M-Theory and Quantum Geometry

M-Theory and Quantum Geometry Proceedings of the NATO Advanced Study Institute on Quantum Geometry, Held in Akureyri, Iceland, on August 9-20, 1999

By: Larus Thorlacius (Editor), Thordur Jonsson (Editor)

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Hardcover | 30 September 2000

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The fundamental structure of matter and spacetime at the shortest length scales remains an exciting frontier of basic research in theoretical physics. A unifying theme in this area is the quantization of geometrical objects. The majority of lectures at the Advanced Study Institute on Quantum Ge ometry in Akureyri was on recent advances in superstring theory, which is the leading candidate for a unified description of all known elementary par ticles and interactions. The geometric concept of one-dimensional extended objects, or strings, has always been at the core of superstring theory but in recent years the focus has shifted to include also higher-dimensional ob jects, so called D-branes, which play a key role in the non-perturbative dynamics of the theory. A related development has seen the strong coupling regime of a given string theory identified with the weak coupling regime of what was previ ously believed to be a different theory, and a web of such" dualities" that interrelates all known superstring theories has emerged. The resulting uni fied theoretical framework, termed M-theory, has evolved at a rapid pace in recent years.

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Preface	p. xi
D Branes in String Theory, I
Introduction	p. 1
Perturbative String Theory	p. 2
Conformal Field Theory Formulation	p. 11
T-Duality	p. 20
Classical Solutions Of The Low-Energy String Effective Action	p. 28
Bosonic Boundary State	p. 30
Fermionic Boundary State	p. 39
Classical Solutions From Boundary State	p. 46
Interaction Between a p and a p' Brane	p. 48
Moduli Spaces of Calabi-Yau Compactifications
Introduction	p. 61
A short story about string theory, F-theory and M-theory	p. 61
String Theory	p. 61
Calabi-Yau compactifications	p. 63
String Dualities	p. 64
F-Theory	p. 66
M-Theory	p. 67
Three Triplets of Dualities	p. 68
The q = 16 triplet	p. 68
The q = 8 triplets	p. 72
Calabi-Yau manifolds	p. 83
The M(atrix) Model of M-Theory
Introduction	p. 91
Matrix theory from the quantized supermembrane	p. 92
Review of light-front string	p. 95
The bosonic membrane theory	p. 96
The light-front bosonic membrane	p. 98
Matrix regularization	p. 100
The bosonic membrane in a general background	p. 103
The supermembrane	p. 104
Covariant membrane quantization	p. 110
The BFSS conjecture	p. 111
Membrane "instability"	p. 112
M-theory	p. 114
The BFSS conjecture	p. 115
Matrix theory as a second quantized theory	p. 116
Matrix theory and DLCQ M-theory	p. 118
M-theory objects from matrix theory	p. 123
Supergravitons	p. 123
Membranes	p. 125
5-branes	p. 133
Extended objects from matrices	p. 137
Interactions in matrix theory	p. 139
Two-body interactions	p. 140
The N-body problem	p. 156
Longitudinal momentum transfer	p. 160
Matrix theory in a general background	p. 160
T-duality	p. 161
Matrix theory on tori	p. 163
Matrix theory in curved backgrounds	p. 165
Outlook	p. 168
The Holographic Principle
Black Hole Complementarity	p. 179
The Schwarzschild Black Hole	p. 180
Penrose Diagrams	p. 184
Black Hole Thermodynamics	p. 185
The Thermal Atmosphere	p. 189
The Quantum Xerox Principle	p. 190
Information Retention Time	p. 192
Quantum Xerox Censorship	p. 194
Baryon Violation and Black Hole Horizons	p. 195
String Theory at High Frequency	p. 197
The Space Time Uncertainty Relation	p. 199
Entropy Bounds	p. 201
Maximum Entropy	p. 201
Entropy on Light-Like Surfaces	p. 203
Robertson Walker Geometry	p. 205
