Homological Algebra of Mirror Symmetry-英文文献.pdf

HOMOLOGICAL ALGEBRA OF MIRROR SYMMETRY Maxim Kontsevich 4 9 Max-Planck-Institut f¨ur Mathematik, Bonn 9 1 and University of California, Berkeley v o N Mirror Symmetry was discovered several years ago in string theory as a dual- 0 3 ity between families of 3-dimensional Calabi-Yau manifolds (more precisely, com- plex algebraic manifolds possessing holomorphic volume elements without zeroes). 1 The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau v manifolds V, W of dimension n (not necessarily equal to 3) one has 8 1 p q n−p q 0 dim H (V, Ω ) = dim H (W, Ω ) . 1 1 4 Physicists conjectured that conformal field theories associated with mirror 9 varieties are equivalent. Mathematically, MS is considered now as a relation be- / tween numbers of rational curves on such a manifold and Taylor coefficients of m periods of Hodge structures considered as functions on the moduli space of com- o plex structures on a mirror manifold. Recently it has been realized that one can e g make predictions for numbers of curves of positive genera and also on Calabi-Yau - manifolds of arbitrary dimensions. g We will not describe here the complicated history of the subject and will l a not mention many beautiful contsructions, examples and conjectures motivated : v by MS. On the contrary, we want to give an outlook of the story in general terms i and propose a conceptual framework for a possible explanation of the mirror phe- X nomenon. We will restrict ourselves