Gauge Theories, Applied Supersymmetry, Quantum Gravity 9061867339, 9789061867333

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Series B: Theoretical Particle Physics, edited by R. Gastmans, W. Troost and A. Van Proeyen Instituut voor Theoretische Fysica Katholieke Universiteit Leuven Celestijnlaan 200 D B-3030 Leuven

Leuven Notes in Mathematical and Theoretical Physics Volume 6 Series B: Theoretical Particle Physics

Gauge Theories, Applied Supersymmetry, Quantum Gravity Proceedings o f the workshop held at Leuven, July 10 1 4 ,1 9 9 5 ‫־‬

Edited by B. de Wit, A. Sevrin, K. Stelle, M. Tonin, W. Troost, P. van Nieuwenhuizen and A. Van Proeyen

Leuven University Press

1996

LEU V EN NOTES IN M ATHEM ATICAL A N D THEORETICAL PHYSICS 1. M. Fannes , A. V erbeure (eds.),

Mathematical Methods in Statistical. Mechanics. Conference Proceedings, Leuven, 23-24June 1988, Series A: Mathematical Physics 1989, IV - 196 p., ISBN 90 6186 306 6, 600,- BEF 2.

D . PETZ,

An Invitation to the Algebra o f Canonical Commutation Relations, Series A: Mathematical Physics 1990, iv-196 p., ISBN 90 6186 360 0 ,4 0 0 ,- BEF 3.

P. VAN NIEUWENHUIZEN,

Anomalies in Quantum Field Theory: Cancellation o f Anomalies in d=10 Supergravity, Series B: Theoretical Particle Physics 1 9 8 9 ,1 2 0 p., ISBN 90 6186 345 7 ,4 5 0 ,- BEF

4. J. G0VAERTS, Hamiltonian Quantisation and Constrained Dynamics, Series B: Theoretical Particle Physics

1991, vm-371 p., ISBN 90 6186 445 3, 950,- BEF 5. J. M ekisz , Quasicrystals. Microscopic models on nonperiodic structures, Series A: Mathematical Physics

1993, vi-100 p., ISBN 90 6186 573 4,450,- BEF 6. P. D e Wrr, A. Sevrin, K. Stelle, M. Tonin, W. Troost P. V an N ieuwenhuizen, A. V an Proeyen (eds.)

Gauge Theories, Applied Supersymmetry, Quantum Gravity. Proceedings of the workshop held at Leuven, July 10-14,1995, Series B: Theoretical Particle Physics

1996, x-358 p., ISBN 90 6186 733 9,1.150,- BEF 7. W. Troost, A. V an Proeyen

An Introduction to Batalin-Vilkovisky Lagrangian Quantisation Series B: Theoretical Particle Physics F o rth co m in g

© 1996 by Leuven University Press / Presses Universitaires de Louvain / Universitaire Pers Leuven. Blijde-Inkomststraat 5, B-3000 Leuven (Belgium). No part of this book may be reproduced, in any form, by print, photoprint, microfilm or any other means without written permission from the publisher. ISBN 90 6186 733 9 D / 1 9 9 6 /1 8 6 9 / 12

Preface The idea for the project Gauge Theories, supersymmetry, and applied supersymme-

try, endorsed by the European commission under the Science program, started some day at the office of Sergio Ferrara at CERN. The setup was successful, as the group of 16 institutes of 8 different countries came to close collaborations. Sergio’s foresight in including ’applied supersymmetry’ became clearer over the years, as many of the group’s activities focused on very modern developments in quantum field theory, centred around dualities and mirror symmetries which boomed world-wide since 1994. The workshops which have been organised in the context have shown this evolution. When the project was approved, the first item on the activity list was the organisation of a conference to bring all participants together. One of the main aims of this meeting, which was held in the international venue of Trieste in may 1993, was to renew and improve contacts with partners of 16 different institutes and 8 different countries, in order to improve the efficiency in the research activities towards realising the common goals as set out in the approved research proposal. The meeting was organised by the participating group of Sissa, guided by P. Frè and R. Iengo. Because of the nature of the meeting, the range of subjects treated was vast. A second meeting was held, in the same spirit, organised by A. Ceresole, R. D’Auria and M. Rasetti of the Politécnico di Torino, in September 1994 in Torino.

Even though the original proposal did not explicitly

mention the organisation of such a repeat meeting, the financial restraint exercised in the organisation, combined with the universal positive appreciation of participants, made it mandatory to have, in the third and final year of the original project, a third meeting to discuss results obtained by the different groups and look towards the future. In view of the truly world-wide character and open nature of the scientific enterprise, the proposal to extend the scope of this network conference, a five day workshop to be held in Leuven, found general approval among network representatives. With the help of Alexander Sevrin, we arranged immediately to divide the time equally between Network speakers (one speaker appointed by each institute) and external speakers. For the latter, earlier connections to the network participants played an important role. This resulted in a very homogeneous group, which included to a large extent the former supergravity practitioners. An old habit of this group was taken up: the soccer match. An N = 4 -‫־‬ team was pitted against an N = 2 team, and won . . . 4 against 2, of course. Dualities had become the hottest application of supersymmetry, and the very warm atmosphere at the workshop was a pleasure for all. The activities that were held mirror the different dualities of purposes of the participants at the meeting. Of course the seasoned researchers who wanted to discuss their latest result with colleagues could find plenty of opportunities here. But also the students were not forgotten: every day, the meeting was opened with a lecture introducing one of the “hot topics”. These lectures contributed in no small measure to the success of the meeting. They successfully gatliered the full audience (aren’t we all students?), instigating it with a spirit of learning and

v

teaching that was to spill over to numerous research talks as well. The official w orking day was closed by a discussion session, in which elementary questions on the foundations were encouraged. It was prepared by students who took notes of the morning lectu re, and by an experienced chairman. Peter van Nieuwenhuizen, the inventor of this setu p , refused all questioners who approached his age. The appreciation for these lectu res, and for the seminars, was so great that, although this was not planned originally, th e organisers felt that they had to meet the wish of many to make them available m ore permanently, in the form of proceedings. Thanks to the efforts of Bernard de W it, K elly Stelle and Mario Tonin it became a realistic project which could be finished in a short time. And so this book was made. The reader will find here the lecture notes and seminar summaries. Three sets of notes are closely connected to the dominant theoretical themes of the m eeting. T h ese are by Robbert Dijkgraaf on Four-manifolds and Topological Field theory, by Stefan T h eisen on Mirror Symmetry and Duality, by Lerche on N=2 Supersijmmetric Gauge Theory. The fourth set, on Phenomenological aspects of Supersymmetry by H ans-Pcter N illes, emphasises the link with the physics of the standard model of elementary particles. W e are sad that a fifth set, by Peter Townsend on Black holes, could not be included in th e present volume: interest in this work was so great that someone took it from his h o m e, the whole computer included. The lecture notes are supplemented with an accou n t o f the research seminars, 25 in all. Finally, some words of thanks. To the K.U.Lcuvcn, which provided the p ractical facilities for lecturing, computing, and accommodation. To the sponsoring in stitu tio n s, the EEC and the NFWO, the Flemish research council. The latter provided financial aid both through a direct grant and through the funds of the “O nderzoeksgenieenschap theoretische en wiskundige natuurkundc”, of the Flemish Community adm inistered by the Institute for Theoretical Physics of the KUL. We also want to express our appreciation to Andre Verbeure, the institute’s director, and to all our graduate stu d en ts, especially Piet Claus, Alex Deckmyn, Ruud Siebelink, Piet Termonia, François Vanderseypen, Stefan Vandoren and Bart Vereecke, who contributed in the organ isation . Also the help of the K.U.Leuven in providing accomodation was appreciated. B u t th e final and warmest words of thanks must go to our secretary, Christine D el rove.

Her

practical sense and attention to numerous ‘forgotten’ little things was ap preciated by all participants, and most of all by the organisers.

