Unity from Duality

by Bachas, C.; Bilal, A.; Douglas, M.; Nekrasov, N.; David, F.

• ISBN13: 9783540002765
• ISBN10: 3540002766
• Format: Hardcover
• Copyright: 2003-02-01
• Publisher: Springer Verlag

Summary

In a distilled and pedagogical fashion, the contributions to this volume of the famous summer school in Les Houches cover the recent developments in supersymmetric string theory, the gauge theory/string theory correspondence and string duality. Further chapters deal with quantum gravity and D-brane geometry. Black hole mechanics and cosmology are treated too, as well as the AdS-CFT correspondence. The book is a comprehensive introduction to the recent developments in string/M-theory and quantum gravity. It addresses graduate students in physics and astrophysics.

Table of Contents

Lecturers	p. xi
Participants	p. xiii
Préface	p. xvii
Preface	p. xxi
Contents	p. xxv
Supergravity	p. 1
Introduction	p. 3
Supersymmetry in various dimensions	p. 4
The Poincaré supersymmetry algebra	p. 5
Massless supermultiplets	p. 7
D = 11 supermultipets	p. 8
D = 10 supermultiplets	p. 9
D = 6 supermultiplets	p. 13
Massive supermultiplets	p. 17
Central charges and multiplet shortening	p. 19
On spinors and the R-symmetry group HR	p. 22
Supergravity	p. 28
Simple supergravity	p. 28
Maximal supersymmetry and supergravity	p. 32
D = 11 supergravity	p. 35
Dimensional reduction and hidden symmetries	p. 38
Frames and field redefinitions	p. 43
Kaluza-Klein states and BPS-extended supergravity	p. 46
Nonmaximal supersymmetry: Q = 16	p. 53
Homogeneous spaces and nonlinear sigma models	p. 55
Nonlinearly realized symmetries	p. 56
Geometrical quantities	p. 61
Nonlinear sigma models with homogeneous target space	p. 66
Gauged nonlinear sigma models	p. 69
Gauged maximal supergravity in 4 and 5 dimensions	p. 70
On E7(7)/SU(8) and E6(6)/USp(8) cosets	p. 72
On ungauged maximal supergravity Lagrangians	p. 74
Electric-magnetic duality and E7(7)	p. 79
Gauging maximal supergravity; the T-tensor	p. 84
Supersymmetry in anti-de Sitter space	p. 89
Anti-de Sitter supersymmetry and masslike terms	p. 93
Unitary representations of the anti-de Sitter algebra	p. 97
The superalgebras OSp(N 4)	p. 110
Superconformal symmetry	p. 116
The superconformal algebra	p. 119
Superconformal gauge theory and supergravity	p. 122
Matter fields and currents	p. 126
Supersymmetric Gauge Theories	p. 137
Introduction	p. 141
Supersymmetric quantum mechanics	p. 143
Symmetry and symmetry breaking	p. 151
A nonrenormalisation theorem	p. 152
A two variable realization and flat potentials	p. 154
Geometric meaning of the Witten index	p. 158
Landau levels and SUSY QM	p. 159
Conformal quantum mechanics	p. 161
Superconformal quantum mechanics	p. 164
Review of supersymmetric models	p. 165
Kinematics	p. 165
Superspace and chiral fields	p. 167
Kähler potentials	p. 169
F-terms	p. 170
Global symmetries	p. 170
The effective potential	p. 172
Supersymmetry breaking	p. 172
Supersymmetric gauge theories	p. 173
Phases of gauge theories	p. 179
Supersymmetric gauge theories/super QCD	p. 180
The classical moduli space	p. 182
Quantum moduli spaces	p. 185
Quantum moduli space for 0 < NF < NC	p. 185
Integrating in	p. 186
Quantum moduli space for NF ≥ NC	p. 190
NF = NC	p. 190
NF = NC + 1	p. 191
Higgs and confinement phases	p. 192
Infra-red duality	p. 194
Superconformal invariance in d = 4	p. 202
Comments on vacuum energies in scale invariant theories	p. 207
Supersymmetric gauge theories and string theory	p. 210
Branes in string theory	p. 210
Branes in IIA and IIB string theories	p. 211
The effective field theory on branes	p. 213
