Superfiltered $A_\infty$deformations of the exterior algebra, and local mirror symmetry
Abstract
The exterior algebra $E$ on a finiterank 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 ChoHongLau, 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 ChoHongLau we prove a local form of homological mirror symmetry: the Floer $A_\infty$algebra of a monotone Lagrangian torus is quasiisomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.01096
 Bibcode:
 2019arXiv191001096S
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Quantum Algebra;
 18G55;
 14F05;
 53D37;
 14J33;
 16E40
 EPrint:
 v2 Added computation of Hochschild cohomology algebra. Various minor corrections and clarifications. 28 pages, comments welcome