Lie bialgebras and the cyclic homology of $A_\infty$ structures in topology

TL;DR

This paper explores the cyclic homology of Calabi-Yau $A_inite$ categories, revealing their structure as equivariant topological conformal field theories with involutive Lie bialgebras, and applies these findings to string topology and Fukaya categories.

Contribution

It demonstrates that the cyclic homology of Calabi-Yau $A_inite$ categories forms an equivariant topological conformal field theory with a Lie bialgebra structure, linking it to string topology and symplectic topology.

Findings

Cyclic homology contains an involutive Lie bialgebra.

Establishment of a Lie bialgebra homomorphism to contact homology.

Applications to Fukaya categories and string topology.

Abstract

$A_\infty$ categories are a mathematical structure that appears in topological field theory, string topology, and symplectic topology. This paper studies the cyclic homology of a Calabi-Yau $A_\infty$ category, and shows that it is naturally an equivariant topological conformal field theory, and in particular, contains an involutive Lie bialgebra. Applications of the theory to string topology and the Fukaya category are given; in particular, it is shown that there is a Lie bialgebra homomorphism from the cyclic cohomology of the Fukaya category of a symplectic manifold with contact type boundary to the linearized contact homology of the boundary.