A double Poisson algebra structure on Fukaya categories

Abstract Let M be an exact symplectic manifold with c 1 ( M ) = 0 . Denote by Fuk ( M ) the Fukaya category of M . We show that the dual space of the bar construction of Fuk ( M ) has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology HC • ( Fuk ( M ) ) , which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.