A double Poisson algebra structure on Fukaya categories

Let MM be an exact symplectic manifold with c1(M)=0c1(M)=0. Denote by Fuk(M) the Fukaya category of MM. 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.