A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops

Abstract:

Let $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a topological space. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $\mathbb {H}_*(LM):=H_{*+d}(LM)$. Getzler (1994) has defined a structure of Batalin-Vilkovisky algebra on the homology of the pointed double loop space of $X$, $H_*(\Omega ^2 X)$. Let $G$ be a topological monoid with a homotopy inverse. Suppose that $G$ acts on $M$. We define a structure of Batalin-Vilkovisky algebra on $H_*(\Omega ^2BG)\otimes \mathbb {H}_*(M)$ extending the Batalin-Vilkovisky algebra of Getzler on $H_*(\Omega ^2BG)$. We prove that the morphism of graded algebras \[ H_*(\Omega ^2BG)\otimes \mathbb {H}_*(M)\rightarrow \mathbb {H}_*(LM)\] defined by Félix and Thomas (2004), is in fact a morphism of Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected compact Lie group, we compute the Batalin-Vilkovisky algebra $\mathbb {H}_*(LG;\mathbb {Q})$.