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Mirror Symmetry
Kentaro Hori, University of Toronto, ON, Canada, Sheldon Katz, University of Illinois at Urbana-Champaign, IL, Albrecht Klemm, Humboldt-University, Berlin, Germany, Rahul Pandharipande, Princeton University, NJ, Richard Thomas, Imperial College, London, England, Cumrun Vafa, Harvard University, Cambridge, MA, Ravi Vakil, Stanford University, CA, and Eric Zaslow, Northwestern University, Evanston, IL
A co-publication of the AMS and Clay Mathematics Institute.

Clay Mathematics Monographs
2003; 950 pp; hardcover
Volume: 1
ISBN-10: 0-8218-2955-6
ISBN-13: 978-0-8218-2955-4
List Price: US$137
Member Price: US$109.60
Order Code: CMIM/1
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See also:

Dirichlet Branes and Mirror Symmetry - Paul S Aspinwall, Tom Bridgeland, Alastair Craw, Michael R Douglas, Mark Gross, Anton Kapustin, Gregory W Moore, Graeme Segal, Balazs Szendroi and PMH Wilson

Advances in String Theory: The First Sowers Workshop in Theoretical Physics - Eric Sharpe and Arthur Greenspoon

Mirror Symmetry and Algebraic Geometry - David A Cox and Sheldon Katz

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).


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


"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 Kähler 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
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