String Topology for Lie Groups

TL;DR

This paper describes the Batalin-Vilkovisky algebra structure on the homology of free loop spaces of compact Lie groups, providing explicit computations for SO(n) with rational and mod 2 coefficients.

Contribution

It offers a direct description of the BV algebra for compact Lie groups, connecting it to homology of G, based loops, and suspension, with explicit calculations for SO(n).

Findings

Explicit BV algebra description for compact Lie groups.

Computed BV algebra for SO(n) over rational numbers.

Computed BV algebra for SO(n) over integers mod two.

Abstract

In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer is phrased in terms of the homology of G, the homology of the space of based loops on G, and the homology suspension. The result is applied to compute the Batalin-Vilkovisky algebra associated to the special orthogonal groups SO(n) with coefficients in the rational numbers and in the integers modulo two.