Softcover ISBN:  9780821819470 
Product Code:  SMFAMS/1 
List Price:  $34.00 
MAA Member Price:  $30.60 
AMS Member Price:  $27.20 

Book DetailsSMF/AMS Texts and MonographsVolume: 1; 1999; 120 ppMSC: Primary 14; Secondary 32; 81; 53;
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 CalabiYau 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 CalabiYau threefold determines the GromovWitten 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 GromovWitten 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 CalabiYau case. The book concludes with the first “naive” Givental computation, which is a mysterious mathematical justification of the computation of Candelas, et al.ReadershipGraduate students and research mathematicians interested in algebraic geometry; mathematical physicists.

Table of Contents

Cover

Title page

Contents

Introduction

CalabiYau 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

Back Cover


Additional Material

Reviews

This book … might yet give, even to the nonspecialist, some basic orientation in the complicated and rapidly developing world of mirror symmetry.
European Mathematical Society Newsletter 
Without any doubt, the English version of this panoramic introduction to the phenomenon of mirror symmetry will find a much larger number of interesting readers than the French original could do, and that is what this beautiful text really deserves.
Zentralblatt MATH


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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 CalabiYau 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 CalabiYau threefold determines the GromovWitten 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 GromovWitten 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 CalabiYau case. The book concludes with the first “naive” Givental computation, which is a mysterious mathematical justification of the computation of Candelas, et al.
Graduate students and research mathematicians interested in algebraic geometry; mathematical physicists.

Cover

Title page

Contents

Introduction

CalabiYau 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

Back Cover

This book … might yet give, even to the nonspecialist, some basic orientation in the complicated and rapidly developing world of mirror symmetry.
European Mathematical Society Newsletter 
Without any doubt, the English version of this panoramic introduction to the phenomenon of mirror symmetry will find a much larger number of interesting readers than the French original could do, and that is what this beautiful text really deserves.
Zentralblatt MATH