Chern-Simons theory, analytic continuation and arithmetic

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.

AMS-LaTeX, 22 pages with 6 figures

References in corpus (8)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• On the formation of singularities of solutions of nonlinear differential systems in antistokes directions
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Perturbative 3-manifold invariants by cut-and-paste topology
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• A dilogarithmic formula for the Cheeger-Chern-Simons class
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Cited by in corpus (14)

• Analytic Continuation Of Chern-Simons Theory
• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• Asymptotics of the colored Jones function of a knot
• Resurgence in complex Chern-Simons theory
• Peacock patterns and new integer invariants in topological string theory
• Analyticity of the Free Energy of a Closed 3-Manifold
• Resurgence of the Kontsevich-Zagier power series
• Knots, Perturbative Series and Quantum Modularity
• A resurgence analysis of the $SU(2)$ Chern-Simons partition functions on a Brieskorn homology sphere $\Sigma (2,5,7)$
• Quantum Knot Invariants
• Resurgent Analysis of SU(2) Chern-Simons Partition Function on Brieskorn Spheres $\Sigma (2,3,6n+5)$
• Rationality of the SL(2,C)-Reidemeister torsion in dimension 3
• Resurgence of the fractional polylogarithms
• Exact computation of the n-loop invariants of knots