Chern-Simons theory, analytic continuation and arithmetic

TL;DR

This paper explores the conjectural arithmetic and analytic properties of quantum invariants of knots and 3-manifolds, proposing a framework for their analytic continuation and resurgence behavior.

Contribution

It introduces conjectures linking quantum invariants to arithmetic resurgence, unifying perturbative and nonperturbative invariants into a new analytic framework.

Findings

Proposal of power series P and NP for invariants

Conjecture of their arithmetic resurgence properties

Discussion of evidence supporting the conjectures

Abstract

The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in a neighborhood of zero, and we postulate their arithmetic resurgence. By the latter term, we mean analytic continuation as a multivalued analytic function in the complex numbers minus a discrete set of points, with restricted singularities, local and global monodromy. We point out some key features of arithmetic resurgence in connection to various problems of asymptotic expansions of exact and perturbative Chern-Simons theory with compact or complex gauge group. Finally, we discuss theoretical and experimental evidence for our conjecture.