W alter T r o o st Toinc Van P ro eyen January 1996

VI

Network participation The institutes involved in the science network, and the speakers representing them at this conference, are given in the following table. K.U. Leuven Paris ENS Utrecht London Imperial College Neuchatel Ludwig-Maximilians München Technische Universitat München Frascati Torino university Torino politécnico Milano Padova SISSA Trieste Napoli Genova Nordita Copenhagen

Stefan Vandoren —

Costas Sfctsos Kris Thielemans —

Jan Louis Stefan Stieberger —

Jeannette Nelson Anna Ccresole Luciano Girardello Dima Sorokin Marco Billó Gerardo Cristofano Carlo Becchi .Joergen Rasmussen

In a new proposal in the framework of ”Training and Mobility of Researchers”, this group is extended with 3 more institutes, at this conference also treated as part of the network: V.U. Brussel Groningen Humboldt Univ. Berlin

Alex Sevrin Harm-Jan Boonstra Christian Preitschopf

Contents1 Preface......................... Net work part ici pat ion

.V Vii

Lectures Lectures on four-manifolds and topological gauge theories R. Dijkgraaf..........................................................................................................................

I

Mirror symmetry and duality

A. Klcmm and S. Theism .................................................................................... Notes on N =2 supersymmetric gauge theory IT. Lerchc..............................................................................................................................

1yj

Phenomenological aspects of supersvmmetrv H. P. Xilies ....................................*.............' .............................................................................. ‫ן‬

Seminars Black holes in superstring theory R. E. Kallosh......................................................................................................................... ... The Gribov problem, Cech-De Rham cohomology and the instability of BRS sym m etry in 2‫־‬dimensional topological gravity C. Bccchi and C. Imbimbo............................................................................................... 7‫ן ן‬ Redefining B-twisted topological sigma models F. Dc Jonghc, P. Tcrmonia, IP. Troost and S. Vandortn..........................................! 2}) On T-dualitv and supersymmetry K. Sfetsos................................................................................................................................... 139 On Calabi-Yau compactifications of D = 11 Supergravity .4. Ceresole................................................................................................................................ 149 Strong/weak coupling duality and dyonic solutions in string theory H. J. Boonstra.......................................................................................................................... 157 Space-time symmetries in duality symmetric models P. Pasti, D. Sorokin and M. Tonin.................................................................................... 167 Instabilities in strong magnetic fields in string theory E. Kiritsis and C. Kounnas........................................................................................ 177

... ‫׳‬

Magnetic backgrounds in closed superstring theory .4..4. Tseytlin ............................................................................................................................. 187 S-duality in N =4 supersymmetric gauge theories

L. Girardello, A. Giveon, M. Porrati and A. Zaffaroni................................................ 199 1The texts of the lecture by P. Townsend, and the talks by P. Sorba, II. Verlinde and S. Yankielowicz are absent.

vm

Aspects of duality in N = 2 .string vacua V. Kaidanovsky, ,/, Louis anil ,S'. Theism ................................................................... 205 Moduli dependence of Perturbative Gauge ( Couplings in four-dimensional heterotic string compactifications P. Mayr anil S. Sliihcrycr............................. 211 Oclonions and supersymmetry ('.II. Pri ilsclioyf................................................................................................................. 225 New spacetime-supersymmetric methods for the supers! ring N. Be rko oils .............................................................................................................................................................................2•Id R-syminctry of heterotic N = 2, 1)=1 supergravities M. miió .................................................................................................................................. 21:1 Super /»-form charges and a reformulation of the supennembrane action in eleven ditnensions /!‫’׳‬. Sczgin................................................................................................................................ 253 Exact quantisation of 2+1 gravity on the torus ./. F. Nelson........................................................................................................................... 203 Quantum Hall fluid in the presence of dissipation: transport properties, compressibility and quantum coherence G. Cristofano, D.Giuliano and G. Maiclla.....................................................................275 Black hole in a periodic Kasncr universe D. Korolkin andII.Nicolai ................................................................................................. 283 Quantum ( 2,2 ) supergravity

M.T. Grisaru and M.E. Wehlau

289

Gauging Wess-Zumino-Witten Models /1. Scorin.............................................................................................................................. 299 Gauging conformal algebras with relations between the generators A*. Thielemans and S. Vando ren ......................................................................................309 Conformal blocks for admissible representations in SL(2) current algebra J.L. Petersen, J. Rasmussen and M. Yu ...................................................................... 319 Twistors and supersymmetry P.S. Hoiue.............................................................................................................................329 The equivalence between the operator approach and the path integral approach for quantum mechanical non-linear sigma models

Jan de Boer, Bas Peelers, Kostas Skenderis and Peter van Nieuwenhuizen...................................................................................................339

Participants

353

L E C T U R E S O N F O U R -M A N IF O L D S A N D T O P O L O G IC A L G A U G E T H E O R IE S

ROBBERT DIJKGRAAF

Department of Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands ABSTRACT I give an elementary introduction to the theory of four-manifold invariants and its relation with topological field theory. I review the recent developments in the theory of Donaldson and Seiberg-Witten invariants.

1. A L ectu rer’s A p o lo g y Why another set of lectures on four-manifolds and topological field theory? Apart from the contractual obligations imposed upon the lecturer there might be some justification in the following. The field of four-dimensional topological field theory has seen some remarkable advances in the last year. This has brought the topic of four-manifolds and Donaldson theory to the attention of a much bigger audience in the physics community. More or less as a service to those who are interested but feel intimidated to read the mathematical literature or the recent research papers, I present an elementary introduction to the field in these lectures. I am very much aware of the limited scope of the material as treated here. I will have to refer to the literature for further reading. Luckily, there are some very good books [3, 4, 5] and recently some excellent review papers have appeared that I can warmly recommend [1, 11). These lectures are organised as follows. I will first review in §2 some standard material about four-dimensional geometry. One could call these classical invariants. The introductory chapter in the book by Donaldson and Kronheimer is a good place to read more about this. I will then proceed in §3 by sketching the ideas of Donaldson theory and Seiberg-Witten theory. Since these invariants are closely related to quantum field theory, one can regard these as quantum invariants. Finally in §4 I provide a very brief introduction to topological field theories in general.

2 . C lassical Invariants It is a remarkable fact that in many aspects dimension four is as distinctive from a mathematical perspective as it is from a physical perspective. Many theorems that can be proven in generality for manifolds of dimension d > 5, such as the generalized Poincare conjecture (Smale 1961), meet fundamental difficulties in the case of four dimensions. With d > 5 generic and d < 3 often simple enough ‘to do by hand’, this leaves d = 4 as the most challenging arena in differential topology. It has been a beautiful development that the richness of four-dimensional physics, with asymptotically free

1

non-abclian gauge theories* can be brought to bare upon these difficult m athem atical problems. The classification problem of 4-manifolds is very much dilfcrcnt in nature than, say, the classification of surfaces. Compact, orient able surfaces arc topologically classified by their genus g t M. That the situation is not that simple in four dimensions wc can already see by considering the most important invariant of any manifold — lh e fundamental group t!. It is a classical theorem in topology that any finitely representable group (i.c. a group that can be represented by a finite number of generators satisfying a finite number of relat ions — not a very severe restriction) can appear as the fundam ental group of a four dimensional manifold. There exist a surgery algorithm that constructs the required manifold starting from the generators and the relations. Markov’s solution of the word problem shows that the question whether two finitely representable groups are actually isomorphic is undecidable. That is, there is no computing algorithm that can decide this question within a guaranteed finite time. This theorem has therefore rather dramatic consequences for the classification problem of manifolds in dim ensions d > 4. There is simply not a conceivable ‘list’ of four-manifolds. This fundamental problem can however easily be circumvented by assuming that the four-manifold X is simply connected n(A ') = o.

( 2 . 1)

This we will often assume in the following.

2.1. The intersection form After the fundamental group, the next important invariant is the second cohom ology group H2(X). We recall that De Rham cohomology groups Hk(X) of a compact manifold X are defined as the space of closed differential forms of degree k modulo exact forms

da = 0,

a ~ a + d/3.

( 2 .2 )

The Betti numbers 6* of X are the dimensions of these groups

bk = dim Hk.

(2.3)

To understand better the role played by the second cohomology of a four-manifold, it might be instructive to consider first the analogous situation for two-dim ensional surfaces, if only because it is so much more easily visualized. For any two-dimensional surface X we can consider the first homology group H \(X ) of homology cycles or equivalenty the dual cohomology group / / 1(A') of 1-cocycles. We can think of a 1-cocycle physically as a flat abelian gauge field A, F = dA = 0. T he pairing between a cycle C € //1 and a cocycle A € Hl is the Wilson line

/ A.

Jc

(2.4)

For a genus g surface Hl has rank 6! = 2g. It is naturally a symplectic space by the intersection form

Q W ) = j io A /? . 2

(2.5)

Fig. 1: A canonical homology basis for a surface of genus g.

