A chain level Batalin-Vilkovisky structure in string topology and decorated cacti

TL;DR

This paper constructs a chain-level Batalin-Vilkovisky structure in string topology using decorated cacti, linking loop space homology with Hochschild cochains and introducing new combinatorial operads.

Contribution

It introduces decorated cacti as a new combinatorial model for a dg operad acting on chain complexes in string topology.

Findings

Chain complex of free loop space admits an action of a dg operad.

Defined a chain level Gerstenhaber structure on Hochschild cochains.

Established compatibility of structures in string topology and differential graded algebras.

Abstract

We show that a model of chain complex of the free loop space of a $C^\infty$-manifold, which is proposed in arxiv:1404.0153, admits an action of a certain dg operad. This is a chain level structure under the Chas-Sullivan BV structure on loop space homology. Our dg operad is a variant of the cacti operad, and we introduce combinatorial objects called "decorated cacti" to define it. We also define a chain level Gerstenhaber structure on Hochschild cochains of any differential graded algebra. Applied to the dga of differential forms, this structure is compatible with our chain level structure in string topology.