Quantization of conic Lagrangian submanifolds of cotangent bundles.

Several recent works make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold M and the symplectic geometry of the cotangent bundle of M is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle.

Given a compact exact Lagrangian submanifold of the cotangent bundle of M, we can add a variable and associate with it a conic Lagrangian submanifold of the cotangent bundle of MxR, say L. We will see that it is possible to build a sheaf on MxR with microsupport L, in a canonical way. We recover from this construction an earlier result of Abouzaid, which says that the projection to M induces a homotopy equivalence between L and M.