Rational BV-algebra in String Topology
Yves Felix, Jean-Claude Thomas

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 be a 1-connected closed manifold and be the space of free loops on . In \cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of , . When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between and the shifted . We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of .
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Geometric and Algebraic Topology
