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

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.

v3 Improved exposition and corrected some mistakes in light of referee comments. Accepted version, to appear in JLMS

References in corpus (4)

• Quantum Structures for Lagrangian Submanifolds
• On the Fukaya category of a Fano hypersurface in projective space
• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
• The closed-open string map for $S^1$-invariant Lagrangians