On categorification of Stokes coefficients in Chern-Simons theory

Abstract

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.

Figures (27)

• Figure 1: The black ray originating at $\xi=\mathrm{CS}_\mathbb{\alpha}$ shows the contour in (\ref{['thimble-integral-borel']}) corresponding to the Lefschetz thimble, valid for the case $\delta_\bbalpha\geq 0$. The blue contour shows the Hankel-type contour needed instead in the case $\delta_\bbalpha<0$, with $B^\bbalpha(\xi)$ replaced by $\widetilde{B}^\bbalpha(\xi)$ in the integrand. The red crosses mark singularities at critical values of the Chern-Simons functional.
• Figure 2: The path of analytic continuation in the $\xi$-plane from the neighborhood of $\mathrm{CS}_\bbalpha$ to the neighborhood of $\mathrm{CS}_\bbbeta$, in the case when there are other critical values of the Chern-Simons functional (shown as red crosses) on the straight line connecting $\mathrm{CS}_\bbalpha$ with $\mathrm{CS}_\bbbeta$.
• Figure 3: An example of a plumbing graph.
• Figure 4: Dehn surgery diagram corresponding to the plumbing graph shown in Figure \ref{['fig:plumbing-graph']}, with $g_i=0,\,\forall i$.
• Figure 5: Left: the contour $\gamma_I$ in the $v_I$-plane, such that $\gamma=\bigtimes_{I\in H} \gamma_I$ in (\ref{['Z-hat-to-LG-model']}). Right: a homologically equivalent contour. The red crosses schematically denote the positions poles in $v_I$ of the integrand in (\ref{['Z-hat-to-LG-model']}), located potentially at integer values.