A reduction of the string bracket to the loop product

TL;DR

This paper demonstrates that under certain conditions, the string bracket in string topology reduces to the loop product combined with the BV operator, linking algebraic and geometric structures.

Contribution

It establishes a reduction of the string bracket to the loop product and BV operator for BV exact manifolds and classifying spaces, connecting homological properties with geometric operations.

Findings

String bracket reduces to loop product and BV operator for BV exact manifolds.

Lie brackets on loop cohomology of classifying spaces exhibit the same reduction.

BV exactness holds for simply-connected spaces with positive weights.

Abstract

The negative cyclic homology for a differential graded algebra over the rational field has a quotient of the Hochschild homology as a direct summand if the $S$-action is trivial. With this fact, we show that the string bracket in the sense of Chas and Sullivan is reduced to the loop product followed by the BV operator on the loop homology provided the given manifold is BV exact. The reduction is indeed derived from the equivalence between the BV exactness and the triviality of the $S$-action. Moreover, it is proved that a Lie bracket on the loop cohomology of the classifying space of a connected compact Lie group possesses the same reduction. By using these results, we consider the non-triviality of string brackets. Another highlight is that a simply-connected space with positive weights is BV exact. Furthermore, the higher BV exactness is also discussed featuring the cobar-type Eilenberg-Moore spectral sequence.