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Gaetan Borot, University of Geneva

Matrix models, non-perturbative topological recursion, and knot invariants

At least heuristically, the asymptotics of convergent matrix integrals are described by a non-perturbative version of the topological recursion, which is not in general an expansion in powers of the coupling constant $g_s$. However, if a Boutroux and a quantification condition hold (these notions will be explained), one recovers a perturbative expansion from the non-perturbative answer provided $g_s$ is quantified, and the non-perturbative effects just result in renormalizations at all orders by derivatives of theta functions. In the second part of the talk, I will explain how this framework can be used in knot theory. I will describe a Chern-Simons matrix model computation for torus knots invariants inspired by a recent work of Brini, Eynard and Marino. By generalizing to the case of hyperbolic knots (even though no matrix model is known in this case), this leads us to a conjecture (completing a former one of Dijkgraaf, Fuji and Manabe) for the all-order asymptotic expansion of the Jones polynomial of hyperbolic knots.