Table Of ContentÉCOLE DE PHYSIQUE DES HOUCHES – UJF & INPG – GRENOBLE
a NATO Advanced Study Institute
LES HOUCHES
SESSION LXXVI
30 July – 31 August 2001
Unity from duality: Gravity, gauge theory and strings
L’unité de la physique fondamentale :
gravité, théorie de jauge et cordes
Edited by
C. BACHAS, A. BILAL, M. DOUGLAS,
N. NEKRASOV and F. DAVID
Springer
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Preface
The 76thsessionofthe Les HouchesSummer Schoolin TheoreticalPhysics
was devoted to recent developments in string theory, gauge theories and
quantum gravity.
As frequently stated, Superstring Theory is the leading candidate for
a unified theory of all fundamental physical forces and elementary parti-
cles. This claim, and the wish to reconcile general relativity and quantum
mechanics,haveprovidedthemainimpetus forthe developmentofthe the-
ory over the past two decades. More recently the discovery of dualities,
and of important new tools such as D-branes, has greatly reinforced this
point of view. On the one hand there is now good reason to believe that
the underlying theory is unique. On the other hand, we have for the first
timeworking(thoughunrealistic)microscopicmodelsofblackholemechan-
ics. Furthermore, these recent developments have lead to new ideas about
compactification and the emergence of low-energy physics.
Whilepursuingthegoalofunificationwehavealsowitnessedadramatic
return to the “historic origins” of string theory as a dual model for meson
physics. Indeed, the study of stringy black branes has uncovered a surpris-
ing relation between string theory and large-N gauge dynamics. This was
cristallizedintheAdS/CFTcorrespondence,whichhasrevivedtheoldhope
forastringdescriptionofthestronginteraction. TheAdS/CFTcorrespon-
dence is moreovera prime illustrationof the centralrole of string theory in
modern theoretical physics. Much like quantum field theory in the past, it
provides a fertile springboard for new tools, concepts and insights, which
should have ramifications in wider areas of physics and mathematics.
The main lectures of the Les Houches school covered most of the re-
cent developments, in a distilled and pedagogical fashion. Students were
expected to have a good knowledge of quantum field theory, and of basic
string theory at the level, for instance, of the first ten chapters of Green,
Schwarz and Witten. The emphasis was on acquiring a working knowledge
of advanced string theory in its present form, and on critically assessing
open problems and future directions.
ThelecturesbyBernarddeWitwereacomprehensiveintroductiontosu-
pergravitiesin different dimensions andwith variousnumbers of supersym-
metries. Topics covered include the allowed low-energy couplings, duality
symmetries, compactifications and supersymmetry in curved backgrounds.
xxii
Part of this is older material not easily accessible in the literature, and
presented here from a modern perspective.
EliezerRabinovicilecturedonsupersymmetricgaugetheories,reviewing
earlier and more recent results for N = 1, 2 and 4 supersymmetries in four
dimensions. Theseresultsinclude the structureofthe effective lagrangians,
non-renormalization theorems, dualities, the celebrated Seiberg-Witten so-
lution and brane engineering of effective gauge theories.
M-theoryandstringdualities wereintroducedinthe lecturesbyAshoke
Sen. He reviewed the conjectured relations between the five perturbative
string theories, the maximal N = 1 supergravity in eleven dimensions and
their compactifications. He summarized our present-day knowledge of the
still elusive fundamental or “M theory”, from which the above theories
derive as special limits. More recent topics include non-BPS branes, where
duality is of limited (but not zero) use.
Philip Candelas gave a pedagogical introduction to the important sub-
ject of Calabi Yau compactifications. He first reviewed the older material,
and then discussed more recent aspects, including second quantized mirror
symmetry,conifoldtransitionsandsomeintriguingrelationstonumberthe-
ory. Unfortunately a written version of his lectures could not be included
in this volume.
The holographicgauge/stringtheory correspondencewas the subject of
the lectures by Juan Maldacena and by Igor Klebanov. Maldacena intro-
ducedtheconjecturedequivalencebetweenstringtheoryinthenear-horizon
geometries of various black branes and gauge theories in the large Ncolor
limit. He focused on the celebrated example of N = 4 four-dimensional
super Yang Mills dual to string theory in AdS5 ×S5, and gave a critical
review of the existing evidence for this correspondence. He also discussed
analogous conjectures in other spacetime dimensions, in particular those
relevant to the study of stringy black holes, and of the still elusive little
string theory.
