Brane structures in microlocal sheaf theory

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