Resurgent Analysis for Some 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.

42 pages, 4 figures

References in corpus (4)

• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• Quantum Knot Invariant for Torus Link and Modular Forms
• Large color $R$-matrix for knot complements and strange identities
• A resurgence analysis of the $SU(2)$ Chern-Simons partition functions on a Brieskorn homology sphere $\Sigma (2,5,7)$

Cited by in corpus (4)

• BPS Invariants for a Knot in Seifert Manifolds
• Resurgent Analysis of SU(2) Chern-Simons Partition Function on Brieskorn Spheres $\Sigma (2,3,6n+5)$
• Quantum Modular $\hat{Z}{}^G$-Invariants
• Orientation Reversal and the Chern-Simons Natural Boundary