On categorification of Stokes coefficients in Chern-Simons theory

We consider a finite-dimensional oscillatory integral which provides a "finite-dimensional model" for analytically continued $SU(2)$ Chern-Simons theory on closed 3-manifolds that are described by plumbing trees. This model allows an efficient description of Stokes phenomenon for perturbative expansions in Chern-Simons theory around classical solutions - $SL(2,\mathbb{C})$ flat connections. Moreover, the Stokes coefficients can be categorified, i.e. promoted to graded vector spaces, in terms of this finite-dimensional model. At least naively, the categorification gives BPS spectrum of 5d maximally supersymmetric Yang-Mills theory on the 3-manifold times a line with appropriate boundary conditions. We also comment on necessity of taking into account "flat connections at infinity" to capture Stokes phenomenon for certain 3-manifolds.