Singular Hochschild cohomology and algebraic string operations

Abstract

Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex ${\mathcal{D}}^{\ast }(A,A)$, 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 ${\mathcal{D}}^{\ast }(A,A)$ 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 ${\mathcal{D}}^{\ast }(A,A)$. Moreover, we prove that the cohomology algebra $H^{\ast }({\mathcal{D}}^{\ast }(A,A))$ is a Batalin-Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two Frobenius algebras are quasi-isomorphic as dg algebras then their Tate-Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate-Hochschild complex to string topology.

48 pages, 9 figures, Revisions made based on a referee report. To appear in Journal of Noncommutative Geometry

References in corpus (3)

• Natural operations on the Hochschild complex of commutative Frobenius algebras via the complex of looped diagrams
• Singular Hochschild Cohomology and Gerstenhaber Algebra Structure
• Invariance of the Gerstenhaber algebra structure on Tate-Hochschild cohomology