Q is an integer, unimodular anti-symmetric form. By a standard theorem in symplectic linear algebra there exists a canonical symplectic base a ! , . . . , ,..., satisfying =

( 2.6)

,) = ¾ .

with all other intersections vanishing. The dual homology basis 0, , 6, we can picture as in fig. 1 . If we have a orientation-preserving difieomorphism on the surface, it will act on the first cohomology by a symplectic transformation. That is, there is a map (actually a surjection) Diff+( ; 0 - S ‫־‬p(2 © //^ 2.

(2.65)

So the Hodge and Betti numbers .are related by

6+ = 2h2'°+ 1.

( 2.66)

Since we have a complex structure, we have an extra characteristic class, the first Chern class of the tangent bundle T\ cl(X) = cl(Tx ). (2-67) It is closely related to the canonical bundle which is defined as A a‫ = ׳‬det T'x = A2T;x.

(2.CS)

Its sections are (2 ,0)-forms (holomorphic volume forms) that locally look like

f ( z \ z 2) - d z l Adz2.

(2.69)

This line bundle gives a distinguished divisor, the canonical divisor often also denoted by the symbol K. It is represented by minus the first Chern class of the manifold A' = ct(/v ) = - Cl(X).

(2.70)

We also have

K =

w2

(mod 2).

(2-71)

We can think of the divisor A' as the curve on which a meromorphic (2,0) form has its zeroes and/or poles. If h2,0 > 0 then we can pick a holomorphic form. Note that the second Chern class equals the Euler class, and therefore does not give a new invariant

c2(X) = e.

11

(2.72)

Noto also that the condition 0! = 0 tolls us that tho canonical bundle is t rivial and that the manifold A theroforo is a ( ,alahi-Vau space. In four-dimensions this is equivalent to being hvpcr-Kãhler and there are only two compact examples: the four-torus 7 '1 and A'3. There is also a holomorphic index theorem, Noether’s formula ('!'odd genus, cMndcx) which reads A«‫ ״‬- /,»■' + A « = i (v + ‫ = ) ״‬Jx i (c; + c2.7 3 )

.(‫) ־‬

This can be used to express the length of the canonical class l\ as A’2 = cj = 2\ + 3(t = a

(mod 8).

(2 .7 1 ‫)׳‬

Finally, we want to mention the concept of Kodaira dimension. In the case of Riemann surfaces or algebraic curves there arc many reasons to distinguish the cases of the sphere N*. the torus 7'2 and genus g > 1. For example from the perspective of uniformisation. automorphism groups etc. One way to make this distinction is to look at the number of holomorphic »•differentials, i.c. sections of the tensor powers of the canonical bundle A ", as a function of the weight » € N. If this grows as /10(A'") ~ »*,

(2.75)

we call K the Kodaira dimension. If /1° = 0 we declare />‫ = ־‬-o o . We easily check with the Riemann-Roch theorem in hand that

-O O ,

1. As an example (or exercise) of all the various classical invariants we have introduced in this section let us consider a 4-manifold (complex surface) of the form X — Cg x C), with Cs a Riemann surface of genus g. We find the following invariants: \

=

4 ( 1 - ‫ ( ) ע‬1 -/» ),

K

=

(2g-2)[Ch\ + {2h-2)[C9l -OO g = 0 or h = 0,

0

1 or vice versa,

2

g ,h > l. 12

3. Q uantum Invariants Wc now turn to the manifold invariants first studied by Donaldson. For more information see the books [3, 4, 5]. As wc will see in the next section, these are actually correlation functions in a topological field theory, as was first realised by Witten [ 12]. We also discuss very briefly Sciberg-Witten theory.

3.1. Donaldson theory The starting point in Donaldson theory is non-abclian gauge theory with gauge group 57/(2). (Apart from the closely related case of 5 0 (3 ) here is very little known about the case of a general Lie group.) So let A be a 57/( 2) gauge field on a principal bundle P over the four-manifold X. It will have curvature

F = dA + A7.

(3.78)

Wc denote by A the space of all connections on P. Since the difference between two connections is a 1-form with values in the Lie algebra s«( 2), A is an affine space associated to the vector space fi 1(51/(2)). It naturally decomposes in connected components labeled by the instanton number n of P

n=

Ix

Tr F7 € Z.

(3.79)

87r2

The gauge group

Q = Aut ( P )

(3.80)

acts on the connection in the usual way. In a local trivilisation a gauge transformation is a map h : U C X —> SU{2) and A transform as

h : A -+ h 1Ah + h l dh.

(3.81)

The relevant space we are interested in is actually the quotient space

B := A/Ç0,

(3.82)

where Q° are the gauge transformations that leave one fiber fixed. This group acts freely on «4, so the quotient B is a smooth infinite-dimensional manifold. After we pick a metric g on X , we can consider anti-self-dual (ASD) connections or instantons, satisfying the following non-linear equation

F+ = F + *F = 0.

(3.S3)

The moduli space of ASD connections is defined as the space of solutions to this equation modulo gauge equivalence

M n = {/1 6 A ‫ ן‬F+ = 0}/g°. 13

(3.S4)

Although "v do not emph.wirc this in tho notation, it is important to realise that the definition of A t, dej»ends on the choice of metric 0 . This follows directly by considering the Yang-Mills action functional

s = ^ í J V (F A t F )
0.

(3.S7)

2.

Generically. M ‫ ״‬is an oriented manifold. Its virtual dimension can be computed by an index computation

d := dim M n = 8n - 3(1 —6! + = Sn — ‫( ־‬x + ,. . . vanish. In particular, to zero'th order in o' this means that the background has to be Ricci flat, i.e. Rij = 0. This Ricci-fiatness condition does however get corrections from higher loops in sigma-model perturbation theory; we will comment on this below.

27

Since we are dealing with strings rather than point ])articles, it. is not the cla ssical geometry or even topology of A* which is relevant. Much of the attraction o f strin g theory relies on the hope that the modification of the concept of classical g eo m etry to string geometry at very small scales will lead to interesting effects and ev en tu a lly to an understanding of physios in this range. At scales large compared to the strin g scale, which is related to o ',1 a description in terms of point particles should h e valid and one should recover classical geometry. This limit is referred to as the large rad iu s limit. Possible deformations of the background fields which do respect conformal invariance are parametrised by moduli. The simplest example is the radius of the circle for A’ = Fvtu.j x 5 ' for the bosonic string. In a low-energy (four-dimensional) effective field theory the moduli appear as n eu tral massless scalar fields T, with vanishing potential, i.e. their vacuum exp ectation value (T,) is undetermined. The low-energy effective action

S‫ ״‬, =

J

S x y /j v q T iM W A ; ,...)

(2 )

can be computed in sigma model and/or string perturbation theory. In the ab ove effective action T, are the moduli fields, C the charged matter fields, A “ gau ge fields and R cS ~ t - '*' ~ p-. Here g is the four-dimensional coupling ‘con stant’; the quotation marks indicate that it is in fact field dependent. Sigma-model p ertu rb ation theory is an expansion is powers of a ‫ ׳‬whereas string perturbation theory is an exp an sion in powers of g. The aim is of course to get the non-perturbative effective a ction . This requires to incorporate (i) world sheet instantons which are non-perturbative in the sigma-model coupling constant (~ c_fi Ia ) and (ii) space-time in stan tons w hich are non-perturbative in the string coupling constant ( ~ e-1/9‫ ׳‬. The effective a ction is restricted if we require e.g. the theory to be space-time supersym m etric. O th er restrictions might come from duality symmetries (3). In order to discuss the nature of duality symmetries we use the Let S , ( X , G , B , $ , . . . ) be a string theory (t = 11eterotic, type II ( A a target space A' with background fields G ,£ , $ , . . . . The notion theory is then well captured by paraphrasing Mark Kac’s famous hear the shape of a drum?‫״‬, i.e. are there dual theories such that 5 ,(A-, G, B, ¢ , . . . ) = S , { X \ f f , B \ ¢ , , . . . ) ?

following notation : or B), typ e I) w ith of duality in strin g question “C an you

(3 )

Here the equivalence means that the two theories have not only the sam e n onperturbative spectrum, but also the same correlation functions. One d istin g u ish es between various types of dualities. • T-iuality is non-perturbative in the sigma-model coupling constant a ‫ ׳‬and valid order by order in the string coupling constant p = Re(S). An example for T -d u a lity is the bosonic string compactified on a circle: bosonic string on(IR 24,1 x S 1, /? )= 1

