Bordered algebras and the wrapped Fukaya category

TL;DR

This paper proves an isomorphism between endomorphism algebras in the wrapped Fukaya category of punctured surfaces and star algebras from bordered knot Floer homology, using Hochschild cohomology calculations.

Contribution

It establishes a novel isomorphism linking geometric Fukaya categories with algebraic star algebras, expanding understanding of their structures and relations.

Findings

Star algebras are unique with given generators and relations.

Endomorphism algebras in the wrapped Fukaya category match star algebras.

Hochschild cohomology confirms the deformation and uniqueness of star algebras.

Abstract

This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which the author first constructed in her previous work. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that the star algebras are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.