Flat connections at infinity on knot surgery manifolds

TL;DR

This work investigates flat connections at infinity in $ m SL(2,\mathbb{C})$ Chern-Simons theory on 3-manifolds obtained by knot surgeries, linking their existence to resurgence and non-perturbative path integrals. It develops two counting frameworks—one via Wirtinger presentations of knot groups and another via a degree-theory map tied to the $A$-polynomial and Dehn-filling constraints—to enumerate asymptotic ends and compute their Chern-Simons invariants. Applying these methods to rational surgeries on torus, twist, and double-twist knots, the authors demonstrate abundant flat connections at infinity, provide explicit CS data, and validate the results against plumbing-graph techniques where possible. The findings illuminate how flat connections at infinity could influence non-perturbative contributions and resurgent structures in complex Chern-Simons theory, and point toward a richer, systematic treatment of non-perturbative effects in 3d-3d and $\widehat{Z}$-invariants frameworks.

Abstract

$\rm SL(2,\mathbb{C})$ Chern-Simons theory on a closed 3-manifold is one of the most interesting, yet tractable examples of a QFT. On one hand, its non-perturbative structure is not yet fully understood; on the other, the mathematical structure turns out to be very rich. In this work we explore the new phenomenon of flat connections at infinity on various knot surgery manifolds. Such flat connections can be understood as asymptotic ends in the non-compact moduli space of flat $\rm SL(2,\mathbb{C})$ connections. We focus on the examples of $\pm 1/r$-surgeries on torus, twist and some double twist knot complements in $S^3$. Surprisingly, our findings suggest that flat connections at infinity are abundant even for simple low-crossing knot surgeries. We therefore believe that their presence would shed light on the resurgent nature of the path integral.

Figures (10)

• Figure 1: Wirtinger presentation of the trefoil knot complement $S^3\setminus T(2,3)$
• Figure 2: Plot of $x$-values: $x_{\alpha}$ (blue dots) and zeros of the Alexander polynomial $\{x\,\vert\, \Delta_{3_1}(x^2)=0\}$ (gray dots) for $S^3_{1/2}(T(2,-3)) = \overline{\Sigma(2,3,13)}$
• Figure 3: Plumbing graph representation for $\Sigma(2, 3, 13)$
• Figure 4: Plot of $x$-values: $x_{\alpha}$ (blue dots) and zeros of the Alexander polynomial $\{x\,\vert\, \Delta_{T(3,4)}(x^2)=0\}$ (gray dots) for $S^3_{+1/2}(T(3,4)) = \overline{\Sigma(3,4,23)}$
• Figure 5: Twist knot $K_n$ with $n=2$ and $n=-2$