# Homological Algebra of Mirror Symmetry

@article{Kontsevich1995HomologicalAO,
title={Homological Algebra of Mirror Symmetry},
author={Maxim Kontsevich},
journal={arXiv: Algebraic Geometry},
year={1995},
pages={120-139}
}
• M. Kontsevich
• Published 1995
• Mathematics, Physics
• arXiv: Algebraic Geometry
Mirror symmetry (MS) 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 zeros). 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 $$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$ .
1,195 Citations

#### Paper Mentions

Introduction to Homological Mirror Symmetry
Mirror symmetry states that to every Calabi-Yau manifold $$X$$ with complex structure and symplectic symplectic structure there is another dual manifold $$X^\vee$$, so that the properties of $$X$$Expand
Homological mirror symmetry with higher products
We construct an $A_{\infty}$-structure on the Ext-groups of hermitian holomorphic vector bundles on a compact complex manifold. We propose a generalization of the homological mirror conjecture due toExpand
Mirror Symmetry for hyperkaehler manifolds
We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complexExpand
Local Calabi–Yau manifolds of typeA˜via SYZ mirror symmetry
• Mathematics, Physics
• Journal of Geometry and Physics
• 2019
We carry out the SYZ program for the local Calabi--Yau manifolds of type $\widetilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. Mirror geometryExpand
Local mirror symmetry and type IIA monodromy of Calabi-Yau manifolds
We propose a monodromy invariant pairing Khol(X) H3(X _ ;Z) ! Q for a mirror pair of Calabi-Yau manifolds, (X; X _ ). This pairing is utilized implicitly in the previous calculations of theExpand
Homological mirror symmetry of $\mathbb{F}_1$ via Morse homotopy
• Mathematics, Physics
• 2020
This is a sequel to our paper arXiv:2008.13462, where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category ofExpand
Homological mirror symmetry at large volume
• Mathematics
• 2021
A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as aExpand
Mirror symmetry and deformation quantization
• Physics, Mathematics
• 2002
In homological mirror symmetry (see [Ko1]) and in the theory of D-modulesone meets similar objects. They are pairs (L,ρ) where L is a Lagrangianmanifold, and ρ is a ﬂat bundle on L (local system). InExpand
Mirror symmetry for log Calabi-Yau surfaces I
• Mathematics
• 2011
We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family isExpand
On the homological mirror symmetry conjecture for pairs of pants
The n-dimensional pair of pants is defined to be the complement of n+2 generic hyperplanes in CP^n. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphismExpand

#### References

SHOWING 1-10 OF 29 REFERENCES
Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians
We give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three- fold. Our account involves the interpretation ofExpand
FORMAL (NON)-COMMUTATIVE SYMPLECTIC GEOMETRY
Some time ago B. Feigin, V. Retakh and I had tried to understand a remark of J. Stasheff [S1] on open string theory and higher associative algebras [S2]. Then I found a strange construction ofExpand
A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory
• Physics
• 1991
Abstract We compute the prepotentials and the geometry of the moduli spaces for a Calabi-Yau manifold and its mirror. In this way we obtain all the sigma model corrections to the Yukawa couplings andExpand
Topological field theory and rational curves
• Mathematics, Physics
• 1993
We analyze the quantum field theory corresponding to a string propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear σ-model andExpand
Topological sigma models
A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry,Expand
Mirror Manifolds And Topological Field Theory
These notes are devoted to explaining aspects of the mirror manifold problem that can be naturally understood from the point of view of topological field theory. Basically this involves studying theExpand
Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Physics, Mathematics
• 1994
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomaticExpand
Algebraic Cohomology and Deformation Theory
• Mathematics
• 1988
We should state at the outset that the present article, intended primarily as a survey, contains many results which are new and have not appeared elsewhere. Foremost among these is the reduction, inExpand
Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes
• Mathematics, Physics
• 1994
We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus inExpand
Chern-Simons gauge theory as a string theory
Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, theseExpand