Rational BV-algebra in String Topology

TL;DR

This paper establishes a BV-algebra structure on the Hochschild cohomology of a manifold's cochain algebra, linking it to the homology of free loop spaces and confirming compatibility with the Hodge decomposition.

Contribution

It introduces a BV-algebra structure on Hochschild cohomology that corresponds to the Chas-Sullivan BV-algebra on loop space homology, extending previous results to characteristic zero fields.

Findings

Constructs a BV-algebra on Hochschild cohomology of cochains.

Shows an isomorphism between Hochschild cohomology BV-algebra and shifted loop space homology.

Demonstrates compatibility of BV-operator and product with Hodge decomposition.

Abstract

Let $M$ be a 1-connected closed manifold and $LM$ be the space of free loops on $M$. In \cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_\ast(LM; \bk)$. When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on $\hH^\ast(C^\ast (M); C^\ast (M))$ which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $\hH^\ast (C^\ast (M); C^\ast (M)) $ and the shifted $ H_{\ast+m} (LM; {\bk})$. We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of $H_\ast (LM) $.