Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition
Oct 4, 2017Citations per year
Abstract: (Springer)
We study the change in the resurgent asymptotic properties of a trans-series in two parameters, a coupling g and a gauge index N, as a system passes through a large N phase transition, using the universal example of the Gross-Witten-Wadia third-order 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 Tracy-Widom 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.Note:
- 46 pages; 20 figures
- Nonperturbative Effects
- 1/N Expansion
- Matrix Models
- expansion: strong coupling
- expansion: weak coupling
- matrix model: unitarity
- critical phenomena
- nonperturbative
- transmutation
- differential equations
References(96)
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