On the Hochschild homology of open Frobenius algebras

Abstract

We prove that the shifted Hochschild homology $H{H}_{\ast }(A,A)[m]$ of a symmetric open Frobenius algebra $A$ of degree $m$ has a natural coBV-algebra structure which is defined at the chain level. As a consequence $H{H}^{\ast }(A,A^{\vee })[-m]$ is a BV algebra. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural BV-algebra structure on $H{H}_{\ast }(A,A)[m]$ which is also defined at the chain level. Hence there is a BV-structure on $H{H}_{\ast }(A,A)[m]$. Moreover we prove that the product and coproduct on $H{H}_{\ast }(A,A)[m]$ satisfy the Frobenius compatibility condition i.e. $H{H}_{\ast }(A,A)[m]$ is an open Frobenius algebras. If $A$ is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology ${\widetilde{HH}}_{\ast }(A)[m-1]$. We conjecture that the product of this BV structure is identical to the Goresky–Hingston [10] product on the cohomology of free loop spaces when $A$ is a commutative cochain algebra model for $M$.