On the Hochschild homology of open Frobenius algebras

TL;DR

This paper establishes that the Hochschild homology of symmetric open Frobenius algebras naturally carries homotopy coBV and BV algebra structures, revealing deep algebraic properties and conjecturing connections to free loop space cohomology.

Contribution

It introduces a homotopy coBV-algebra structure on Hochschild chains of open Frobenius algebras and explores their algebraic and coalgebraic properties, including a conjecture relating to free loop spaces.

Findings

Hochschild homology of open Frobenius algebras has a homotopy coBV-algebra structure.

The Hochschild homology and cohomology are respectively coBV and BV algebras.

Conjecture: the BV product matches the Goresky-Hingston product on free loop space cohomology.

Abstract

We prove that the shifted Hochschild chain complex $C\_*(A,A)[m]$ of a symmetric open Frobenius algebra $A$ of degree $m$ has a natural homotopy coBV-algebra structure. As a consequence $HH\_*(A,A)[m]$ and $HH^*(A,A^\vee)[-m]$ are respectively coBV and BV algebras. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural homotopy BV-algebra structure on $C\_*(A,A)[m]$ hence a BV-structure on $HH\_*(A,A)[m]$. Moreover we prove that the product and coproduct on $HH\_*(A,A)[m]$ satisfy the Frobenius compatibility condition i.e. $HH\_*(A,A)[m]$ is an open Frobenius algebras. If $A$ is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology $\widetilde{HH}\_*(A)[m-1]$. We conjecture that the product of this BV structure is identical to the Goresky-Hingston\cite{GH} product on the cohomology of free loop spaces when $A$ is a commutative cochain algebra model for $M$.