See also the recent work by Shenker on the possibility of a second scale in string theory

[2]• 28

bosonic string on(JR21,1 x S l ,a'/R). This is just a special case of Bushers’ duality; (for a review, see (4]). Another important example, which will be elaborated below, is mirror symmetry: heterotic string( 1R3,! x X; T) = heterotic s tr in g (^ ,! x A *;T *) where X,X* are a mirror pair of Calabi-Yau manifolds and (T, T*) are the moduli of X, X", which are related by the mirror map. These concepts will be treated in more detail below. In general X .and X * have different topology. • S-duality is non-perturbative in the string coupling constant g but valid orderby-order in a'. An example for 5 —duality is heterotic string(IR.3^ x T 6,5 ) = heterotic string(®^^ x T 6,S ') where S' =

,

6 SL(2,7L).

This

duality has been conjectured in [5]. More evidence, especially for the realization on the Bogomol’nyi saturated spectrum, has been given recently in [6]. • U-duality is the unification of T and 5 duality. It is non-perturbative in a' and (j. E.g. for the heterotic string(!!^,! x T6) the conjectured [/-duality group is SO ( 6, 22,Z ) x S I ( 2 ,Z ) [7]. • String ■String ■Duality is a non-perturbative equivalence between string theories whose perturbative definition might be very different. E.g. in four dimensions the N = 4 heterotic string on (IR.3,1 x T 4 x T 2) is conjectured to be equivalent to the N = 4 type II string on (IR3,! x A'3 x T 2). One argument in favour of this equivalence comes from the analysis of the above [/-duality group for both cases

im Similarly, for N = 2 supersymmetric string theories, it has been suggested [9] that h eterotic(!^ ,! x A'3 x T2) is equivalent to type II(IR3,1 x CY-threefolds). Dual pair candidates have first been constructed in [10]; see also the contribution of J. Louis to this volume. These examples of string-string-dualities have the very interesting property that the dilaton of one theory is mapped to a target space modulus in the other. This fact makes space-time non-perturbative predictions possible since world-sheet instanton effects in the type II representation are mapped to space-time instanton effects in the heterotic representation [11]. This has been named second quantised mirror symmetry in ref. [12]. Below ten dimensions the type IIA theory is equivalent to the type IIB theory, but for transformed target space moduli. E.g. type IIA(IR3,1 x A';T) = type IIB (E 3,1 x A'*; T ‘ ). Here the dilaton is not exchanged with a target space modulus, hence the situation is not interesting for the calculation of spacetime nonperturbative effects. The technical virtue of this mirror-duality is that it identifies sectors with nontrivial world sheet instantons contributions with sectors, which axe not world sheet instanton corrected, as will be explained in detail below. In addition to these dualities, which relate different types of string theories, or the same string theory, but at different points of their moduli space, there are duality symmetries between strings and extended objects, such as p-branes, or the more recently discussed D-branes; see [13]. This is the string analogy of the Olive-Montonen electricmagnetic duality in A/ = 4 supersymmetric gauge theory, under which electric charges and magnetic monopoles exchange their roles as fundamental and solitonic objects in the dual formulations of the theory (for a review, sec [14]). Wc thus have to extend the index on Si to include other theories, such as theories with fundamental p-branes

29

or oven eleven-dimensional supergravitv [15] among the possible theories related by duality symmetries to string theories. This leads to the question whether ‘strin g theory' is really a theory of strings [1C]. Even though we do not know the answ er it is clear that we will get there only through an understanding of non-perturbative effects and that string duality symmetries will be the crucial tool, since they m ap a strin g theory at strong coupling to another string theory at weak coupling. We have decided to concentrate in this lecture on one particular duality sym m etry, namely mirror symmetry. This has been well understood by now, at least to the p oint where it can be used as a working tool to compute non-perturbative w orld-sheet effects for strings on Calabi-Yau manifolds. Results from mirror symmetry have also been of use in checking the duality between certain type IIA and certain heterotic string theories. We will come back to this issue at the end of this lecture. The general setting for mirror symmetry as it will be discussed here is prim arily the heterotic string compactificd on a Calabi-Yau manifold. The adoption for typ e II string theory is then immediate; we will comment on the type II case at places. Let us thus turn to a discussion of the relevant facts of Calabi-Yau com pactifications. One of the basic facts of string theory is the existence of a critical dim ension w hich, for the heterotic and type II string, is ten. To reconcile this with the observed fourdimensionality of space-time, one makes the compactification ansatz that the tendimensional space-time through which the string moves has the direct product form .1/10 = -1/4 x -Y where A’ is a six-dimensional compact internal manifold, and 71/.! is four-dimensional Minkowski space. In the following we want to concentrate on the case where A* is a so-called Calabi-Yau manifold. The original motivation for this was that this leads for the heterotic string to N = 1 supersymmetry in the four uncom pactificd dimensions [17] [IS] [19].

Def.2: A Calabi-Yau manifold is a compact Kãhler manifold with trivial first Chcrn class. The condition of tririal first Chern class on a compact Kãhler manifold is, by Y au’s theorem, equivalent to the statement that they admit a unique Ricci flat K ãhler m etric.3 The necessity is easy to see, since the first Chern class c! (.Y) is represented by the 2-form A-p where p is the Ricci form, which is the 2-form associated to the Ricci tensor of the Kãhler metric: p = Rijdz' Adz1. Locally, it is given by p = —idd log det(( g-,j)). One of the basic properties of Chern classes is their independence of the choice of Kãhler metric; i.e. p[g') = p(g) + da. If now p[g) = 0, cj(A") has to be trivial. T h at this is also sufficient was conjectured by Calabi and proven by Yau [20]. Ricci flatness also implies that the holonomy group is contained in SU( 3) (rather than 17(3); the 17(1) part is generated by the Ricci tensor /?,; = Rijkk)• If the 2 Here and below we restrict ourselves to the three complex-dimensional case. 3 Note th at this does not imply that every metric on a CY manifold is Ricci flat. In fact, computing higher loop a-modei contributions to the beta-function

da

and finds th a t confor-

mal invariance does not require that the a-model metric be Ricci flat, but there are higher derivative corrections to the vacuum Einstein equations on the CY manifold. Y au’s theorem does however ensure that there is a unique Ricci flat metric whose associated K ãhler form is in the same cohomology class.

30

holonomy is SU( 3) one has precisely N = 1 space-time supersymmetry in the case of a heterotic compactification and N = 2 space-time supersymmetry in the case of type II compactification. Other dim([; = 3 manifolds with c! = 0 but smaller holonomy groups arc the six-dimensional torus T° and K 3 x T2, which lead to ;V = 4 and N = 2 space-time supersymmetry, respectively, in a heterotic compactification. Another consequence of the c! = 0 condition is the existence of a unique nowhere vanishing covariantly constant holomorphic three form, which we will denote by ft = Çlijkdz' A dz> A dzk ( i , j , k = 1,2,3). Here z' are local complex coordinates of the CY space. Since ft is a section of the canonical bundle1, vanishing of the first Chern class is equivalent to the triviality of the canonical bundle. Before entering a discussion of the moduli space of CY manifolds, which is central in the discussion of Calabi-Yau compactifications and in p;1rticular of mirror symmetry, we take a look at the Hodge numbers of CY manifolds. According to Hodge theory, on a compact Káhler manifold one has a decomposition of cohomology groups 4 5*

Hk(X,£) = ®P+q=kH™(X)

(4)

If we define the Hodge numbers /!7‫’׳‬, (AT) = dim Hp'q(X), then the fc’th Betti number bk(X) = d\mHk(X,C) is

M -Y)= £

*‫(׳־■־׳‬-V)

(5)

/)+7=^ The Hodge numbers of a Calabi-Yau manifold are conveniently arranged in the Hodge diamond

4 The canonical bundle is the highest (degree dim(p(Ar)) exterior power of the holomorphic cotangent bundle T*. 5 Note th at on an arbitrary compact manifold, any fc-form

a

admits a decomposition

a = Y l y + q - k a ^P'q^ w^1cre is a (p, are elements of H l { X , T x ) which are related to the harmonic (2,1) forms via the unique element of H3{X): ( 6ft)j = 9p p - ^ , 6)‫ ״‬ft)fc/j, ||fi||2 = Note that while the former couplings arc purely topological the latter do depend on the complex structure (through Í2). Both types of cubic couplings are totally symmetric. Note also that by the discussion above there is a one-to-one correspondence between charged matter fields and moduli: 27 ( 1,1 ) moduli and 27 < - 2 , 1 ) (0) = 1. In addition, there are /r • 1 p eriods u‫׳‬,(z ) which are linear in logarithms (log‫כ‬,): they can be chosen to be of the general form u’i(z) = tt>o(z) log ‫ב‬, + u’,(z) with ti0 = (0);‫׳‬. These are. in fact the only so lu tio n s to the Picard-Fuchs equations needed for the mirror map, which is

< ,=

«’.(z) ~ log Z{ + power series in z w‫׳‬o(z)

( 21 )

We want to stress again that here ‫ ;כ‬are coordinates on the structure m od u li sp a ce of A * in a neighbourhood of the large complex structure point and /, arc co o rd in a tes on the Kãhler moduli space of A* where /, —►ioo corresponds to the large rad iu s limit, i.e. going far out in the Kãhler cone. Inverting the mirror m ap. w hich can always be done locally, leads to a multiple power series r, = c,(q ) where q = e2n,)|p =0

\ D ( 3)t0‫(״‬z,/>)|p=0 /

43

u0‫(׳‬z)n(z).

