String topology of classifying spaces

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose of this paper is to study the singular homology $H_{\ast }(\mathcal{L}BG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\mathcal{L}BG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^{\ast }(\mathcal{L}BG)$. We also prove an algebraic version of this result by showing that the Hochschild cohomology $H{H}^{\ast }(S_{\ast }(G),S_{\ast }(G))$ of the singular chains of $G$ is a Batalin-Vilkovisky algebra.