A bordered HF- algebra for the torus

TL;DR

This paper introduces a weighted A-infinity algebra for the torus, providing a combinatorial and abstract framework that connects to the wrapped Fukaya category and underpins bordered Heegaard Floer homology for three-manifolds with torus boundary.

Contribution

It presents a new combinatorial and abstract construction of a weighted A-infinity algebra for the torus, linking it to the wrapped Fukaya category and bordered Floer homology.

Findings

Defines a weighted A-infinity algebra for the torus

Provides a combinatorial and abstract characterization

Establishes a relationship with the wrapped Fukaya category

Abstract

We describe a weighted $A_\infty$-algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.