Van Den Bergh isomorphisms in String Topology

TL;DR

This paper establishes Van Den Bergh isomorphisms linking Hochschild (co)homology of chains on a topological group with string topology homology, revealing a BV algebra structure and proposing a new derived bracket characterization.

Contribution

It proves Van Den Bergh type isomorphisms for chains on classifying spaces, connecting Hochschild (co)homology with string topology BV algebras, and introduces a new derived bracket characterization.

Findings

Hochschild cohomology is isomorphic to Hochschild homology shifted by the manifold dimension.

Gerstenhaber algebra structure on Hochschild cohomology is a BV algebra.

Linear isomorphism between Hochschild cohomology and string topology homology.

Abstract

Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let ${\Bbbk}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{*+d}(LM)$ is a Batalin-Vilkovisky algebra. Let $G$ be a topological group such that $M$ is a classifying space of $G$. Denote by $S_*(G)$ the (normalized) singular chains on $G$. Suppose that $G$ is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism $$ HH^{-p}(S_*(G),S_*(G))\cong HH_{p+d}(S_*(G),S_*(G)). $$ Therefore, the Gerstenhaber algebra $HH^{*}(S_*(G),S_*(G))$ is a Batalin-Vilkovisky algebra and we have a linear isomorphism $$HH^{*}(S_*(G),S_*(G))\cong H_{*+d}(LM).$$ This linear isomorphism is expected to be an isomorphism of Batalin-Vilkovisky algebras. We also give a new characterization of Batalin-Vilkovisky algebra in term of derived bracket.