Igor Klebanov then concentrated on this duality in the phenomenolog-
ically more interesting contexts of certain N = 1 and 2 supersymmetric
gauge theories in four dimensions. He reviewed the relevant geometries on
thesupergravityside,whichincludenon-trivialfluxesandfractionalbranes,
and discussed the gravity duals of renormalizationgroup flow, confinement
andchiralsymmetrybreaking. Theseresultshaverevivedandmadesharper
the old ideas about the “master field” of large N gauge theory.
ThelecturesofMichaelGreendealtwithsomefineraspectsofstringdu-
alities and of the gauge theory/string theory correspondence. He discussed
higherderivativecouplingsineffectivesupergravityactions,focusinginpar-
ticular on the contributions of instantons both in string theory and on the
xxiii
YangMillsside. HislecturesalsoincludedsomeintroductorymaterialonD-
branes. Unfortunatelyawrittenversionofhislecturescouldnotbeincluded
in this volume.
AndrewStromingergaveadetailedintroductiontoquantumgravityina
deSitterspacetime. Hediscussedinparticularwhetherideasofholography,
that have worked well in anti de Sitter, could also be applied in this case.
Thiswasoneofthemorespeculativesubjectsintheschool,butafascinating
one not the least because astrophysical observations seem to indicate that
we actually live in an accelerating universe.
Finally Michael Douglas gave three lectures on D-brane geometry, and
in particular on the problem of classifying all N = 1 string-theory vacua,
whileAlexanderGorskydiscussedN =1andN =2supersymmetricgauge
theories and their relation to integrable models. Nikita Nekrasov lectured
on open strings and non-commutative gauge theories.
Some more advanced and/or topical subjects were covered in the ac-
companyingseriesofseminars. SeminarspeakersincludedLaurentBaulieu,
Mirjam Cvetiˇc, Frank Ferrari, Dan Freedman, Bernard Julia, Peter Mayr,
Soo-Jong Rey, Augusto Sagnotti, Samson Shatashvili, and one of the orga-
nizers(C.B.). Therewasalsoalivelyweeklystudentseminaranddiscussion
sessions,whichcontributedgreatlytothelivelyandstimulatingatmosphere
of the school. Some of the seminar speakers have kindly accepted to con-
tribute to the present volume.
In the yearthat has elapsed since the end of the schoolthere have been
further developments in the subject. The pp-waves,which arise as Penrose
limits of near-horizon geometries, offer for instance a new line of attack on
the important problem of solving string theory in Ramond-Ramond back-
grounds. Such developments and others will no doubt make, one day, the
presentvolumeobsolete. Thisis ofcoursenoreasonfor regret–to the con-
trary we hope that this may happen sooner rather than later, and that the
participants of this school will help shape the (non-recognisable?) future
form of M theory.
Acknowledgements
This Les Houches school and the present proceedings were made possible
due to the contributions of many individuals and several different funding
sources.
Funding sources included the NATO Scientific Affairs Division through
its Advanced Study Institute program, the European Union through its
High-Level Scientific Conferences program, the Commissariat `a l’E´nergie
Atomique, the Universit´e Joseph Fourier de Grenoble, and two European
xxiv
IHP Networks: “Superstring Theory” (HPRN-CT-2000-00122) and “The
Quantum Structure of Spacetime” (HPRN-CT-2000-00131).
Among the many people contributing to the success of the school, we
should mention
- the board of the School and in particular Franc¸ois David, who has
worked tirelessly at all different stages (funding applications, admis-
sions,runningthesession,preparationofproceedings)exceedingoften
the organizers in zeal and energy;
- the secretariesMmes G.D’Henry,I.Leli`evreandB.Rousset(andthe
other personnel of the school), who helped solve administrative and
everyday problems;
and last but not least
- the lecturers, for their efforts in presenting hard material in a clear
and pedagogical fashion, and also for writing up their lecture notes.
