The Resurgent Structure of Quantum Knot Invariants
Abstract
The asymptotic expansion of quantum knot invariants in complex ChernSimons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of qseries with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear qdifference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the DimofteGaiottoGukov 3Dindex, and thus is given by a counting of BPS states. We illustrate our conjectures explicitly by matching theoretically and numerically computed integers for the cases of the 4_{1} and the 5_{2} knots.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 August 2021
 DOI:
 10.1007/s00220021040760
 arXiv:
 arXiv:2007.10190
 Bibcode:
 2021CMaPh.386..469G
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Geometric Topology
 EPrint:
 minor corrections, 25 pages, 4 figures