Homological Algebra of Mirror Symmetry
Maxim Kontsevich
TL;DR
The work outlines a unified homological framework for Mirror Symmetry, proposing that a deep equivalence exists between symplectic geometry (embodied by Fukaya’s A∞-category) and complex geometry (via derived categories of coherent sheaves). Central to this is the idea that Gerstenhaber-type deformations, Hochschild cohomology, and extended moduli spaces organize the data of Gromov–Witten invariants and period integrals into a common algebraic structure, yielding a Homological Mirror Conjecture that subsumes the traditional numerical predictions. The paper surveys axioms, variations of Hodge structures, and explicit examples (notably elliptic curves and quintic 3-folds), arguing that categorified Mirror Symmetry should reproduce the enumerative and period data observed in the physical theory. It emphasizes that the derived Fukaya category and its A∞-structure provide the natural language for these dualities, with equivalences to derived categories of coherent sheaves on mirror manifolds and with duality principles (e.g., Serre duality) playing a crucial role. The proposed program aims at a broad, algebraic understanding of Mirror Symmetry through homological methods, including extensions to higher genus and more general Calabi–Yau settings.
Abstract
This is my talk at ICM, Zurich 1994. It contains a short introduction, two basic examples and a refined version of the Mirror Conjecture formulated in terms of homological algebra.