Costas Bachas
Adel Bilal
Michael Douglas
Nikita Nekrasov
Franc¸ois David
CONTENTS
Lecturers xi
Participants xiii
Pr´eface xvii
Preface xxi
Contents xxv
Lecture 1. Supergravity
by B. de Wit 1
1 Introduction 3
2 Supersymmetry in various dimensions 4
2.1 The Poincar´e supersymmetry algebra. . . . . . . . . . . . . . . . . 5
2.2 Massless supermultiplets . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 D=11 supermultipets . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 D=10 supermultiplets . . . . . . . . . . . . . . . . . . . . 9
2.2.3 D=6 supermultiplets . . . . . . . . . . . . . . . . . . . . . 13
2.3 Massive supermultiplets . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Central charges and multiplet shortening. . . . . . . . . . . . . . . 19
2.5 On spinors and theR-symmetrygroup HR . . . . . . . . . . . . . 22
3 Supergravity 28
3.1 Simple supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Maximal supersymmetry and supergravity . . . . . . . . . . . . . . 32
3.3 D=11 supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Dimensional reduction and hidden symmetries . . . . . . . . . . . 38
3.5 Frames and field redefinitions . . . . . . . . . . . . . . . . . . . . . 43
3.6 Kaluza–Klein states and BPS-extendedsupergravity . . . . . . . . 46
3.7 Nonmaximal supersymmetry: Q=16 . . . . . . . . . . . . . . . . 53
4 Homogeneous spaces and nonlinear sigma models 55
4.1 Nonlinearly realized symmetries . . . . . . . . . . . . . . . . . . . . 56
4.2 Geometrical quantities . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Nonlinear sigma models with homogeneous target space . . . . . . 66
4.4 Gauged nonlinear sigma models . . . . . . . . . . . . . . . . . . . . 69
xxvi
5 Gauged maximal supergravity in 4 and 5 dimensions 70
5.1 On E7(7)/SU(8) and E6(6)/USp(8) cosets . . . . . . . . . . . . . . 72
5.2 On ungauged maximal supergravity Lagrangians . . . . . . . . . . 74
5.3 Electric–magnetic duality and E7(7) . . . . . . . . . . . . . . . . . 79
5.4 Gauging maximal supergravity; the T-tensor . . . . . . . . . . . . 84
6 Supersymmetry in anti-de Sitter space 89
6.1 Anti-deSitter supersymmetry and masslike terms. . . . . . . . . . 93
6.2 Unitary representations of theanti-deSitter algebra . . . . . . . . 97
6.3 The superalgebras OSp(N|4) . . . . . . . . . . . . . . . . . . . . . 110
7 Superconformal symmetry 116
7.1 The superconformal algebra . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Superconformal gauge theory and supergravity . . . . . . . . . . . 122
7.3 Matter fields and currents . . . . . . . . . . . . . . . . . . . . . . . 126
Lecture 2. Supersymmetric Gauge Theories
by D.S. Berman and E. Rabinovici 137
1 Introduction 141
2 Supersymmetric quantum mechanics 143
2.1 Symmetryand symmetry breaking . . . . . . . . . . . . . . . . . . 151
2.2 A nonrenormalisation theorem . . . . . . . . . . . . . . . . . . . . 152
2.3 A two variable realization and flat potentials . . . . . . . . . . . . 154
2.4 Geometric meaning of the Witten index . . . . . . . . . . . . . . . 158
2.5 Landau levels and SUSYQM . . . . . . . . . . . . . . . . . . . . . 159
2.6 Conformal quantummechanics . . . . . . . . . . . . . . . . . . . . 161
2.7 Superconformal quantummechanics . . . . . . . . . . . . . . . . . 164
3 Review of supersymmetric models 165
3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.2 Superspaceand chiral fields . . . . . . . . . . . . . . . . . . . . . . 167
3.3 K¨ahler potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.4 F-terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.5 Global symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.6 The effectivepotential . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.7 Supersymmetrybreaking. . . . . . . . . . . . . . . . . . . . . . . . 172
3.8 Supersymmetricgauge theories . . . . . . . . . . . . . . . . . . . . 173
4 Phases of gauge theories 179
xxvii
5 Supersymmetric gauge theories/super QCD 180
5.1 The classical moduli space . . . . . . . . . . . . . . . . . . . . . . . 182
