Mirror Symmetry

by Voisin, Claire; Cooke, Roger

• ISBN13: 9780821819470
• ISBN10: 082181947X
• Format: Paperback
• Copyright: 1999-08-01
• Publisher: Amer Mathematical Society

Summary

This is the English translation of Professor Voisin's book reflecting the discovery of the mirror symmetry phenomenon. The first chapter is devoted to the geometry of Calabi-Yau manifolds, and the second describes, as motivation, the ideas from quantum field theory that led to the discovery of mirror symmetry. The other chapters deal with more specialized aspects of the subject: the work of Candelas, de la Ossa, Greene, and Parkes, based on the fact that under the mirror symmetry hypothesis, the variation of Hodge structure of a Calabi-Yau threefold determines the Gromov-Witten invariants of its mirror; Batyrev's construction, which exhibits the mirror symmetry phenomenon between hypersurfaces of toric Fano varieties, after a combinatorial classification of the latter; the mathematical construction of the Gromov-Witten potential, and the proof of its crucial property (that it satisfies the WDVV equation), which makes it possible to construct a flat connection underlying a variation of Hodge structure in the Calabi-Yau case. The book concludes with the first "naive" Givental computation, which is a mysterious mathematical justification of the computation of Candelas, et al.

Table of Contents

Calabi-Yau manifolds "Physical" origin of the conjecture

The work of Candelas-de la Ossa-Green-Parkes

The work of Batyrev

Quantum cohomology

The Givental construction

Bibliography

Table of Contents provided by Publisher. All Rights Reserved.