Resurgent Analysis for Some 3-manifold Invariants

TL;DR

This paper applies resurgence analysis to 3-manifold invariants, revealing that abelian flat connections encode all non-abelian data, and connects homological blocks with analytically continued Chern-Simons partition functions.

Contribution

It demonstrates that abelian flat connections' contributions contain complete information about non-abelian connections in Chern-Simons theory for certain 3-manifolds.

Findings

Abelian flat connections encode all non-abelian flat connection data.

Homological blocks are analytic continuations of full Chern-Simons partition functions.

Resurgent analysis links different flat connection contributions in 3-manifold invariants.

Abstract

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial.