Rational BV-algebra in String Topology

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) $.

💡 Research Summary

The paper investigates the BV‑algebra structure that arises in string topology, focusing on the free loop space (LM = \operatorname{Map}(S^{1},M)) of a simply‑connected closed manifold (M). Building on the seminal work of Chas and Sullivan, which introduced a BV‑algebra on the singular homology (H_{*}(LM;\mathbf{k})) when the coefficient field (\mathbf{k}) has characteristic zero, the authors aim to realize this structure algebraically via Hochschild cohomology.

First, they recall that the Hochschild cohomology (\widehat{HH}^{}(C^{}(M);C^{}(M))) of the cochain algebra (C^{}(M)) carries a canonical Gerstenhaber algebra structure (cup product and Lie bracket). To promote this Gerstenhaber algebra to a full BV‑algebra, they construct a degree (-1) operator (\Delta) using the cyclic bar construction of (C^{*}(M)) together with Connes’ (B)‑operator. The operator (\Delta) satisfies the defining BV identity \