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SMF/AMS Texts and Monographs 1999; 120 pp; softcover Volume: 1 ISBN10: 082181947X ISBN13: 9780821819470 List Price: US$32 Member Price: US$25.60 Order Code: SMFAMS/1 See also: Strings 2001  Atish Dabholkar, Sunil Mukhi and Spenta R Wadia CalabiYau Varieties and Mirror Symmetry  Noriko Yui and James D Lewis Mirror Symmetry and Algebraic Geometry  David A Cox and Sheldon Katz  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. Titles in this series are copublished with Société Mathématique de France. SMF members are entitled to AMS member discounts. Readership Graduate students and research mathematicians interested in algebraic geometry; mathematical physicists. 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 Table of Contents



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