(27)

We can always take U as the period vector, simply by rescaling ÍÍ by 1/ 0‫יוז‬

U n d er

this change of Kalder gauge the Yukawa (57*) Yukawa couplings change as k If we now use the mirror conjecture we have to interpret the second row in FI as the special coordinates in the Kahler moduli space on A* and the third row as OiF with F the prepotential for A*',1(A‘). This leads to a concise expression for th e fu lly instant on corrected (275) couplings on A'(34):

*«MO = dtA j

‫ ■ ״ ” ! ״‬o W f M /‫=י‬ L0

(2 8 )

u’0(r (0 )

After presenting all this rather formal material, let us now dem onstrate how it is p u t to practice on a particular example. We take the hypcrsurface in weighted p ro jectiv e space F [6 .2 ,2 .1 .1 ). For explanation of notation and other useful facts ab ou t C alab iYau manifolds, see e.g. (19). We have chosen this example since it appears as th e ty p e II model of one of the dual type IIA/hetcrotic pairs of [IO].16 This particular ex a m p le had already been discussed in the context of mirror symmetry in [33)(31), w here m an y details and other examples can be found. To satisfy the Calabi-Yau condition, c! = 0, wc need a degree 12 polynom ial to d efin e the hypersurface, We will denote it by A '12[6,2,2,l,l). The defining p olyn om ial is Po = r2 + z\ + =6 + 12‫ ב‬+ z12

(2 9 )

where po = 0 defines the hypersurface. The subscript ‘0' refers to the fact th a t we can perturb the polynomial by monomial perturbations which are also of degree 12. It is not difficult to find all possible, up to homogeneous coordinate tran sform ation s, 126 monomial perturbations of degree 12. The coefficients of these m onom ials in th e general 12th order polynomial are coordinates in the complex structure m oduli sp ace. This Calabi-Yau manifold does however admit 128 complex structure d eform ation s, two of which have no algebraic description in terms of a monomial perturbation of th e defining polynomial po (see, however, [77]). This model thus has /12,1 = 128. A lso, on e can show that h1,1 = 2. One of the harmonic two forms is just the Kahler form in d u ced from the Kãhler form of the projective space in which the CY manifold is em b ed d ed . The second harmonic (1,1) form is due, via Poincaré duality, to the exceptional d ivisor which arises by resolving the singular curve in the embedding space. The mirror can now be defined by orbifolding with an appropriate discrete group of phase symmetries or, which is equivalent in this example, as the hypersurface assodated to the second member of a pair of reflexive polyhedra, in the language o f toric geometry. Here the situation is simple as the mirror can again be expressed as th e vanishing locus of a transverse polynomial in the same embedding space and o f th e same degree as the original CY, namely p* = Po - a0z!Z2Z3Z4Z5 - « 1(24 zs)6 16 See also J. Louis1 contribution to this volume.

44

(3 0 )

The two tnonorninl perturbations are the only ones which survive the orbifolding by the discrete phase symmetry. To show that the manifold X * so defined also has /1*•1( *) = 128 takes more effort. One has to resolve the singularities introduced by the orbifolding and count the harmonic (1,1) forms, which are, via Poincare duality, associated to the exceptional divisors that have to be introduced in the process of resolution of the singularities. The manifold p* = 0 is transverse for generic values of the moduli. It, however, fails to be transverse if the moduli satisfy the relation A = (1 - x! )2 - x2x 2 where v/e have introduced the coordinates x! = 1728x! = 1728^ and i! = 4x2 = 4 7 ‫ ־‬. These 1* 0‫״‬ coordinates turn out to parametrise the vicinity of the large complex structure point, which is x! = X2 = 0. Since this model has /12,1 = 2, there are six periods. If we expand them in the vicinity of the large complex structure point, there will be one power series solution, the fundamental period, two solutions with terms linear in logarithms, two solutions quadratic in logarithms and one solution cubic in logarithms. The fundamental period is easily computed as17,18

t0»)0‫״‬,(11) = 00 —u.

59

whore \Z \2 = U2 + Y 2. The important point is that the BPS bound (3.1) is saturated by a certain class of excitations, namely the “BPS-statcs” that obey Q\rp) = 0 . T h e idea is that if a state obeys this condition semi-classically, it obeys it also in the exact quantum theory, because the number of degrees of freedom of a “short” (or “chiral” ) multiplet that obeys (?|0 = (‫ז‬/‫ י‬is smaller as compared to the degrees of freedom o f a generic supersymmetry multiplet, and the number of degrees of freedom is su p p osed not to jump when switching on quantum corrections. In particular, since ,t H ooftPolyakov monopoles do satisfy the BPS bound semi-classically, they m ust ob ey it in the exact theory as well. FYom semi-classical considerations we can also learn th a t the monopoles lie in JV = 2 hvpermultiplcts, which have maximum spin £.

For AT= 2 supersymmetric Yang-Mills theories, the central charge takes the form Z = qa + g a o ,

(3 .2 )

where (g , ç) are the (magnetic,electric) quantum numbers of the BPS state under consideration. Above, ap is the ”magnetic dual” of the electric Higgs field a and belongs to the vector multiplet (Ap, Ho,p) that contains the dual, magnetic photon, A'q . By studring the electric-magnetic duality transformation, under which the ordinary electric gauge potential .4‫ *׳‬transforms into A^, it turns out [1] that in the N = 2 Yang-Mills theory the dual variable ap is simply given by: ‫ ״‬D = £ r ( < 3 .3 )

. ( ‫)־‬

That is, the general idea would be that at the singularity at u = A2, one has a ^ 0 but ap = 0, such that (by (3.2)) a monopole hypermultiplet with charges ( g , q ) = (± 1 ,0 ) would be massless. On the other hand, one would have that u = 0 does not imply that a = 0 in the exact theory, such that, in contrast to the classical theory, no gauge bosons with charges (0, ±2) become massless. This in particular would imply that the classical relation u = 2a2 holds only asymptotically in the weak-coupling region. The point is now to view ap(u) as a variable that is on a equal footing as a(u); it just belongs to a dual gauge multiplet that couples locally to magnetically charged excitations, in the same way that a couples locally to electric excitations (such as Hr~). A priori, it would not matter which variable we use to describe the theory, and which variable we actually use will rather depend on the region in M q that we are looking at. More specifically, in the original semi-classical, “electric” region near u = oc, the preferred local variable is a, and an appropriate lagrangian is given by (2.4). As mentioned above, the instanton sum converges well for large a ~ \ f u / 2 . However, if we try to extend T { a ) to a region far enough away from u = oo, we will leave the domain of convergence of the instanton sum, and we cannot really sake any more much sense of T . That is, in attempting to globally extend the *flective lagrangian description outside the semi-classical coordinate patch, we face :he problem of suitably analytically continuing T . The point is that even though we annot have a choice of T that would be globally valid anywhere on M q (it would »nflict with positivity, cf., (2.6)), we can resum the instanton terms in T in terms of »ther variables, to yield another form of the lagrangian that converges well in another egion of M q. The reader might already have guessed that while a is the preferred 60

variable near u = oo, it is ap that is the preferred variable in the “magnetic” str coupling coordinate patch centered at u= A2. More precisely, near u to have the following, dual form of the effective lagrangian:

* * • ‫ = => ״‬k V - ^ V 1 ° s [

*‫י‬

‫\׳‬ T

M

f

\

(u),a(u) at sem iclassical infinity, / a D( 2/ \ £ /

^ \ ( ‫ ״‬u log(u /A 2) ) \

\ a(u) / ~ v

y/2u

/

we infer that for a loop around u = 00:

= (‫־‬o’ -1) ■

(3-5)

As for the strong coupling singularities at u = ± A 2, we choose a different strategy: we know on general grounds that the monodromy of a dyon with charges (g, q ) th at becomes massless at a given singularity is given by:

V-9

1-99

(3.G)

This can be seen in various ways, one of which will be explained later at the end o f section 4. The global consistency condition on how to patch together the local, p ertu rb ative data is then simply

M + \1 ‫י‬

— Moo ‫י‬

(3 . ‫) ו‬

since v.‫׳‬e can smoothly pull the monodromy paths 7 around the Riem ann sphere ( 1/0 is an arbitrary base point):

62

7«

F ig.5: Monodromy paths in the u-piane.

