Algebraic string bracket as a Poisson bracket

TL;DR

This paper constructs a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology, extending Goldman's results and applicable to complex geometric structures.

Contribution

It introduces a generalized algebraic framework for the string bracket as a Poisson bracket, broadening its application to various geometric contexts.

Findings

Lie algebra representation of algebraic string bracket constructed

Extension of Goldman's results to algebraic and geometric settings

Applicable to de Rham complex and Dolbeault resolution on Calabi-Yau manifolds

Abstract

In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi-Yau manifold.