A chain level Batalin-Vilkovisky structure in string topology via de Rham chains

TL;DR

This paper constructs a chain-level Batalin-Vilkovisky algebra structure on the free loop space of a manifold using de Rham chains, providing a refined algebraic model that enhances understanding of string topology.

Contribution

It introduces a novel chain-level refinement of the BV algebra in string topology via a cyclic dg operad of de Rham chains, bridging differential forms and singular chains.

Findings

Defines a cyclic dg operad of de Rham chains for free loops

Constructs a chain model of the free loop space with BV structure

Recovers the string topology BV algebra on homology

Abstract

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented $C^\infty$-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space which admits an action of a chain model of the framed little disks operad, recovering the string topology BV algebra structure on the homology level.