Effective D = 4 dimensional systems with N = 2 supersymmetry	p. 216
An effective D = 4, N = 1, U(NC) gauge theory with matter	p. 226
More pieces of information	p. 229
Obtaining the dual field theory	p. 233
Concluding remarks	p. 235
Final remarks	p. 237
An Introduction to Duality Symmetries in String Theory	p. 241
Introduction	p. 243
A brief review of perturbative string theory	p. 245
Thespectrum	p. 246
Interactions	p. 251
Compactification	p. 254
Notion of duality symmetries in string theory	p. 255
Duality symmetries: Definition and examples	p. 255
Testing duality conjectures	p. 258
Analysis of low energy effective field theory	p. 260
Type I-SO(32) heterotic duality in D = 10	p. 261
Self-duality of heterotic string theory on T6	p. 264
Duality between heterotic on T4 and type IIA on K3	p. 268
SL(2, Z) self-duality of type IIB in D = 10	p. 272
Other examples	p. 274
Precision test of duality: Spectrum of BPS states	p. 276
SL(2,Z) S-duality in heterotic on T6 and multi-monopole moduli spaces	p. 280
SL(2,Z) duality in type IIB on S1 and D-branes	p. 286
Massless solitons and tensionless strings	p. 296
Interrelation between different duality conjectures	p. 299
Combining non-perturbative and T-dualities	p. 299
Duality of dualities	p. 299
Fiberwise duality transformation	p. 301
Recovering higher dimensional dualities from lower dimensional ones	p. 304
Duality in theories with less than sixteen supersymmetry generators	p. 305
Construction of a dual pair of theories with eight supercharges	p. 306
Test of duality conjectures involving theories with eight supercharges	p. 309
M-theory	p. 312
M-theory in eleven dimensions	p. 312
Compactification of M-theory	p. 315
Les Houches Lectures on Large N Field Theories Gravity	p. 323
General introduction	p. 325
The correspondence	p. 330
The field <$>\leftrightarrow<$> operator correspondence	p. 336
Holography	p. 338
Tests of the AdS/CFT correspondence	p. 341
The spectrum of chiral primary operators	p. 342
The field theory spectrum	p. 342
The string theory spectrum and the matching	p. 348
Matching of correlation functions and anomalies	p. 351
Correlation functions	p. 353
Two-point functions	p. 355
Three-point functions	p. 359
Wilson loops	p. 362
Wilson loops and minimum surfaces	p. 363
Other branes ending on the boundary	p. 368
Theories at finite temperature	p. 369
Construction	p. 369
Thermal phase transition	p. 372
D-Branes on the Conifold and <$>{\cal N} = 1<$> Gauge/Gravity Dualities	p. 383
Introduction	p. 385
D3-branes on the conifold	p. 388
Dimensions of chiral operators	p. 391
Wrapped D3-branes as "dibaryons"	p. 393
Other ways of wrapping D-branes over cycles of T1,1	p. 394
The RG cascade	p. 397
Matching of the ß-functions	p. 400
The chiral anomaly	p. 402
The anomaly as a classical effect in supergravity	p. 403
The anomaly as spontaneous symmetry breaking in AdS5	p. 405
Deformation of (KS) the conifold	p. 410
The first-order equations and their solution	p. 412
SO(4) invariant expressions for the 3-forms	p. 413
Infrared physics	p. 414
Dimensional transmutation and confinement	p. 414
Tensions of the q-strings	p. 416
Chiral symmetry breaking and gluino condensation	p. 418
De Sitter Space	p. 423
Introduction	p. 425
Classical geometry of de Sitter space	p. 427
Coordinate systems and Penrose diagrams	p. 428
Schwarzschild-de Sitter	p. 435
Geodesies	p. 436
Quantum field theory on de Sitter space	p. 437
Green functions and vacua	p. 437
Temperature	p. 440
Entropy	p. 443
Quantum gravity in de Sitter space	p. 446
Asymptotic symmetries	p. 446
De Sitter boundary conditions and the conformai group	p. 447
Calculation of the Brown-York stress tensor	p. 451
