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.

Part 1. Mathematical Preliminaries

Chapter 1. Differential Geometry

Chapter 2. Algebraic Geometry

Chapter 3. Differential and Algebraic Topology

Chapter 4. Equivariant Cohomology and Fixed-Point Theorems

Chapter 5. Complex and K¨ahler Geometry

Chapter 6. Calabi–Yau Manifolds and Their Moduli

Chapter 7. Toric Geometry for String Theory

Part 2. Physics Preliminaries

Chapter 8. What Is a QFT?

Chapter 9. QFT in d = 0

Chapter 10. QFT in Dimension 1: Quantum Mechanics

Chapter 11. Free Quantum Field Theories in 1 + 1 Dimensions

Chapter 12. N = (2, 2) Supersymmetry

Chapter 13. Non-linear Sigma Models and Landau–Ginzburg Models

Chapter 14. Renormalization Group Flow

Chapter 15. Linear Sigma Models

Chapter 16. Chiral Rings and Topological Field Theory

Chapter 17. Chiral Rings and the Geometry of the Vacuum Bundle

Chapter 18. BPS Solitons in N=2 Landau–Ginzburg Theories

Chapter 19. D-branes

Part 3. Mirror Symmetry: Physics Proof

Chapter 20. Proof of Mirror Symmetry

Part 4. Mirror Symmetry: Mathematics Proof

Chapter 21. Introduction and Overview

Chapter 22. Complex Curves (Non-singular and Nodal)

Chapter 23. Moduli Spaces of Curves

Chapter 24. Moduli Spaces Mg,n(X, β) of Stable Maps

Chapter 25. Cohomology Classes on Mg,n and Mg,n(X, β)

Chapter 26. The Virtual Fundamental Class, Gromov–Witten Invariants, and Descendant Invariants

Chapter 27. Localization on the Moduli Space of Maps

Chapter 28. The Fundamental Solution of the Quantum Differential Equation

Chapter 29. The Mirror Conjecture for Hypersurfaces I: The Fano Case

Chapter 30. The Mirror Conjecture for Hypersurfaces II: The Calabi–Yau Case

Part 5. Advanced Topics

Chapter 31. Topological Strings

Chapter 32. Topological Strings and Target Space Physics

Chapter 33. Mathematical Formulation of Gopakumar–Vafa Invariants

Chapter 34. Multiple Covers, Integrality, and Gopakumar–Vafa Invariants

Chapter 35. Mirror Symmetry at Higher Genus

Chapter 36. Some Applications of Mirror Symmetry

Chapter 37. Aspects of Mirror Symmetry and D-branes

Chapter 38. More on the Mathematics of D-branes: Bundles, Derived Categories, and Lagrangians

Chapter 39. Boundary N = 2 Theories

Chapter 40. References

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

About the Author

Ravi Vakil is the Robert Grimmett Professor of Mathematics at Stanford, and President of the American Mathematical Society 2025-27. He is an algebraic geometer, whose work touches on topology, string theory, applied mathematics, combinatorics, number theory, and more. He received his undergraduate degree from the University of Toronto in 1992 and his Ph.D. from Harvard in 1997, and taught at Princeton and MIT before moving to Stanford in 2001. His research awards include the Centennial Fellowship from the American Mathematical Society, the Coxeter-James Prize from the Canadian Mathematical Society, a CAREER grant from the National Science Foundation, and the Presidential Early Career Award for Scientists and Engineers.

He also has received the Dean's Award for Distinguished Teaching and was the Bass University Fellow in Undergraduate Education. He co-founded the web resource MathOverflow and the San Francisco educational institution Proof School, which serves grades 6–12. He remains on the boards of both.

He serves on the Mathematical Sciences Education Board of the National Academies of Sciences, Engineering, and Medicine. He also is on the board of the nonprofit National Math Stars, which aims to ensure mathematically extraordinary students from all communities have the resources they need to reach the frontiers of math and science.