# 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 »

- Authors:

- 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:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- 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)},

issn = {1029-8479},

number = 11,

volume = 2017,

place = {United States},

year = {2017},

month = {11}

}