String Compactification with <$>{\cal N} = 1<$> Supersymmetry
Introduction and the question of vacuum selection	p. 457
From ten dimensional geometry to four dimensional effective field theory	p. 461
D-branes with stringy corrections	p. 466
Quantum corrections	p. 470
Towards the low energy theory of everything	p. 472
Lectures on Open Strings, and Noncommutative Gauge Theories	p. 477
Introduction	p. 479
Background independence	p. 480
Dolan-Nappi solutions	p. 483
Intersecting branes	p. 485
T-duality	p. 485
BPS algebras	p. 486
Noncommutative U(1) instantons	p. 486
Higher dimensional instantons	p. 489
Fermions in the Y background	p. 490
Fermions in the instanton background	p. 490
The Dirac field in the monopole background	p. 491
Non-trivial backgrounds	p. 492
Example: ¿ = Z2	p. 493
Conclusions and outlook	p. 494
Condensates Near the Argyres-Douglas Point in SU(2) Gauge Theory with Broken <$>{\cal N} = 2<$> Supersymmetry	p. 497
Introduction	p. 499
Matter and gaugino condensates	p. 501
Dyon condensates	p. 504
Monopole condensate	p. 505
Charge and dyon condensates	p. 506
The Argyres-Douglas point: How well the theory is defined	p. 508
Conclusions	p. 509
Quantum Field Theory with Extra Dimensions	p. 513
Introduction	p. 515
The fields	p. 516
The action at level	p. 518
Special Holonomy Spaces	p. 523
Introduction	p. 525
Resolution via transgression	p. 527
Motivation	p. 527
Resolved M2-brane	p. 528
Resolved D2-brane	p. 529
Other examples	p. 529
Special holonomy spaces, harmonic forms and resolved branes	p. 530
Harmonic forms for the Stenzel metric	p. 531
Stenzel metric	p. 531
Harmonic middle-dimension (p, q) forms	p. 532
Old G2 holonomy metrics and their harmonic forms	p. 533
Resolved cones over S2 × S4 and S2 × CP2	p. 533
Resolved cone over S3 × S3	p. 534
New Spin(7) holonomy metrics and their harmonic forms	p. 534
The old metric and harmonic 4-forms	p. 534
The new Spin(7) holonomy metric	p. 535
Applications: Resolved M2-branes and D2-branes	p. 536
New G2 holonomy metrics	p. 537
Classification of G2 holonomy spaces with S3 × S3 orbits	p. 537
Conclusions and open avenues	p. 541
Four Dimensional Non-Critical Strings
Introduction	p. 549
Many paths to the gauge/string duality	p. 550
Confinement	p. 550
Large N	p. 551
D-branes	p. 552
Non-criticalstrings	p. 554
Four dimensional non-critical strings	p. 559
Four dimensional CFTs as Kazakov critical points	p. 562
Instantons and large N	p. 564
A toy model example	p. 566
Exact results in 4D string theory	p. 569
Furtherinsights	p. 571
Full proofs	p. 571
Non-perturbative non-Borel summable partition functions	p. 572
Open problems	p. 572
U-Opportunities: Why is Ten Equal to Ten?
Classifications	p. 577
Real forms of Lie algebras	p. 581
Chaos controlled by symmetry	p. 583
Exact Answers to Approximate Questions -Noncommutative Dipoles	p. 587
Introduction and conjectures	p. 589
Flying noncommutative dipole	p. 591
Open wilson lines: How and why?	p. 593
Open Wilson lines	p. 593
Generalized star products	p. 595
Free and interacting OWLs	p. 597
Free OWLs	p. 598
Interacting OWLs	p. 600
Closed strings out of open strings	p. 604
Open strings as miniature dipoles	p. 604
Witten's *w-product is Moyal's *m-product	p. 605
Closed strings as OWLs	p. 607
Open-String Models with Broken Supersymmetry	p. 611
Broken supersymmetry and type-0 models	p. 613
Scherk-Schwarz deformations and brane supersymmetry	p. 618
Brane supersymmetry breaking	p. 620
On a Field Theory of Open Strings, Tachyon Condensation and Closed Strings	p. 627
Exceptional Magic	p. 643
G2	p. 648
Spin(7)	p. 657
Topological twist	p. 661
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