In recent years, the old idea that gauge theories and string theories are equivalent has been implemented and developed in various ways, and there are by now various models where the string theory / gauge theory correspondence is at work. One of the most important examples of this correspondence relates Chern-Simons theory, a topological gauge theory in three dimensions which describes knot and three-manifold invariants, to topological string theory, which is deeply related to Gromov-Witten invariants. This has led to some surprising relations between three-manifold geometry and enumerative geometry. This book gives the first coherent presentation of this and other related topics. After an introduction to matrix models and Chern-Simons theory, the book describes in detail the topological string theories that correspond to these gauge theories and develops the mathematical implications of this duality for the enumerative geometry of Calabi-Yau manifolds and knot theory. It is written in a pedagogical style and will be useful reading for graduate students and researchers in both mathematics and physics willing to learn about these developments.
Author(s): Marcos Marino
Series: International Series of Monographs on Physics'', 131
Publisher: Oxford University Press, USA
Year: 2005
Language: English
Commentary: missing font
Pages: 210
PREFACE......Page 8
ACKNOWLEDGEMENTS......Page 11
Contents......Page 12
I: MATRIX MODELS, CHERN–SIMONS THEORY, AND THE LARGE N EXPANSION......Page 14
1.1 Basics of matrix models......Page 16
1.2 Matrix model technology I: saddle-point analysis......Page 24
1.3 Matrix model technology II: orthogonal polynomials......Page 31
2.1 Chern–Simons theory: basic ingredients......Page 38
2.2 Perturbative approach......Page 42
2.3 Canonical quantization and surgery......Page 49
2.4 Framing dependence......Page 57
2.5 Results on Wilson loops and knot invariants......Page 59
2.6 U(∞) representation theory......Page 64
2.7 The 1/N expansion in Chern–Simons theory......Page 72
2.8 Chern–Simons theory as a matrix model......Page 75
II: TOPOLOGICAL STRINGS......Page 82
3.1 The N = 2 supersymmetric sigma model......Page 84
3.2 Topological twist......Page 86
3.3 The topological type-A model......Page 91
3.4 The topological type-B model......Page 96
4.1 Coupling to gravity......Page 100
4.2 Relation to compactifications of type II string theory......Page 101
4.3 The type-A topological string......Page 102
4.4 Open topological strings......Page 110
5.1 Non-compact Calabi–Yau geometries: an introduction......Page 120
5.2 Constructing toric Calabi–Yau manifolds......Page 122
5.3 The conifold transition......Page 128
5.4 Examples of closed string amplitudes......Page 131
III: THE TOPOLOGICAL STRING/GAUGE THEORY CORRESPONDENCE......Page 134
6 String theory and gauge theory......Page 136
7.1 Open string field theory......Page 140
7.2 Chern–Simons theory as an open string theory......Page 141
7.3 Matrix model as an open string theory......Page 146
8.1 The conifold transition and the large N duality......Page 156
8.2 Incorporating Wilson loops......Page 157
8.3 Geometric transitions for more general toric manifolds......Page 161
8.4 Matrix models and geometric transitions......Page 166
9.1 Framing of topological open string amplitudes......Page 170
9.2 Definition of the topological vertex......Page 171
9.3 Gluing rules......Page 173
9.4 Derivation of the topological vertex......Page 175
9.5 Useful formulae for the vertex......Page 179
9.6 Some applications......Page 181
9.7 Further properties of the topological vertex......Page 186
10.1 Applications to knot invariants......Page 190
10.2 Applications to N = 2 supersymmetric gauge theory......Page 193
A: Symmetric polynomials......Page 198
References......Page 200
Index......Page 210