One may view equation (3.7) as a condition on the possible massless spectra at u = ± A 2. For matrices of the restricted form ( 3.6), its solution is:

M+ a 1 = Af(1-0) , , M _a* = A/( 1 2 ‫ )־־‬,

(3-8)

which is unique up to irrelevant conjugacy. From this we can read off the allowed (magnetic, electric) quantum numbers of the massless monopoles/dyons. They indeed give back the coefficient of the logarithmic term of T d that we had anticipated in eq. (3.4). If we would consider a situation with more than two strong coupling singularities, we would have to solve an equation like (3.7) with the corresponding product of matrices. However, cursory investigations indicate that such equations for more than three matrices do not have any solution; this can probably be made rigorous, by taking the special (parabolic) form (3.6) of the matrices into account.

4. Solving the monodromy problem The physics problem has thus become a mathematical one, namely simply to find multi-valued functions ( “‫() ״‬u) that display the required monodromies M±A2 0). This is a classical mathematical problem, the “Riemann Hilbert" problem, which is known to have a unique * solution. The RH problem can be accessed from two complementary point of views: either by considering a ,a o as solutions of a differential equation with regular singular points, or from considering a,ap as certain “period” integrals. The latter approach, to be discussed momentarily, allows to more easily implement the right monodromy properties, while the differential equation approach, to be considered later, is more useful for obtaining explicit expressions. f Unique up to multiplication of (*‫() ״‬u) by an entire function; this can however be fixed by considering the asymptotic behavior.

63

Any two of the monodromy matrices M±\*tgenerate the m onodrom y group T a/ , which constitutes the subgroup T0(4) of the modular group 5 1 ( 2 , 2 ) and co n sists of matrices of the form To(4) = {

(

0

= 6 ,(2 , 2 )5 1 6

‫ ״‬mod 4 } . (4 .1 )

This group represents the quantum symmetries of the theory. In particular, w e see that 5 =

0) ("'hich acts as

t

—» —£) is not part of I'm , and this m ean s th at

the theory is not self-dual (in contrast to N = 4 Yang-Mills theory); how ever, other transformations do exist that relate weak and strong coupling sectors. The quantum moduli space can therefore be viewed as

M, = H+/ r 0( 4 ) , where H + is the upper half-plane. Now, motivated by the appearance of a subgroup of the modular group (which is the group of the discontinuous reparam etrizations o f a torus), the idea is that the monodromy problem can be solved in term s o f a toroidal Riemann surface, whose moduli space is precisely M q. Such an elliptic curve indeed exists and can be algebraically characterized by: y 2(x ,u ) = (x 2 —u )2 —A4 4

(4 .2 )

= : J J (x -e ,(u ,A )) 1=1

The point is to interpret the gauge coupling r(a) as the period “m atrix” o f th is torus, and this has the added bonus that manifestly Imr > 0 , by virtue of a m a th em a tical theorem called “Riem aim's second relation” . As such r is defined by a ratio o f period integrals: ^ p (u ) (4 .3 ) r (u ) m(u)

where (4 .4 ) w ith (*‫ = •׳‬V(zu ) ‫ ־‬Here, a,/? are the canonical basis homology cycles o f the toru s, as shown as follows:

Fig.6:

Basis of one-cycles on the torus.

64

From the relation 7‫ = ־‬daop we thus infer that x0 d { u)

dap(u) du

‫)״(לס‬

do(u ) du

(4.5)

That is, the yet unknown functions 0 d ( u ), o( u ), and consequently the prepotential T —f a 0.0 (0 ), are supposed to be obtained by integrations of torus periods. Note that (4.5) implies that we can also write

(4.6)

where

\

dx y(x,u)

(4.7)

(up to normalization and total derivatives) is a particular meromorphic one-form. What remains to be shown is that the periods, derived from the specific choice of elliptic curve given in (4.2), indeed enjoy the correct monodromy properties. The periods (4.4),(4.6) are actually largely fixed by their monodromy properties around the singularities of Mq, and obviously just reflect the monodromy properties of the basis homology cycles, a‫ ׳‬and /?. That is, it suffices to study how the basis cycles a , /3 of the torus transform when we loop around a given singularity. For this, we represent the above torus in a convenient way that is well-known in the mathematical literature: we will represent it in terms of a two-sheeted cover of the branched 1-plane. More precisely, denoting the four zeroes of y2(x ,u ) = 0 by e! = - \J u + A2 ,

t 2 = —\/u — A2

e3 = y/u - A2 ,

e4 = \Ju + A2 ,

we specify the torus in the following way:

Fig.7: Representation of the auxiliary elliptic curve (4.2) in terms of a two-sheeted covering of the branched x-plane. The two sheets are meant to be glued together along the cuts that run between the branch points ei(u). Shown is our choice of homology basis, given

by the cycles a,(}. This picture corresponds to the choice of the basepoint uo > A2 real.

65

The singularities in the quantum moduli space arise when the torus degenerates, and this obviously happens when any two of the zeros ct coincide. T h is can be expressed as the vanishing of the "discriminant‫״‬

Aa = Ò(‫־‬, - tjf = (2A)V‫ ־‬A‘)•

(4.8)

• .

To o b tain th e m onodrom y around A2/ u —♦ 0, one can com pactify th e u —p la n e to P 1, as we did before, and get the m onodrom y at infinity from th e g lo b al re la tio n A i« = M + aj M _ a » (cf., Fig.5). We th u s have reproduced the m onodrom y m atrices associated w ith th e e x ac t quantum m oduli space directly from the th e elliptic curve (4.2), and w h a t th is m e an s is th a t the in teg rated torus periods a o ( u ),a (u ) defined by (4.5) m u st in d eed h av e th e requisite m onodrom y properties. However, before we are going to ex p licitly d e te rm in e these functions in th e next section, let us say some more words on th e g e n eral logic of w hat we have ju st been doing.

66

We have seen in Fig.8 that when we loop around a singularity in the branch points Cj(ti) exchange along certain paths, //, that shrink to zero as c, —♦ c j . Such paths are called “vanishing cycles” and play, quite generally, an important role for the properties of BPS states. Indeed many features of a massless BPS spectrum can directly be studied in terms of the singular homology of an auxiliary Riemann surface, or a /v3 surface or a Calabi-Yau threefold, depending on the physical model under consideration. Specifically, assume that such a path vanishes at a given singularity and has the following expansion in terms of given basis cycles: v

=

gf } + q a

.

(4.10)

T hen, as v = 0, we obviously have 0 = X = g \ Ju

+q

6 \

=

gaD + qa =

so th a t we have at this singularity a massless BPS charges equal to ( g , q). T h at is, we can simply read off less states from the coordinates of the vanishing cycle homology basis, the charges change as well, but this is W hat rem ains invariant is the intersection number

Vi 0 Uj =

Z,

Ja

state w ith (m agnetic,electric) the quantum num bers of mass• ! Obviously, under a change of nothing b u t a duality rotation.

vl • ÍÍ • v = giqj - gjqi € Z ,

(4.11)

where 0 is the intersection product of one-cycles and Q. is the sym plectic (skewsym m etric) intersection metric for the basis cycles. Note th a t this represents the well-known Dirac-Zwanziger quantization condition for the possible electric and magnetic charges, rind we see th at it is satisfied by construction. T he vanishing of the r.h.s. of (4.11) is required for two states to be local w ith respect to each oth er [16,18]. T his m eans th a t only states th at are related to non-intersecting vanishing cycles are m utually local. In our example, the monopole w ith charges (1,0), the dyon w ith charges ( 1 ,- 2 ) and the (massive) gauge boson W + w ith charges (0,2) are all mutually non-local, and thus cannot be simultaneously represented in a local effective lagrangian. Furtherm ore, there is a closed formula for the monodromy around a given singularity associated w ith a vanishing cycle v: the monodromy action on any given cycle, is directly determined in term s of this vanishing cycle by m eans of the “Picard-Lefshetz” formula [19]:

seH1(xtz),

M6 0 ) - 6 seem s to be at the border of observability and further exp erim en ta l resu lts are eagerly awaited.

