On the Hochschild homology of open Frobenius algebras

We prove that the shifted Hochschild chain complex $C\mathrm{\_}\ast (A,A)[m]$ of a symmetric open Frobenius algebra $A$ of degree $m$ has a natural homotopy coBV-algebra structure. As a consequence $HH\mathrm{\_}\ast (A,A)[m]$ and $H{H}^{\ast }(A,A^{\vee })[-m]$ 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 $C\mathrm{\_}\ast (A,A)[m]$ hence a BV-structure on $HH\mathrm{\_}\ast (A,A)[m]$. Moreover we prove that the product and coproduct on $HH\mathrm{\_}\ast (A,A)[m]$ satisfy the Frobenius compatibility condition i.e. $HH\mathrm{\_}\ast (A,A)[m]$ is an open Frobenius algebras. If $A$ is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology $\widetilde{HH}\mathrm{\_}\ast (A)[m-1]$. 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 $M$.