The holomorphic matrix model

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Table of contents

Setting the Stage
I Gauge Theories, Matrix Models and Geometric Transitions
1 Introduction and Overview
2 Effective Actions
2.1 The 1PI effective action and the background field method
2.2 Wilsonian effective actions of supersymmetric theories
2.3 Symmetries and effective potentials
3 Riemann Surfaces and Calabi-Yau Manifolds
3.1 Properties of Riemann surfaces
3.2 Properties of (local) Calabi-Yau manifolds
3.2.1 Aspects of compact Calabi-Yau manifolds
3.2.2 Local Calabi-Yau manifolds
3.2.3 Period integrals on local Calabi-Yau manifolds and Riemann surfaces
4 Holomorphic Matrix Models and Special Geometry
4.1 The holomorphic matrix model
4.1.1 The partition function and convergence properties
4.1.2 Perturbation theory and fatgraphs
4.1.3 Matrix model technology
4.1.4 The saddle point approximation for the partition function
4.2 Special geometry relations
4.2.1 Rigid special geometry
4.2.2 Integrals over relative cycles
4.2.3 Homogeneity of the prepotential
4.2.4 Duality transformations
4.2.5 Example and summary
5 Superstrings, the Geometric Transition and Matrix Models
5.1 Superpotentials from string theory with fluxes
5.1.1 Pairings on Riemann surfaces with marked points
5.1.2 The superpotential and matrix models
5.2 Example: Superstrings on the conifold
5.3 Example: Superstrings on local Calabi-Yau manifolds
6 B-Type Topological Strings and Matrix Models
7 Conclusions
II M-theory Compactifications, G2-Manifolds and Anomalies
8 Introduction
9 Anomaly Analysis of M-theory on Singular G2-Manifolds
9.1 Gauge and mixed anomalies
9.2 Non-Abelian gauge groups and anomalies
10 Compact Weak G2-Manifolds
10.1 Properties of weak G2-manifolds
10.2 Construction of weak G2-holonomy manifolds with singularities
11 The Hoˇrava-Witten Construction
12 Conclusions
III Appendices
A Notation
A.1 General notation
A.2 Spinors
A.2.1 Clifford algebras and their representation
A.2.2 Dirac, Weyl and Majorana spinors
A.3 Gauge theory
A.4 Curvature
B Some Mathematical Background
B.1 Useful facts from complex geometry
B.2 The theory of divisors
B.3 Relative homology and relative cohomology
B.3.1 Relative homology
B.3.2 Relative cohomology
B.4 Index theorems
C Special Geometry and Picard-Fuchs Equations
C.1 (Local) Special geometry
C.2 Rigid special geometry
D Topological String Theory
D.1 Cohomological field theories
D.2 N = (2, 2) supersymmetry in 1+1 dimensions
D.3 The topological B-model
D.4 The B-type topological string
E Anomalies
E.1 Elementary features of anomalies
E.2 Anomalies and index theory
IV Bibliography