5.2 Quantummoduli spaces . . . . . . . . . . . . . . . . . . . . . . . . 185
5.3 Quantum moduli space for 0<NF <NC . . . . . . . . . . . . . . 185
5.4 Integrating in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.5 Quantum moduli space for NF ≥NC . . . . . . . . . . . . . . . . . 190
5.6 NF =NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.7 NF =NC+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.8 Higgs and confinement phases . . . . . . . . . . . . . . . . . . . . . 192
5.9 Infra-red duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.10 Superconformal invariance in d=4 . . . . . . . . . . . . . . . . . . 202
6 Comments on vacuum energies in scale invariant theories 207
7 Supersymmetric gauge theories and string theory 210
7.1 Branes in string theory. . . . . . . . . . . . . . . . . . . . . . . . . 210
7.2 Branes in IIAand IIBstring theories . . . . . . . . . . . . . . . . . 211
7.3 The effective field theory on branes . . . . . . . . . . . . . . . . . . 213
7.4 Effective D=4 dimensional systems with N =2 supersymmetry . 216
7.5 An effective D=4, N =1, U(NC) gauge theory with matter . . . 226
7.6 More pieces of information. . . . . . . . . . . . . . . . . . . . . . . 229
7.7 Obtaining thedual field theory . . . . . . . . . . . . . . . . . . . . 233
7.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8 Final remarks 237
Lecture 3. An Introduction to Duality Symmetries
in String Theory
by A. Sen 241
1 Introduction 243
2 A brief review of perturbative string theory 245
2.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
2.3 Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
3 Notion of duality symmetries in string theory 255
3.1 Duality symmetries: Definition and examples . . . . . . . . . . . . 255
3.2 Testing duality conjectures . . . . . . . . . . . . . . . . . . . . . . 258
xxviii
4 Analysis of low energy effective field theory 260
4.1 TypeI–SO(32) heterotic duality in D=10 . . . . . . . . . . . . . 261
4.2 Self-duality of heterotic string theory on T6 . . . . . . . . . . . . . 264
4.3 Duality between heterotic on T4 and typeIIA on K3 . . . . . . . . 268
4.4 SL(2,Z) self-duality of typeIIBin D=10 . . . . . . . . . . . . . 272
4.5 Otherexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
5 Precision test of duality: Spectrum of BPS states 276
5.1 SL(2,Z) S-duality in heterotic on T6 and multi-monopole moduli
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
5.2 SL(2,Z) duality in typeIIB on S1 and D-branes . . . . . . . . . . 286
5.3 Massless solitons and tensionless strings . . . . . . . . . . . . . . . 296
6 Interrelation between different duality conjectures 299
6.1 Combining non-perturbativeand T-dualities. . . . . . . . . . . . . 299
6.2 Duality of dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
6.3 Fiberwise duality transformation . . . . . . . . . . . . . . . . . . . 301
6.4 Recoveringhigher dimensional dualities
from lower dimensional ones . . . . . . . . . . . . . . . . . . . . . . 304
7 Duality in theories with less than sixteen supersymmetry
generators 305
7.1 Construction of a dualpair of theories with eight supercharges . . 306
7.2 Test of duality conjectures involvingtheories
with eight supercharges . . . . . . . . . . . . . . . . . . . . . . . . 309
8 M-theory 312
8.1 M-theory in eleven dimensions . . . . . . . . . . . . . . . . . . . . 312
8.2 Compactification of M-theory . . . . . . . . . . . . . . . . . . . . . 315
Lecture 4. Les Houches Lectures on Large N Field Theories
and Gravity
by J. Maldacena 323
1 General introduction 325
2 The correspondence 330
2.1 The field ↔ operator correspondence . . . . . . . . . . . . . . . . . 336
2.2 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
3 Tests of the /CFT correspondence 341
3.1 The spect(cid:0)ru(cid:1)m(cid:2)of chiral primary operators . . . . . . . . . . . . . . 342
3.1.1 The field theory spectrum . . . . . . . . . . . . . . . . . . . 342
3.1.2 The string theory spectrumand thematching . . . . . . . . 348
3.2 Matching of correlation functions and anomalies . . . . . . . . . . 351
