The K\"unneth theorem for the Fukaya algebra of a product of Lagrangians

Given a compact Lagrangian submanifold $L$ of a symplectic manifold $(M,\omega )$, Fukaya, Oh, Ohta and Ono construct a filtered $A_{\mathrm{\infty }}$-algebra $\mathcal{F}(L)$, on the cohomology of $L$, which we call the Fukaya algebra of $L$. In this paper we describe the Fukaya algebra of a product of two Lagrangians submanifolds $L_1\times L_2$. Namely, we show that $\mathcal{F}(L_1\times L_2)$ is quasi-isomorphic to $\mathcal{F}(L_1){\otimes }_{\mathrm{\infty }}\mathcal{F}(L_2)$, where ${\otimes }_{\mathrm{\infty }}$ is the tensor product of filtered $A_{\mathrm{\infty }}$-algebras defined in arXiv:1404.7184. As a corollary of this quasi-isomorphism we obtain a description of the bounding cochains on $\mathcal{F}(L_1\times L_2)$ and of the Floer cohomology of $L_1\times L_2$.