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

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 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 themore » double-scaling limit trans-series. Furthermore, 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.« less

Authors:
 [1];  [1]
  1. Univ. of Connecticut, Storrs, CT (United States)
Publication Date:
Research Org.:
Univ. of Connecticut, Storrs, CT (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
OSTI Identifier:
1501886
Grant/Contract Number:  
SC0010339
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 97 MATHEMATICS AND COMPUTING; Nonperturbative Effects; 1/N Expansion; Matrix Models

Citation Formats

Ahmed, Anees, and Dunne, Gerald V. Transmutation of a trans-series: the Gross-Witten-Wadia phase transition. United States: N. p., 2017. Web. doi:10.1007/jhep11(2017)054.
Ahmed, Anees, & Dunne, Gerald V. Transmutation of a trans-series: the Gross-Witten-Wadia phase transition. United States. doi:10.1007/jhep11(2017)054.
Ahmed, Anees, and Dunne, Gerald V. Thu . "Transmutation of a trans-series: the Gross-Witten-Wadia phase transition". United States. doi:10.1007/jhep11(2017)054. https://www.osti.gov/servlets/purl/1501886.
@article{osti_1501886,
title = {Transmutation of a trans-series: the Gross-Witten-Wadia phase transition},
author = {Ahmed, Anees and Dunne, Gerald V.},
abstractNote = {We study the change in the resurgent asymptotic properties of a trans-series in two parameters, a coupling g2 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. Furthermore, 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.},
doi = {10.1007/jhep11(2017)054},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2017,
place = {United States},
year = {2017},
month = {11}
}

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