Homological mirror symmetry with higher products

Abstract

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 to Kontsevich. Namely, we conjecture that for mirror dual Calabi-Yau manifolds $M$ and $X$ there exists an $A_{\infty }$-functor from Fukaya's symplectic $A_{\infty }$-category of $M$ to the $A_{\infty }$-derived category of $X$ which is a homotopy equivalence on morphisms. We verify the part of this conjecture concering triple products for elliptic curves.

AMSLatex, 13 pages, the final version

Cited by in corpus (28)

• D-branes, Categories and N=1 Supersymmetry
• Computation of Superpotentials for D-Branes
• Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings
• LHC Signatures of Resonant CP Violation in a Minimal Supersymmetric Higgs Sector
• Obstructed D-Branes in Landau-Ginzburg Orbifolds
• Matrix Factorizations And Mirror Symmetry: The Cubic Curve
• D-branes on Calabi-Yau Manifolds and Superpotentials
• $A_{\infty }$-structures on an elliptic curve
• Generalized complexes and string field theory
• D-brane categories
• A-infinity structure and superpotentials
• Open-closed homotopy algebra in mathematical physics
• Instanton Geometry and Quantum A-infinity structure on the Elliptic Curve
• D-Branes in Topological String Theory
• Homotopy L-infinity spaces
• Generating the superpotential on a D-brane category: I
• Homological mirror symmetry on noncommutative two-tori
• Calabi-Yau structures on categories of matrix factorizations
• Homological Mirror Symmetry in Dimension One
• Classical Yang-Baxter equation and the $A_{\infty }$-constraint
• Rapidly converging series for the Weierstrass zeta-function and the Kronecker function
• Products of harmonic forms and rational curves
• Triple Massey products on curves, Fay's trisecant identity and tangents to the canonical embedding
• Remarks on mirror symmetry of Donaldson-Thomas theory for Calabi-Yau 4-folds
• Moduli Space in Homological Mirror Symmetry
• A_{infinity}-structures, Brill-Noether loci and the Fourier-Mukai transform
• Fukaya categories of two-tori revisited
• Generating functions of planar polygons from homological mirror symmetry of elliptic curves