±oo (using that ou(x) "'f dy(fT + >.W)(y)u(x) and similar for ov)

1

1

ou

= aa(. +(.au+ 2aw + 15 >.av + 10 a>.v

ov

=

Eav + 3aw +as)..+ (2>.aa + 9aMP + 15a2>.a+ 10a3 >.)u

+8(28-\ + >.8)uu ,

(4.24)

where ( ) _ 36071' 6rind[h, bJ vx c 6b(x) ·

(4.25)

The algebra on the gauge fields and the effective currents is closed

[6(f.t),6(1'.2))

=

6(Ea

= e281'.1 -

[6(et),6(>.2))

=

6(-Xa

= 2>.28E1- E18A2)

[6(-Xt), 6(-\2)] = 6( t:a =

t: 18t:2)

3~ (2aa AtA2 -

3a2>.1a>.2 + 38>.ta2>.2 - 2-Xtaa >.2)

8 + 15 (-\28-\1- At8A2)u).

(4.26)

576 As we will see later on, it is precisely the choice eq. (4.23) for hextra.h which will emerge from a constrained S/(3, R) theory. With this choice, the A anomaly is not the minimal one: (4.27)

A useful check on this result is the analysis of the Wess-Zumino conditions fo · consistent anomalies, which are indeed satisfied (compare with our analysis in

(10]). Using the chain rule for 6f;nd and eqs. (4.16), (4.17), (4.18), (4.23) and (4.27) we find the final form of the Ward identities in the limit c ~ ±oo:

tJu tJv

=

D1 h + [ 110 v 8 +

1~ (av)] b,

[3va+(av)]h+D 2 b,

(4.28)

where D1 and D2 are the 3rd and 5th order Gelfand-Dickey operators given by D1

D2

= ff + 2 ua + u' , = a 5 +10uff+15u'a2 +9u"a+2u"'+16u 2 a+16uu',

(4.29}

and the primes denote a. This is the situation for c ~ ±oo. Solving eqs. (4.28) for f[h,b] yields the induced action in this limit, which we will denote by r oo[h, b]. Later in this section we will explicitely compute this action, using a constrained Sl(3, R) WZW model. Before coming to this, let us first comment on the finite c corrections to the Ward identities. The induced action of W3 gravity can be expanded in even powers of the field b. The full result for the b-independent terms in Cnd[h, b] (1-loop diagrams) is given by (OJ [ h, bl = 24C71' find

J

a

1 d2 X a2 h 1 1 _ h~ ah.

( 4.30)

The exact result for the terms quadratic in b (2-loop diagrams) reads c 71' [2) b] == 720 rind[h,

Jay -VY,

(4.31)

where V = tJ - ha and the scalar field Y is given by (4.32)

sn The leading '3-loop diagrams' (i.e., the terms with four b fields and no external h fields) is given by (4] _ __c_ _! 2{3c_!. rind[h - 0, b] - 60. 6! '~~" [/] + 5. 7! '~~" [II],

(4.33)

where [I] and [J J] are given by

[I]= jd2x

(2b~b-3ab;b+3a2 b~b-2aab~b)~(2b~b+3ab~b), (4.34)

where (4.35) Since the quantity {3c is of order 1 for large c, one sees here explicitely the onset of a 1/c expansion for the induced action as in eq. {1.2). As such, the Ward identities will be modified for finite c. Indeed, from eq. {4.33) we see that structure [J] gives the three-loop contribution to eq. ( 4.28) while structure [J J] is subleading in the 1/c expansions and modifies the large c Ward identities for the induced action:

au= 8v

=

1

D 1 h+ 30 [3v8+2(8v)Jb,

[3v8+(av)]h+D2b

8 8aQbb8) + o + 435f3 ( 2 ab 81 Qbb+b

(b3h->1 ,bs , ....)

(4.36)

We will now explicitely compute rind[h, b) in the limit c -+ ±oo, denoted by r oo[h, bJ. We follow essentially the same method as in the previous section, but this time we start from an Sl(3, R) ~odel. Our analysis closely follows that in [14]; note that some partial results were also obtained in [35, 36, 37]. We choose the following basis for Sl(3, R):

(4.37) where e;; are 3 X 3 matrices, (e;;)ld

= Dif,Djl•

We impose the following constraints u+3 )

0 0

.

