A Double Poisson Algebra Structure on Fukaya Categories

TL;DR

This paper establishes a double Poisson algebra structure on Fukaya categories of symplectic manifolds with trivial first Chern class, linking noncommutative geometry and string topology.

Contribution

It introduces a differential graded noncommutative Poisson structure on the dual of the bar construction of Fukaya categories, a novel algebraic structure in symplectic geometry.

Findings

Lie algebra structure on cyclic cohomology of Fukaya categories

Connection to Kontsevich's noncommutative symplectic geometry

Relation to Chas and Sullivan's string topology

Abstract

Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $\mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $\mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology $\mathrm{HC}^\bullet(\mathrm{Fuk}(M))$, which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.