On the Hochschild homology of open Frobenius algebras

Abstract

We prove that the shifted Hochschild homology HH_*(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 HH^*(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 HH_*(A,A)[m] which is also defined at the chain level. Hence there is 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 [10] product on the cohomology of free loop spaces when A is a commutative cochain algebra model for M .