Van den Bergh isomorphisms in string topology

Abstract

Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let $\mathbb{k}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H\_{+d}(\mathrm{L}\mkern-1.5mu M )$ 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 $$ \mathrm{HH}^{-p}(S\_(G),S\_(G))\cong \mathrm{HH}{p+d}(S(G),S\_(G)). $$ Therefore, the Gerstenhaber algebra $\mathrm{HH}^{}(S\_(G),S\_\*(G))$ is a Batalin–Vilkovisky algebra and we have a linear isomorphism $$ \mathrm{HH}^{}(S\_(G),S\_(G))\cong H\_{+d}(\mathrm{L}\mkern-1.5mu M ). $$ This linear isomorphism is expected to be an isomorphism of Batalin–Vilkovisky algebras. We also give a new characterization of Batalin–Vilkovisky algebra in terms of the derived bracket.