Singular Hochschild cohomology and algebraic string operations

Abstract

Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex \mathcal{D}^{*}(A,A) , called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex \mathcal{D}^*(A,A) computes the singular Hochschild cohomology of A . We construct a cyclic (or Calabi–Yau) A -infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L -infinity algebra structure extending the classical Gerstenhaber bracket, on \mathcal{D}^*(A,A) . Moreover, we prove that the cohomology algebra H^*(\mathcal{D}^*(A,A)) is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.