Singular hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex 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 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 D.A; A/. Moreover, we prove that the cohomology algebra H .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.