Bousso's Generalization	p. 206
The AdS/CFT Correspondence and the Holographic Principle	p. 210
AdS Space	p. 210
Holography in AdS Space	p. 211
The AdS/CFT Correspondence	p. 212
The Infrared Ultraviolet Connection	p. 214
Counting Degrees of Freedom	p. 215
AdS Black Holes	p. 216
The Horizon	p. 217
The Flat Space Limit	p. 218
The Flat Space Limit	p. 219
High Energy Gravitons Deep in the Bulk	p. 220
Kaluza Klein Modes	p. 222
Born-Infeld Actions and D-Brane Physics
D-Brane Solitons and the Born-Infeld Action	p. 227
Born-Infeld Dynamics of Branes in Flat Space	p. 231
Branes in Curved Space and the Gauge Theory Connection	p. 234
Born-Infeld Analysis of the Baryon Vertex	p. 240
Applications of the AdS/CFT Correspondence	p. 245
Summary	p. 252
Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces
Introduction	p. 255
A Simple Example of Conformal Quantum Mechanics	p. 257
Conformally Invariant N-Particle Quantum Mechanics	p. 259
Superconformal Quantum Mechanics	p. 261
A Brief Diversion on Supergroups	p. 261
Quantum Mechanical Supermultiplets	p. 263
Osp(1 2)-Invariant Quantum Mechanics	p. 264
D(2, 1; [alpha])-Invariant Quantum Mechanics	p. 267
The Quantum Mechanics of a Test Particle in a Reissner-Nordstrom Background	p. 271
Quantum Mechanics on the Black Hole Moduli Space	p. 274
The black hole moduli space metric	p. 274
The Near-Horizon Limit	p. 276
Conformal Symmetry	p. 277
Discussion	p. 278
Differential Geometry with Torsion	p. 279
Large-N Gauge Theories
Introduction	p. 285
O(N) Vector Models	p. 286
Four-Fermi Interaction	p. 287
Bubble graphs as zeroth order in 1/N	p. 290
Scale and Conformal Invariance of Four-Fermi Theory	p. 292
Nonlinear sigma model	p. 297
Large-N factorization in vector models	p. 300
Large-N QCD	p. 300
Index or ribbon graphs	p. 301
Planar and non-planar graphs	p. 305
Topological expansion and quark loops	p. 312
Large-N[subscript c] factorization	p. 315
The master field	p. 317
1/N[subscript c] as semiclassical expansion	p. 319
QCD in Loop Space	p. 321
Observables in terms of Wilson loops	p. 322
Schwinger-Dyson equations for Wilson loop	p. 324
Path and area derivatives	p. 326
Loop equations	p. 330
Relation to planar diagrams	p. 332
Loop-space Laplacian and regularization	p. 333
Survey of non-perturbative solutions	p. 337
Wilson loops in QCD[subscript 2]	p. 338
Large-N Reduction	p. 342
Reduction of scalar field	p. 342
Reduction of Yang-Mills field	p. 347
R[superscript d]-symmetry in perturbation theory	p. 349
Twisted reduced model	p. 350
Introduction to Random Surfaces
Introduction	p. 355
Random paths	p. 356
Lattice paths	p. 357
Dynamically triangulated paths	p. 359
Branched polymers	p. 362
Extrinsic properties	p. 362
Intrinsic properties	p. 366
Dynamicaly triangulated surfaces	p. 368
Definitions	p. 368
Basic properties	p. 369
The string tension	p. 370
Further results	p. 372
Lattice surfaces	p. 373
Definitions	p. 373
Critical behaviour	p. 375
Conclusion	p. 378
Lorentzian and Euclidean Quantum Gravity--Analytical and Numerical Results
Introduction	p. 382
Lorentzian gravity in 2d	p. 385
The discrete model	p. 385
The continuum limit	p. 389
Topology changes and Euclidean quantum gravity	p. 394
Baby universe creation	p. 394
The fractal dimension of Euclidean 2d gravity	p. 401
Euclidean quantum gravity	p. 403
Some generalities	p. 403
Dynamical triangulations	p. 404
The functional integral	p. 407
Inclusion of matter fields	p. 408
Numerical setup	p. 410
Monte Carlo method and ergodic moves	p. 410
Observables in 2d Euclidean gravity	p. 417
Comments on the 2d results	p. 428
Dynamically triangulated quantum gravity in d ] 2	p. 433
Generalization to higher dimensions	p. 433
Numerical results in higher dimensions	p. 438
Outlook	p. 442
Index	p. 451
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M-Theory and Quantum Geometry