(4.38)

578

Six of the eight components of the Ward identities 8u - (A, u] = oA can be used to express the fields At, A2, A::l: 2, A+l and A+3 in terms of A- 1, A- 3 and their conjugates u+ 1 , u+3 A-2

A1 A2

= -oA-3 + A-1 2 A- 3 - 3oA- 1 """u+l A- 3 ) = -~{8 3 2 A- 3 - ~oA- 1 - u+ 1 A- 3 } = -~(8 3 2

~(Q3A- 3 -3o2 A- 1 -o(u+lA-3 )}+u+lA- 1

+u+3A- 3

A+t

=

A+2

A- 3 - ~82 A- 1 = ~(Q3 3 2

o(u+t A-3)) + u+3A- 3

A+3

= ~(Q4A-3 - ~Q3A- 1 -

o 2 (u+l A- 3)} + o(u+ 3 A- 3) + u+ 3 A- 1 •

3

(4.39) The remaining two identities are non-trivial and read

=

(8- 2oA- 1 - r1- 1o)u+ 1 - (2A- 3 o + 3oA- 3 )u+3

=

_!_(2A- 3Q3 + 9oA- 3 o2 + 1582 A- 3 8 + 1083 A- 3 )u+l 12

-~o(u+lu+l)A- 3 - ~u+lu+loA- 3 3

3

+(8- r1- 1o- 3oA- 1 )u+3,

{4.40)

where (4.41)

Comparing eq. (4.40) with the W3 Ward identities for c -+ ±oo in eq. (4.28), one finds that they are identical if one identifies u

1 1 = --u+ 2 ' h

= ..4-1,

(4.42}

and puts "{ 2 = -2/5. The action f oo[h, b) for induced W3 gravity can now be obtained from the S/{3, R) action in exactly the same way as we obtained the action r(h] for induced

579

pure gravity from S/(2, R). We find

where we should put c

k

= 24.

(4.44)

The latter identification agrees with the leading term of a W3 KPZ formula [32, 9]. We see that the induced action for W3 gravity takes the form of the Legendre transform of a constrained WZW model. In order to obtain an explicit form for the action we introduce a Gaussian parametrization for Sl(3, R):

From u

= fJgg- 1 , we can solve the constraints eq. (4.38), h

=

8/3 8JI

rPl

=

_!(a!I)-lfJ2/1 3

r/>2

=

183/38/1- 8hfJ3!I -382/38/I- 8/382/1

rP3

= 84>2 + r/>l

_

!(a(ahn-1a2(ah) 3 8/I 8ft

A1 3

= (8(:~:))-2(8/I)-1

A2 3

= a(:~:)( aJI)-1.

(4.46)

From A = [)gg- 1 and eq. (4.42) we obtain

h 6

-

~

8hfJ3!I b_lab

=

8(8") -

'Y fJ3 /38!1-

fJ(~)

3 fJ2/3fJ/J- fJ/3fJ2/1

=

'Y

-1

(fJhoft- fJJ18h) (82 /38ft- 82 ft8hf

2 (4.47)

The effective currents u and v are u

= -~[a 2

(a.At + a.A2) + (a.At) 2+ (a.A2) 2+ (a>.t) (a.A2)] 2

At

.A2

.At

.A2

.At

.A2

A different parametrization, which stays closer to the Polyakov parametrization, is given by

(4.49) In linearized form, this parametrization was already found in [10]. In these variables one has that h

=

(I+ , 2Ytt

(~~ + a~a,u)

-~a (In [(a/?(1 + , 2 9)]) b- ~ab b

=

1

(o-~a) 9

7 (a/) 2 (1 +~g),

(4.50)

where (4.51)

Pure gravity now simply corresponds to putting g = 0. We can now substitute the Gaussian decomposition into the action in (4.43). There is now no cancellation between the kinetic term of the WZW model and the uA correction term and one finds the following surprisingly simple expression for the induced action

581

_(~~1) (~~2) (aA~1 + ~~2)]} .

(4.52)

By using the expressions in eq. (4.46), this can be further reduced to an expression in terms of / 1 and h only. An expression of r in terms of h and b seems hard to obtain, as it is not clear to us how the relations (4.47) or (4.50) can be inverted explicitly. From ou = OT7 + ["l,u], we can express "1 1 ,"1 2 ,q±2,"1+1 and "1+ 3 in terms of q- 1,q- 3 ,u+l and u+3. The result can, of course, immediately be read off from the fact that 8u- [A, u] = 8A and 6u- [q, u] = 8q have a similar structure. For the transformation rules of the effective currents u and v we find 1 1 ou = ~f + t.8u + 28t.u + 15 AOv + 10 OAv,

ov = e8v + 38ev + EJ5 A+ (2A~ + 98A82 + 1582 AO + 10~ A)u +8(28A + A8)uu,

(4.53)

where -1

-218'1 -3

(.

