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Homological Algebra of Mirror Symmetry (1994)
| Venue: | in Proceedings of the International Congress of Mathematicians |
| Citations: | 521 - 3 self |
BibTeX
@INPROCEEDINGS{Kontsevich94homologicalalgebra,
author = {Maxim Kontsevich},
title = {Homological Algebra of Mirror Symmetry},
booktitle = {in Proceedings of the International Congress of Mathematicians},
year = {1994},
pages = {120--139}
}
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Abstract
Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on Calabi-Yau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Keyphrases
mirror symmetry homological algebra complex structure several year hodge structure rational curve hodge number many beautiful contsructions taylor coefficient complex algebraic manifold modulus space complicated history mirror manifold calabi-yau manifold conformal field theory positive genus 3-dimensional calabi-yau manifold arbitrary dimension dual calabi-yau manifold holomorphic volume element mirror variety
