Cyclic Homology of Fukaya Categories and the Linearized Contact Homology

TL;DR

This paper establishes a deep algebraic structure on the cyclic cohomology of Fukaya categories for certain symplectic manifolds and relates it to linearized contact homology, connecting symplectic topology with string theory.

Contribution

It demonstrates that the cyclic cohomology of Fukaya categories forms an involutive Lie bialgebra and constructs a Lie bialgebra homomorphism from linearized contact homology to this cyclic cohomology.

Findings

Cyclic cohomology of Fukaya categories has involutive Lie bialgebra structure

Constructed a Lie bialgebra homomorphism from linearized contact homology to cyclic cohomology

Links symplectic topology with string topology and conformal field theory

Abstract

Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.