Resurgence and Large N: the Gross-Witten-Wadia and Gross-Neveu Models

Abstract

I present a detailed study of parametric resurgence in the U(N) lattice gauge theory, also known as the Gross-Witten-Wadia model. I show how trans-series expansions in different sectors transmute into each other as they pass through the large N phase transition. This transition is well-studied in the immediate vicinity of the transition point, where it is characterized by a double-scaling limit Painleve II equation, and also away from the transition point using the pre-string difference equation. Here I present a complementary analysis of the transition at all coupling and all finite N, in terms of a differential equation, using the explicit Tracy-Widom mapping of the Gross-Witten-Wadia partition function to a solution of a Painleve 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. A surprising result is uncovered: the strong coupling expansion is convergent and yet there is a non-perturbative trans-series completion. I show how the different instanton terms `condense' at the transition point to match with the double-scaling limit trans-series. I 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. In addition, I study the Ginzburg-Landau expansion of the Gross-Neveu model. As the order of the expansion increases, the crystal phase resembles the exact crystal phase more and more closely.

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