Deformation Theory of Bordered Algebras for Riemann Surfaces

Abstract

To a pointed matched circle, we introduce an associative, graded algebra related to bordered $HF^-$ and the wrapped Fukaya category of genus $g$ surfaces. We prove existence and uniqueness of a (graded) $A_{\infty}$ and weighted $A_{\infty}$ deformation of the associative algebra. We prove these results using Hochschild cohomology and Koszul duality results but describe a concrete (combinatorial) realization of these algebras as well.