7 . S u p e r s y m m e t r ic g ra n d u n ifica tio n Experim ental findings give at the moment the following picture; w ith a top quark m ass between 150 and 200 GeV the strong coupling constant cn»(Mz) = 0-12 ± 0.01 and Q‫ ״‬n{M z) = 1/128 the weak mixing angle is sin 2 6\\‫ {׳‬M z ) = 0.231G ± 0.0 0 0 3 . T h is leads to gauge coupling unification at a scale A / \ 2 — ‫ ״‬x 10 1 0 all such solutions would describe a trivial fiat space configuration with all supersymmetries restored. Now we know many examples of four-dimensional electrically charged black holes and monopoles which in the limit when their ADM mass tends to zero rem ain non-trivial solutions with only half of supersymmetry unbroken. The gravitational part o f the configuration is defined by the parameters of the vector m ultiplets Q vec w hich serve as sources and which do not varnish even when the graviphoton charge |Q gr| = M vanishes. The mass formulas in all cases are constructed from various charges of th e vector fields and vacuum expectation values of the scalars in the vector m ultiplets. W e w ill present the known massless solutions for N =4 supersymmetry below. We will exp lain th e m ost unusual properties of massless solutions.

2. D uality sym m etric quantization of superstrings Investigation of BPS states is the key feature of recent activities in string theory. However, these states have a dual status. To obtain the corresponding solutions on e is using purely classical concepts of space and tim e and classical fields, including th e m etric of extrem e black holes or fundamental strings and membranes. On the oth er hand, one is attributing a certain quantum-mechanical meaning to these states. O ne w ould like to get a coherent description of these “geom etries-states” from the point o f view o f string theory. As a step towards understanding the quantum -m echanical m ean in g o f the states describing various geometries, we suggest to find the place o f th e B P S sta tes in the quantization of Siegel’s *-symmetries. It is known that the gauge *-symm etries on the world-line of the su p erp articlc, on the world-sheet of the Green-Schwarz superstring and in the w orld-volum e o f th e supermembrane share the following important property: the spinorial p aram eter o f th e gauge transformation has to be somehow broken into two parts. T h e reason for th is is the fact that the classical gauge transformations of the anticoinm uting coord in ates 0 are SO = ‫ך‬ and the equations of motion is II2 = 0. Therefore the co m m u tin g g h o sts field C , where * = C'A, A being the anticoinm uting parameter of the B R S T sy m m etry ,

108

lias to satisfy some algebraic condition which projects out one half of the spinor. Any spinor can be constructed out of two parts 6' = Ck+ Ck where C\ = XkC and Ck = \ kC where the projectors X k and X k have the following properties: XkXk =

0,

XkXk = Xk

,

XkXk - X k -

(1)

The ghost field which will provide a consistent quantization has to satisfy an algebraic condition [7] c = C\. = X k C , XkC = C É = 0 , (2 ) which was called a truncation condition. The K-symmetric superstring was quantized years ago in the flat background in the generalized semi-light-cone gauge by the author and by A. Morozov [7, 8]. Semi-lightcone gauge is the gauge where the two-dimensional metric is in the conformal gauge and the spinor in the light-cone gauge 0 = 0+‫ ך‬. Generalized semi-light-cone gauge is the one in which the projectors are constructed out of two null vectors n2 = 0, m2 = 0, 2?n • n = 1, X k(g-i.c.) = M

,

X k ( g ■ 1■0 ■) = ' W

•

(;})

The formalism of ref. [7, 8] was further developed by Grisaru, Nishino, and Zanon [9] and by Candiello, Lcchner, and Tonin [10] for the heterotic sigma model in a curved background. The most recent results are described in (10) where also the reference to previous work can be found. Both groups have found the constraints on possible backgrounds where the heterotic string can be quantized consistently in the generalized light-cone gauge. The reason for such constraints on the background comes from the fact that the algebraic condition on spinor does not admit an arbitrary background. Whereas the exact form of the constraint was not clearly identified in a unique way1 it was however firmly established that some constraints must exist for the consistency of quantization. Thus the procedure of consistent quantization consisted of 2 steps: i) choice of the gauge-fixing (and/or ghost truncation) in the flat background first, for example, rfiifc.0 = 0, ii) allowing only the backgrounds whose curvature satisfies R^\smxn6 = 0. At the time when these constraints were discovered the BPS configurations were not very much studied yet. It remained unnoticed that the constraints from quantization are satisfied, in particular, by the background of extreme electric black holes. We will describe such backgrounds later. Now we know many such configurations with unbroken supersymmetries and it is natural to ask whether they provide a consistent background for the quantization. The answer to this question is positive: any BPS background allows a consistent quantization. However, the choice of the gauge-fixing is defined by the background. In particular, the light-cone gauge is suitable for the quantization in the background of electrically charged states. It is, however, inconsistent for the magnetic backgrounds. And vice versa, magnetic gauges are inconsistent for electrically charged backgrounds. The reason is that each background has its own half-size Killing spinor which has to be used in quantization. 1The constraints look slightly different in different papers.

109

The uetr procedure of quantization which is suggested in [1] has the follow in g im portant features. First we choose the particular class of the backgrounds w hose b oson ic part has one half of unbroken supersymmetries. This is a constraint w hich is satisfied by our backgrounds and will guarantee the consistency of q u antization . Instead of making a choice of the algebraically constrained spinor in a flat background first and afterwards look for backgrounds which admit such spinors, we suggest to q u a n tize in the backgrounds which admit Killing spinors. The basic observation is the follow ing. Consistent quantization of truncated «-symmetry is possible in the backgrounds with one half of unbroken supersymmetry. The integrability condition for the existence

of Killing spinors of the bosonic part of the background is the consistency condition for the quantization. This defines the curved superspacc in which quantized « -sy m m e tr ic objects exist. The supercharge of the gravitational supersymmetric theory was defined by T eitelboim [11] in asymptotically flat spaces as the surface integral in term s of th e gra v itin o field of the configuration, solving the field equations:

Q = l

JdZ

d Z ^ V x .

(4)

The surface over which the integration has to be performed depends on th e ch oice of configuration. In all cases it is the same surface the integration over w hich defines the ADM mass of a given system or the ADM mass per unit area (len gth ). T h e onshell backgrounds with one-half of supersymmetry unbroken in bosonic sectors have th e vanishing supersymmetry variation of the gravitino, where the param eters are K illing spinors. Q k=i d ^ Y ^ x tk = JdL

o.

(5)

For anti-Killing spinors the supercharge is not vanishing. For the black hole m ultiplets it defines the so-called superhair of the black hole:

«Ssuperhair = Qt = 0 ) the Lagrangian is degenerate since the field b has 3'e~SM ,)Q(c + tvb + ( 1 - ״‬/ 0 ) ,1‘‫ ׳‬c ‫ ־‬s ‘*(‘)i i ( c + ft)6 + ( l - « ) ‫ •״‬. ••■)

+ / ’ ‘J17.í■'■ + 2 r f c .r í'.,.^ r %' have ghost num ber on e and i p ' an d h.ave g h o st num ber m inus one. W ith these assign em en ts, th e part S° o f th e a ctio n s till c o n ta in s gh osts and an tigh osts, i.e. it is not the classical action , w hich o n ly d e p e n d s o n th e classical fields and not on th e ghosts. Secondly, th e fields A'‘ are m e r e ly la g r a n g ia n m u ltip liers since th ey do not transform under B R S T . So th e o n ly c la ssic a l field w ou ld b e A'**, b u t then the classical action should on ly depend on A'1*. It is clea r t h a t in th is form ulation the B R S T interpretation is obscure. W e w ill now see th a t s o m e o f th e se p rob lem s disappear when including the auxiliary sector.

