This thorough and detailed exposition is the result of an intensive month-long course sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives. The material will be particularly useful for those wishing to advance their understanding by exploring mirror symmetry at the interface of mathematics and physics.

This one-of-a-kind volume offers the first comprehensive exposition on this increasingly active area of study. It is carefully written by leading experts who explain the main concepts without assuming too much prerequisite knowledge. The book is an excellent resource for graduate students and research mathematicians interested in mathematical and theoretical physics.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

Readership

Graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.

Reviews

"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

• Differential geometry
• Algebraic geometry
• Differential and algebraic topology
• Equivariant cohomology and fixed-point theorems
• Complex and Kähler geometry
• Calabi-Yau manifolds and their moduli
• Toric geometry for string theory

• 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

• Proof of mirror symmetry

• 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

• 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