Transmutation of a trans-series: the Gross-Witten-Wadia phase transition

Abstract

Abstract:We study the change in the resurgent asymptotic properties of a trans-series in two parameters, a coupling g 2 and a gauge index N , as a system passes through a large N phase transition, using the universal example of the Gross-Witten-Wadia thirdorder phase transition in the unitary matrix model. This transition is well-studied in the immediate vicinity of the transition point, where it is characterized by a double-scaling limit Painlevé II equation, and also away from the transition point using the pre-string difference equation. Here we present a complementary analysis of the transition at all coupling and all finite N , in terms of a differential equation, using the explicit TracyWidom mapping of the Gross-Witten-Wadia partition function to a solution of a Painlevé III equation. This mapping provides a simple method to generate trans-series expansions in all parameter regimes, and to study their transmutation as the parameters are varied. For example, at any finite N the weak coupling expansion is divergent, with a non-perturbative trans-series completion; on the other hand, the strong coupling expansion is convergent, and yet there is still a non-perturbative trans-series completion. We show how the different instanton terms 'condense' at the transition point to match with the double-scaling limit trans-series. We also define a uniform large N strong-coupling expansion (a non-linear analogue of uniform WKB), which is much more precise than the conventional large N expansion through the transition region, and apply it to the evaluation of Wilson loops.

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Resurgence, Painlevé equations and conformal blocks

“…Singularities can change their character, which affects the form of the expansion about that point. In this Section we illustrate this phenomenon with the example of the Gross-Witten-Wadia (GWW) unitary matrix model [99][100][101], which is characterized by the Painlevé III equation [103,104] (and in the double-scaling limit by the Painlevé II equation [99,100,102]). Like PVI, the PIII equation has a regular fixed singularity at t = 0, but after the merging of the other two singularities, PIII has an irregular fixed singularity at t = ∞.…”