Lagrangian Correspondence for Microlocal Sheaves

Lagrangian correspondences between symplectic manifolds are generalizations of symplectomorphisms and are expected to give the morphisms in the 2-category of symplectic manifolds under geometric compositions. For the (wrapped) Fukaya categories of certain exact symplectic manifolds, by the Kunneth formula, exact Lagrangian correspondences define bimodules over the categories. We will consider the topological model of wrapped Fukaya categories of Weinstein manifolds in terms of microlocal sheaves and show that the geometric composition of (exact but immersed) Lagrangian correspondences agrees with the algebraic composition of bimodules. This is joint work in preparation with David Nadler and Vivek Shende.