Higher brackets on cyclic and negative cyclic (co)homology

TL;DR

This paper unifies and extends the algebraic structures on negative cyclic (co)homology, embedding string topology brackets into a noncommutative calculus framework, revealing BV and e_3-algebra structures under certain conditions.

Contribution

It introduces a unifying framework for brackets on cyclic (co)homology within noncommutative calculus, linking string topology and Hochschild cohomology structures.

Findings

Establishes a noncommutative differential calculus on cyclic complexes.

Identifies BV algebra structure on Hochschild cohomology.

Derives e_3-algebra structure when the BV bracket vanishes.

Abstract

The purpose of this article is to embed the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e_3-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.