Brane structures in microlocal sheaf theory

Abstract

Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^{\ast }M$, asymptotic to a Legendrian submanifold $\Lambda \subset T^{\infty }M$. We study a locally constant sheaf of $\infty$-categories on $L$, called the sheaf of brane structures or ${\mathrm{Brane}}_L$. Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $\Gamma (L,{\mathrm{Brane}}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $\Lambda$.