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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.
Table des matières
1 D Branes in String Theory, I.- 1 Introduction.- 2 Perturbative String Theory.- 3 Conformal Field Theory Formulation.- 4 T-Duality.- 5 Classical Solutions of The Low-Energy String Effective Action.- 6 Bosonic Boundary State.- 7 Fermionic Boundary State.- 8 Classical Solutions From Boundary State.- 9 Interaction Between a p and a p' Brane.- 2 Moduli Spaces of Calabi-Yau Compactifications.- 1 Introduction.- 2 A short story about string theory, F-theory and M-theory.- 3 The q = 16 triplet.- 4 The q = 8 triplets.- A Calabi-Yau manifolds.- 3 The M(Atrix) Model of M-Theory.- 1 Introduction.- 2 Matrix theory from the quantized supermembrane.- 3 The BFSS conjecture.- 4 M-theory objects from matrix theory.- 5 Interactions in matrix theory.- 6 Matrix theory in a general background.- 7 Outlook.- 4 The Holographic Principle.- 1 Black Hole Complementarity.- 2 Entropy Bounds.- 3 The AdS/CFT Correspondence and the Holographic Principle.- 4 The Flat Space Limit.- 5 Born-Infeld Actions and D-Brane Physics.- 1 D-Brane Solitons and the Born-Infeld Action.- 2 Born-Infeld Dynamics of Branes in Flat Space.- 3 Branes in Curved Space and the Gauge Theory Connection.- 4 Born-Infeld Analysis of the Baryon Vertex.- 5 Applications of the AdS/CFT Correspondence.- 6 Summary.- 6 Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces.- 1 Introduction.- 2 A Simple Example of Conformal Quantum Mechanics.- 3 Conformally Invariant N-Particle Quantum Mechanics.- 4 Superconformal Quantum Mechanics.- 5 The Quantum Mechanics of a Test Particle in a Reissner-Nordström Background.- 6 Quantum Mechanics on the Black Hole Moduli Space.- 7 Discussion.- A Differential Geometry with Torsion.- 7 Large-N Gauge Theories.- 1 Introduction.- 2 0(N) Vector Models.- 3 Large-N QCD.- 4 QCD in LoopSpace.- 5 Large-N Reduction.- 8 Introduction to Random Surfaces.- 1 Introduction.- 2 Random paths.- 3 Branched polymers.- 4 Dynamicaly triangulated surfaces.- 5 Lattice surfaces.- 6 Conclusion.- 9 Lorentzian and Euclidean Quantum Gravity ¡ª Analytical and Numerical Results.- 1 Introduction.- 2 Lorentzian gravity in 2d.- 3 Topology changes and Euclidean quantum gravity.- 4 Euclidean quantum gravity.- 5 Numerical setup.- 6 Dynamically triangulated quantum gravity in d > 2.- 7 Outlook.
Résumé
Proceedings of the NATO Advanced Study Institute on Quantum Geometry, held in Akureyri, Iceland, on August 9-20, 1999
