Homological mirror symmetry with higher products

TL;DR

This paper constructs an $A_{ olinebreak ext{infinity}}$-structure on Ext-groups of vector bundles on complex manifolds, proposes a generalization of homological mirror symmetry involving $A_{ olinebreak ext{infinity}}$-categories, and verifies part of this conjecture for elliptic curves.

Contribution

It introduces a new $A_{ olinebreak ext{infinity}}$-structure on Ext-groups and proposes a generalized homological mirror symmetry conjecture involving $A_{ olinebreak ext{infinity}}$-functors.

Findings

Constructed an $A_{ olinebreak ext{infinity}}$-structure on Ext-groups.

Proposed a generalized homological mirror symmetry conjecture.

Verified the conjecture's triple product case for elliptic curves.

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.