Brane structures from the perspective of microlocal sheaf theory

Abstract: In this talk, I will present the following application of microlocal sheaf theory in symplectic topology. For every closed exact Lagrangian L in the cotangent bundle of a manifold M, we associate a locally constant sheaf of categories on L, which we call Brane_L, whose fiber is the infinity-category of k-modules, for k any ring spectrum. I will discuss the relation of Brane_L with the usual brane structures in Floer theory, and its connection to the J-homomorphism in stable homotopy theory. I will also present a purely topological approach to the Nadler-Zaslow theorem based on Brane_L, namely, there is a fully faithful functor from local systems on L to constructible sheaves on M, when L admits a brane structure. *This is joint work with David Treumann.