=

'1

A

=

'"'( '1

-1 -3

(4.54)

'

which agrees with eqs. (4.24). Combining this with 6A (4.42) we obtain

oh

=

8e + e8h- 8eh + 310

= [)q +

[q, A] and eqs.

(2A~- 38A82 + 382 A8- 2~A) b

+158 (A8b- b8A)u ob

=

e8b- 28eb + 8A + 2A8h - 8Ah,

(4.55)

in agreement with (4.18) and (4.23). Combining the solution of the constraints with the fact that (ogg- 1 ) = '1 yields 8/1

o!J

=

JA8fi8ln [(8 !J8fi - 8 f18!J)]

eofi + '"'fA02 /I-

~8A8/I- 2

f8/3+'"'fA8 2 h-

~8A8h- 2;

2

A8/38ln [(82138/t- 8 2 /1oh)] .(4.56)

After the change of variables eq. (4.49), we find

of

=

e8f- '"'(A82 / -

2

~'"'(8-\of- ~'"'t,\ 1 :~g

582

(4.57)

We will now study the quantization of the W3 gauge fields which leads to the effective action. At first sight, the presence of subleading terms in the induced action seems to cause problems. However, we found evidence, which we will show in a moment, that the subleading terms in the induced action donot appear in the effective action, since they are cancelled by further terms that arise from quantum loops of the gauge fields h and b. We will argue that the effective 1PI action will be equal, up to multiplicative renormalizations, to the leading term in the 1/c expansion of the induced action. Before we proceed, we first introduce a (c-independent) reference functional fL(h, bj, which is such that the leading terms induced action can be written as (4.58)

where the dots indicate terms of order 1, 1/c, etc . The first few terms in fL[h, b] are fL [ h,I b

11

= 211"

11

d2 X h fJ3 [j h + 60 11"

85 + · ·· d2 X b{jb

.

(4.59)

We have that fL[h,b] satisfies the Ward identities eq. (4.28) with

(4.60) A related functional WL[u, v] is obtained from rL[h, bJ by a Legendre transformation WL[u, v]

11

= rL[h(u, v),b(u,v)]-;

1 tfx (hu + 30bv)'

(4.61)

where h(u,v) and b(u,v), which we denote by hL(u,v) and bL(u,v) for later reference, are determined through the relations in (4.60) and we have that

(4.62) The induced action can now be written in terms of these reference functionals

583

+2532. 71 ;1 [II1+0(b4h->1 ,b6 , ... ) +0(~). c

(4.63)

where the consecutive lines are the terms of order c, 1 and 0(1/c), respectively. The generating functional W[t, w] of connected Green's functions is defined by

(4.64) The Legendre transform of W[t, wJ yields the lPI or effective action. In terms of diagrams, the complete perturbative evaluation of W[t, w] involves two independent loop expansions, one with matter loops due to the path integral over 4> and the second with gauge field loops due to the integral over h and b. The net result can be analyzed as follows in terms of a 1/c expansion for large c (which is the weak coupling regime). In the same spirit as in section 2, we approximate the path integral (4.64) by the saddle-point contribution. This leads to the leading term in W[t, w], which is simply the Legendre transform of the induced action. This tree result should then be corrected by further terms coming from diagrams with hand b loops. We now observe that the kinetic terms for h and bin the induced action are proportional to c, such that 1/c plays the role of Planck's constant in the path integral (4.64), while the interaction terms in the induced action are of order c or subleading by extra powers of 1/c (see (4.30), (4.31), (4.33) and (4.63)). From this it follows that the the loop-corrections to the saddle-point result are suppressed by a strictly positive power of 1/c as compared to the leading terms in the saddle-point result. The 1/c expansion of the induced action rind[h,b] in (4.63) leads to a 1/c expansion of the saddle-point contribution to W[t, w] when the latter is viewed as a function of tfc and w/c, i.e.,

W[t,w]

= (4.65)

Through the orders c and c0 , this saddle-point contribution can be written as f;nd[h, b]-

~

1

=

1: (1+ s: + ...)

z!w) =

3!0

(1 + 3:c6 + ...) .

(4.78)

Obviously, the way we arrived at this proposal is rather cumbersome and one would expect that more streamlined derivations should be possible. We are convinced that the non-trivial cancellations that occured in our computations are a true sign of the integrability of this quantum field theory. We remark that the result for kc is consistent with the formula kc =-

4~ (so-c+ J