3.2. Off shell twisting W h en th e au xiliary fields are included, th e B R S T tran sform ation rules are :

6Xr =

+ C'

6F{ = i(d+? - d - V )

= I f ‘•

S’!•‘‘ = “

SC = ~ c L A * = —^d+ X '

F ‘•

6Fr = 0

6X{ = 0 .

( 8)

From th e se exp ression s it is obvious th a t Ó2 = 0. It is proposed in [9] to in te r p r e t 6 as a B R S T op erator o f a so far unspecified gauge sym m etry. T h e a ctio n o f th e

■

(11)

The classical part does not depend on X", and therefore one has a gauge (shift-)sym m etry 8X'm = e'*, and the corresponding ghosts In accordance with the spirit of the B R ST -anti-B R ST scheme [5], one introduces also an antighost 0'*, and its BR ST variation F ,m. Apart from this quartet, there is a second set of fields transforming into each other, viz. F l, 1j>l,£' and A'1. They ensure that one restricts the Ar' to be constant, which was also the case in the 6 picture. Indeed, the BRST gauge fixing condition s£* = —-c L A ' forces these maps to be holomorphic. It is then the anti-BRST operator that kills the anti-holomorphic part of A ‘, since it is anti-BRST exact. There are still two things that are unsatisfactory. First, if F' is interpreted as a classical field, and the classical action is zero, then the gauge symmetry on F' would be an arbitrary shift. Looking at its transformation rule we did not include this shift sym m etry. Secondly the identifications above do not yet exhibit the customary structure of B R S T -anti-B R S T , which exhibits more symmetry between ghosts and antighosts: the an ti-B R S T transformation of the classical fields are usually identical to their BRST transform ation, when replacing ghosts with antighosts. This is not the case for the second set of fields above, since we then also have to interchange d+ and cL. In the N = 2 a-m odel, the starred and unstarred fields occur symmetrically. The tw ist has lifted this sym m etry: the former are all spinless, but V’1 and have helicities 1 and -1 respectively. One can construct 1l'ldx+ and £'dx~, which behave as one forms under holom orphic coordinate transformations. The asymmetry is mirrored in the derivatives in the transformation laws for the second set, which is in accordance with the helicity-assignm ent. At the same time, one can also consider Fl to be a two-form, which can not be distinguished from a scalar in the treatment with a flat metric. The B R S T -a n ti-B R S T sym m etry can be redressed by the following non-local change of field ‘‫ *־‬F o r a r e v i e w o f t h e u s e o f B R S T - a n t i - B R S T s y m m e t r y o f g a u g e t h e o r i e s , w e r e fe r t o (5 ].

133

variables: V■‘

= Ô+X*

e

=0 -‫א‬

p

=3.3+//1.

(12)

AU th e fields on th e right hand sid e arc scalars. R em ark th a t th e J a c o b ia n o f th is tran sform ation is equal to unity, at least form ally, sin ce th e c o n tr ib u tio n s from th e ferm ions can cel again st th e bosons. For th e new variables w e can ta k e th e tra n sfo rirm tio n rules ,1

s‫ א‬:

S \‫־‬

sH' = ip'

sir

(13)

to rep rod u ce th e so far U n e x p la in e d ” rules in (1 0 ). T h e}‫ ׳‬now co rresp o n d to a sh ift s y m m e tr y for th e field H', in trod u cin g th e gh ost field T h e a n tig h o s t is p ' y a n d A'1 c o m p lete s th e q u artet. It is clear th a t we have uncovered a m a n ifest B R S T a n t i- B R S T sy m m etry . T h e a ctio n , w hen w ritten in term s of th e new field s, is of c o u r se s till B R S T exact: on e sim p ly w rites th e ex a ct term in e q . ( l l ) in term s o f th e n ew v a r ia b le s. T h is allow s th e follow ing in terp retation . O ne sta rts from tw o cla ssic a l fie ld s, X'* an d H'. T h e classical actio n is zero, and th e sy m m etries are sh ift s y m m e tr ie s , w ith g h o s ts £** and \* . T h en o n e in trod u ces an tigh osts ‫ז‬/>'* and />', and L agran ge m u ltip lie r s A"' and ■F**. T h is co m p lete s th e field con ten t of th e theory. N o te th a t th e a c tu a l c o n te n t o f th e resu ltin g T F T d ep en d s h eavily on th e gauge fixing proced u re, as u sual: th e r e are no p h y sica l lo ca l flu ctu a tio n s, b u t global variables m ay rem ain. H avin g changed th e B R S T operator, we now d iscu ss th e im p lic a tio n s o f th is ch a n g e. F irst o f aU, w e in v estig a te w hether we still have a to p o lo g ica l th eo ry in th e s e n s e th a t th e en erg y -m o m en tu m is B R S T ex act for the new B R S T o p erator. A fterw a rd s, w e w ill in v e stig a te w h eth er th e ph ysical con ten t (observables) o f th e th eo ry h as ch a n g e d .

5. T h e en erg y m o m en tu m ten sor T h ere are tw o m etrics in th e m od el : th e world sh eet m etric hap, w h ich w as ta k e n to b e flat, and th e sp a ce tim e K ãhler m etric 5 ,j•. T h e world sh eet m etr ic is e x te r n a l. T h e sp ace tim e m etric is a fu n ction of X * and A '* , w hich are in teg ra tio n v a ria b les in th e p a th in tegral. W e can thus o n ly stu d y th e d ep en d en ce o f th e p a th in teg ra l o n th e hap m etric, b y co m p u tin g th e energy m om en tu m tensor. T h e co m p u ta tio n is a n a lo g o u s to th e L an dau-G inzburg m od el [1] and we find =

- g t]. d +X > 'd +X { + 2igl}.r p W +^ ' = s [ 2 ig ^ i> 'd +X 3')

T f_

=

- gi]. d - X > ‘ d .X < + 2i g a - C V . ? • = s [2ig ij. ^ d - X j ']

Tf_

=

0.

(14)

A fter th e d erivation , w e have taken th e m etric to b e flat. T h ese are th e r e fo r e th e relevant operators for variations of correlation fu n ction s around a flat m e tr ic . W e se e th a t, alth ou gh th e action is B R S T ex a ct, th e ( + + ) co m p o n en t o f th e en erg y m o m e n tu m tensor is on ly a n ti-B R S T ex a ct. T h is is because th e B R S T o p erator d e p e n d s on th e 134

metric and one cannot com m ute the BRST variation and the derivative v/.r.t. the metric [6]. To prove m etric independence of correlation functions, one needs not only BRST invariance, but also the Ward identity for the anti-BRST operator. What is needed is that the physical operators are BRST invariant, and that their anti-BRST variation is BRST exact. For a more com plete argument, see [1]. The proper cohom ological formulation is, that one first determines the s cohomology, a space of equivalence classes. The operator s is well defined and nilpotent in that space, so that its cohom ology can be used as our characterisation of physical states3. This characterisation is not arbitrary, but more or less forced upon us by the requirement that the energy m om entum tensor is trivial.We now investigate this cohomology.

6. The spectrum The observables of the B twisted th e N S N S a n d HR a n tisy m m e tr ic ten sor fields, resp ectively. T h e scalar m a trix is again g iv e n b y (G) b u t here th e axion is replaced by th e R R scalar C: A = A! + 1A2 = C -+‫ ־‬/t‫־‬- *. T h e field stren g th o f th e four-form field is given by

(D) = d^D^^pà] +

) 14)

‫)־‬

T h e S L (2 ,R ) sy m m etry is ea sily recognized: 7 i —►

,

M

—» cjM wt .

(1 5 )

5The corresponding conjectured discrete symmetry of the full string theory is named {/-duality (8). 6We use formulas given in ( 25) , in which also the relation between type I I A and type I I I ) fields i n the presence of one isometry is given. ‘From now on we will use hats to denote ten-dimensional fields.

162

From now on we will refer to this sym m etry as 5 /,(2 , R ) i f b • We now use a sim ple ansatz for dimensional reduction to D = 6. Our goal is to extend the Z2 sym m etry (12) to an 5 / ( 2 , R) symmetry and we will accomplish this by taking into account two additional scalar fields coming from the internal metric and the four-form field D. Our ansatz for the ten-dimensional fields is - _ -G 9m n — - 6 m neG , 9vv ^ *7/ji / ‫ו‬ II

‫־‬O-

fjup