Stavros Garoufalidis
Abstract: The purpose of the paper is to introduce an arithmetic resurgence conjecture for two power series that encode exact and perturbative Chern-Simons invariants of knotted 3-dimensional objects. By arithmetic resurgence we mean analytic continuation of a power series 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.
Key words: Chern-Simons theory, analytic continuation, resurgence, arithmetic resurgence, quasi-unipotent monodromy, Gevrey series of mixed type, quasi-unipotent monodromy, Quantum Field Theory, TQFT, knots, 3-manifolds, Habiro ring, $q$-factorials, Rogers dilogarithm, Witten's conjecture, Volume Conjecture, asymptotic expansions, periods, Riemann-Hilbert problem, $G$-functions.
Notes: 17 pages, 5 figures.