A Double Poisson Algebra Structure on Fukaya Categories

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.

27 pages. To appear in Journal of Geometry and Physics

References in corpus (3)

• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry
• Exact Lagrangian submanifolds in simply-connected cotangent bundles

Cited by in corpus (1)

• Shifted double Lie-Rinehart algebras