Superfiltered $A_\infty$-deformations of the exterior algebra, and local mirror symmetry

TL;DR

This paper classifies $A_ abla$-algebra deformations of the exterior algebra using formal functions, extends known results to the $A_ abla$ setting, and proves a local mirror symmetry correspondence involving Lagrangian tori.

Contribution

It extends classification of algebra deformations to $A_ abla$-algebras via formal functions and establishes a local homological mirror symmetry result.

Findings

Classified $A_ abla$-deformations of exterior algebra by formal functions.

Computed Hochschild cohomology of these $A_ abla$-algebras.

Proved a local mirror symmetry equivalence for Lagrangian tori.

Abstract

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.