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SMF/AMS Texts and Monographs - Titles - 1 (1999)

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Mirror Symmetry
Publié en français dans le numéro 2 (1996) de la série Panoramas et Synthèses
Claire Voisin (translated by Roger Cooke)
Société Mathématique de France et AmericanMathematical Society
SMF/AMS Texts and Monographs 1 (1999), 140 pages
(épuisé)

Abstract:
This is the English translation of Professor Voisin's book reflecting thediscovery of the mirror symmetry phenomenon. The first chapter is devotedto the geometry of Calabi-Yau manifolds, and the second describes, as motivation, the ideasfrom quantum field theory that led to the discovery of mirror symmetry.The other chapters deal with more specialized aspects of the subject: thework 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-Witteninvariants of its mirror; Batyrev's construction, which exhibits the mirrorsymmetryphenomenon between hypersurfaces of toric Fano varieties, after a combinatorialclassification of the latter: the mathematical construction of theGromov-Witten potential,and the proof of its crucial property (that it satisfies the WDVV equation), which makes itpossible to construct a flat connection underlying a variation of Hodgestructure in the Calabi-Yau case. The book concludes with the first "naive" Givental computation, which is amysterious mathematical justification of the computation of Candelas, et al.

Keywords: Mirror symmetry, quantum field theory, manifolds

Class. math. : 14D05, 14D07, 14J32, 14M25, 32G13, 32G20, 32L07, 81T30,81T40, 53C23,53C15


ISSN : 1525-2302