Description
Book SynopsisTrade Review“This book, a product of the collective efforts of the lecturers at the School organized ... by the Clay Mathematics Institute, is a valuable contribution to the continuing intensive collaboration of physicists and mathematicians. It will be of great value to young and mature researchers in both communities interested in this fascinating modern grand unification project.” - Yuri Manin, Max Planck Institute for Mathematics, Bonn, Germany
Table of Contents
- Part 1. Mathematical Preliminaries: Differential geometry
- Algebraic geometry
- Differential and algebraic topology
- Equivariant cohomology and fixed-point theorems
- Complex and Kahler geometry
- Calabi-Yau manifolds and their moduli
- Toric geometry for string theory
- Part 2. Physics Preliminaries: What is a QFT?
- QFT in $d=0$
- QFT in dimension 1: Quantum mechanics
- Free quantum field theories 1 + 1 dimensions
- $\mathcal{N} = (2,2)$ supersymmetry
- Non-linear sigma models and Landau-Ginzburg models
- Renormalization group flow
- Linear sigma models
- Chiral rings and topological field theory
- Chiral rings and the geometry of the vacuum bundle
- BPS solitons in $\mathcal{N}=2$ Landau-Ginzburg theories
- D-branes
- Part 3. Mirror Symmetry: Physics Proof: Proof of mirror symmetry
- Part 4. Mirror Symmetry: Mathematics Proof: Introduction and overview
- Complex curves (non-singular and nodal)
- Moduli spaces of curves
- Moduli spaces $\bar{\mathcal M}_{g,n}(X,\beta)$ of stable maps
- Cohomology classes on $\bar{\mathcal M}_{g,n}$ and ($\bar{\mathcal M})_{g,n}(X,\beta)$
- The virtual fundamental class, Gromov-Witten invariants, and descendant invariants
- Localization on the moduli space of maps
- The fundamental solution of the quantum differential equation
- The mirror conjecture for hypersurfaces I: The Fano case
- The mirror conjecture for hypersurfaces II: The Calabi-Yau case
- Part 5. Advanced Topics: Topological strings
- Topological strings and target space physics
- Mathematical formulation of Gopakumar-Vafa invariants
- Multiple covers, integrality, and Gopakumar-Vafa invariants
- Mirror symmetry at higher genus
- Some applications of mirror symmetry
- Aspects of mirror symmetry and D-branes
- More on the mathematics of D-branes: Bundles, derived categories and Lagrangians
- Boundary $\mathcal{N}=2$ theories
- References
- Bibliography
- Index
