A Homotopy Theoretic Proof of the BV Identity in Loop Homology

TL;DR

This paper provides a homotopy theoretic proof of the BV identity in loop homology, connecting algebraic structures with spectral descriptions to deepen understanding of string topology.

Contribution

It offers an explicit homotopy theoretic description of the loop bracket and a homological proof of the BV identity, linking loop operations through spectral methods.

Findings

Homotopy theoretic description of the loop bracket

Homological proof of the BV identity

Connection between loop bracket and BV derivation

Abstract

Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological proof of the BV identity connecting the loop product, the loop bracket, and the BV operator. The proof is based on an observation that the loop bracket and the BV derivation are given by the same cycle in the free loop space, except that they differ by parametrization of loops.