BV rigidity in the string topology of compact Lie groups

Abstract

Let $G$ be a connected compact Lie group and $R$ a commutative ring over which $H_*(G;R)$ is an exterior algebra on odd primitive generators. We prove a rigidity theorem: the Batalin-Vilkovisky (BV) algebra $\mathbb{H}_*(LG;R)$ of Chas and Sullivan on the homology of the free loop space is that of the product of odd spheres with the classical exponents of $G$, tensored with the group ring of the torsion of $\pi_1(G)$. Conversely, over an integral domain the graded algebra $\mathbb{H}_*(LG;R)$ already determines the exponents and the group ring, hence the BV algebra. The isomorphism with the product of spheres is algebraic: for $\mathrm{Sp}(n)$, $n\geq2$, and $\mathrm{SU}(n)$, $n\geq3$, no map of spaces induces it over $\mathbb{Z}$. If moreover $G$ is simply connected, then $\mathbb{H}_*(LG;R)$, as a BV algebra, is the Hochschild cohomology of $H^*(G;R)$ with the BV operator induced by Poincar\'e duality.