Resurgence in complex Chern-Simons theory

TL;DR

The paper develops a comprehensive resurgent framework for SU(2) Chern-Simons theory on closed 3-manifolds, showing that the full partition function can be reconstructed from perturbative data around abelian flat connections via transseries that incorporate non-abelian contributions. It establishes exact Borel transforms for key mock-modular q-series, demonstrates how homological blocks arise as Borel sums, and ties resurgence to the geometry of flat connections on A-polynomial curves and to 4d Yang-Mills instantons on M4. By exploring surgeries, cyclotomic expansions, and 3d N=2 localization, the work provides a unifying view connecting mock modularity, wall crossing, and categorification of WRT invariants, with concrete examples including the Poincaré sphere and Brieskorn spheres. The results illuminate the interplay between abelian data, complex flat connections, and instanton moduli, while offering practical computational tools via Lefschetz thimbles and Borel analysis for quantum invariants of 3-manifolds.

Abstract

We study resurgence properties of partition function of SU(2) Chern-Simons theory (WRT invariant) on closed three-manifolds. We check explicitly that in various examples Borel transforms of asymptotic expansions posses expected analytic properties. In examples that we study we observe that contribution of irreducible flat connections to the path integral can be recovered from asymptotic expansions around abelian flat connections. We also discuss connection to Floer instanton moduli spaces, disk instantons in 2d sigma models, and length spectra of "complex geodesics" on the A-polynomial curve.

Figures (19)

• Figure 1: Graphical representatation of a Stokes phenomenon in the Borel plane. The dashed circle depicts contribution of non-integral terms in (\ref{['Borel-stokes-0']}) which are given by residue of $B^{{\mathbb{\bbalpha}}}(\xi)e^{-k\xi}$ at $\xi=\xi_{{\mathbb{\bbbeta}}}$.
• Figure 2: The Borel plane.
• Figure 3: A numerical approximation to $|\widetilde{BZ}_{\text{pert}} (\xi)|$ for $\xi \in i {\mathbb{R}}$.
• Figure 4: The integration contours that give $\widetilde{\Psi}^{(a)}_p (q)$. $(a)$ Integration contour in $\eta$-plane. The directions in which integrand decays are shown as a hatched domain in the case of generic value $k$ such that $\mathrm{Im}\,k<0$. $(b)$ The image of the contour in $\xi$-plane under the coordinate change. $(c)$ Equivalent contour in $\xi$-plane (we used the fact that the integrand changes sign when $\xi$ goes around the origin. $(all)$ The black dots depict singularities of the integrands.
• Figure 5: The deformation of the contour for generalized Borel resummation (\ref{['Zaverageii']}) (cf. \ref{['Borel-half-sum']}) into contours corresponding to Lefschetz thimbles for $k>0$. Note that logarithmic terms in (\ref{['Borel-general-pole']}) vanish in this case.