The bv algebra in string topology of classifying spaces

TL;DR

This paper computes the Batalin-Vilkovisky algebra structure on the loop cohomology of classifying spaces for compact Lie groups, revealing isomorphisms with Hochschild cohomology in certain cases and exceptions over .

Contribution

It provides explicit calculations of the BV algebra in string topology for classifying spaces, extending known results and identifying exceptions over .

Findings

BV algebra is isomorphic to Hochschild cohomology for odd or zero characteristic.

Over , the isomorphism fails for G=SO(3) and G=G_2.

Explicit computations for various Lie groups and fields.

Abstract

For almost any compact connected Lie group $G$ and any field $\mathbb{F}\_p$, we compute the Batalin-Vilkoviskyalgebra $H^{*+\text{dim }G}(LBG;\mathbb{F}\_p)$ on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra is isomorphicto the Hochschild cohomology $HH^*(H\_*(G),H\_*(G))$. Over $\mathbb{F}\_2$, such isomorphism of Batalin-Vilkovisky algebrasdoes not hold when $G=SO(3)$ or $G=G\_2$.