The Resurgent Structure of Quantum Knot Invariants
Abstract
The asymptotic expansion of quantum knot invariants in complex Chern-Simons 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 q-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear q-difference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the Dimofte-Gaiotto-Gukov 3D-index, 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 41 and the 52 knots.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- August 2021
- DOI:
- 10.1007/s00220-021-04076-0
- arXiv:
- arXiv:2007.10190
- Bibcode:
- 2021CMaPh.386..469G
- Keywords:
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- High Energy Physics - Theory;
- Mathematics - Geometric Topology
- E-Print:
- minor corrections, 25 pages, 4 figures