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

doi:10.1007/jhep11(2017)054

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 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-scalingmore »« less

Authors:

Ahmed, Anees ^[1]; Dunne, Gerald V. ^[1]

Univ. of Connecticut, Storrs, CT (United States)

Publication Date:: 2017-11-09

Research Org.:: Univ. of Connecticut, Storrs, CT (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP)

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

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                    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.

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                    Ahmed, Anees, & Dunne, Gerald V. Transmutation of a trans-series: the Gross-Witten-Wadia phase transition.  United States.  doi:10.1007/jhep11(2017)054.

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                    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.

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@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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DOI: 10.1007/jhep11(2017)054

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Instanton induced large N phase transitions in two- and four-dimensional QCD journal, October 1994

Will a Large Complex System be Stable? journal, August 1972

Pure N = 2 $$ \mathcal{N}=2 $$ super Yang-Mills and exact WKB journal, August 2015

The large-N expansion of unitary-matrix models journal, August 1998

Lectures on non-perturbative effects in large N gauge theories, matrix models and strings: Non-perturbative effects in large N gauge theories, matrix models and strings journal, April 2014

Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory journal, March 2016

The toroidal block and the genus expansion journal, March 2013

Nonperturbative effects and the large-order behavior of matrix models and topological strings journal, January 2008

Supersymmetric U(N) Chern–Simons-Matter Theory and Phase Transitions journal, June 2015

Random Matrix Theory and Chiral Symmetry in QCD journal, December 2000

Quantum field theory in the large N limit: a review journal, October 2003

Non-perturbative effects and the refined topological string journal, September 2014

On Borel summation and Stokes phenomena for rank- $1$ nonlinear systems of ordinary differential equations journal, June 1998

Unitary Matrix Models and PainlevÉ iii journal, December 1996

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations journal, October 2004

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions journal, September 2004

N = ∞ phase transition in a class of exactly soluble model lattice gauge theories journal, June 1980

Random Unitary Matrices, Permutations and Painlevé journal, November 1999

Hyperasymptotics for Integrals with Saddles journal, September 1991

Non-perturbative quantum geometry III journal, August 2016

Loop equations and 1N expansion journal, December 1983

Deconstructing zero: resurgence, supersymmetry and complex saddles journal, December 2016

A boundary value problem associated with the second painlev� transcendent and the Korteweg-de Vries equation journal, January 1980

Non-intersecting Brownian walkers and Yang–Mills theory on the sphere journal, March 2011

Level spacing distributions and the Bessel kernel journal, March 1994

Discrete gap probabilities and discrete Painlev� equations journal, April 2003

Application of the ?-function theory of Painlev� equations to random matrices: PV, PIII, the LUE, JUE, and CUE journal, March 2002

Chiral models: Their implication for gauge theories and large N journal, July 1981

Borel Summability of Perturbative Series in 4D N = 2 and 5D N = 1 Supersymmetric Theories journal, May 2016

Unitary matrix models and phase transition journal, January 1998

Nonperturbative Ambiguities and the Reality of Resurgent Transseries journal, October 2014

Theta dependence, sign problems, and topological interference journal, November 2012

How to resum perturbative series in 3d N = 2 Chern-Simons matter theories journal, July 2016

Instantons as a bridgehead at N = ∞ journal, July 1980

The resurgence of instantons in string theory journal, January 2012

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings journal, December 2008

Two dimensional gauge theories revisited journal, October 1992

Two-dimensional Yang–Mills theory, Painlevé equations and the six-vertex model journal, February 2012

Level-spacing distributions and the Airy kernel journal, January 1994

Tracy-Widom distribution as instanton sum of 2D IIA superstrings journal, September 2014

On the exact evaluation of 〈det Up〉 in a lattice gauge model journal, November 1982

Nekrasov functions and exact Bohr-Sommerfeld integrals journal, April 2010

Resurgent analysis of localizable observables in supersymmetric gauge theories journal, March 2015

Wilson loops in unitary matrix models at finite N journal, July 2017

Exact Scaling Functions for One-Dimensional Stationary KPZ Growth journal, April 2004

On quantum gauge theories in two dimensions journal, October 1991

Topological Strings from Quantum Mechanics journal, April 2016

Some properties of large-N two-dimensional Yang-Mills theory journal, March 1995

Non-perturbative contributions in models with a non-analytic behavior at infinite N journal, March 1981

The longest increasing subsequence in a random permutation and a unitary random matrix model journal, January 1998

A Computational Exploration of the Second Painlevé Equation journal, May 2013

Non-perturbative quantum geometry II journal, December 2014

Langer–Cherry derivation of the multi-instanton expansion for the symmetric double well journal, August 2004

2D gravity and random matrices journal, March 1995

Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems journal, February 2015

Resummations and non-perturbative corrections journal, September 2015

Universal randomness journal, March 2011

Top eigenvalue of a random matrix: large deviations and third order phase transition journal, January 2014

Quantization of Integrable Systems and four Dimensional Gauge Theories conference, January 2012

Phase diagram of q-deformed Yang-Mills theory on S2 at non-zero θ-angle journal, April 2018

Resurgence, Painlevé equations and conformal blocks journal, October 2019

The grin of Cheshire cat resurgence from supersymmetric localization journal, January 2018

Eigenvalue instantons in the spectral form factor of random matrix model journal, March 2019

Schwinger pair production from Padé-Borel reconstruction journal, January 2020

Non-perturbative quantum geometry
journal, February 2014

Instantons in quantum mechanics and resurgent expansions
journal, August 2004

Large N phase transition in continuum QCD2
journal, December 1993

The Similarity Solution for the Korteweg-de Vries Equation and the Related Painleve Transcendent
journal, June 1978

Instanton induced large N phase transitions in two- and four-dimensional QCD
journal, October 1994

Will a Large Complex System be Stable?
journal, August 1972

Pure N = 2 $$ \mathcal{N}=2 $$ super Yang-Mills and exact WKB
journal, August 2015

The large-N expansion of unitary-matrix models
journal, August 1998

Lectures on non-perturbative effects in large N gauge theories, matrix models and strings: Non-perturbative effects in large N gauge theories, matrix models and strings
journal, April 2014

Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory
journal, March 2016

The toroidal block and the genus expansion
journal, March 2013

Nonperturbative effects and the large-order behavior of matrix models and topological strings
journal, January 2008

Supersymmetric U(N) Chern–Simons-Matter Theory and Phase Transitions
journal, June 2015

Random Matrix Theory and Chiral Symmetry in QCD
journal, December 2000

Quantum field theory in the large N limit: a review
journal, October 2003

Non-perturbative effects and the refined topological string
journal, September 2014

On Borel summation and Stokes phenomena for rank- $1$ nonlinear systems of ordinary differential equations
journal, June 1998

Unitary Matrix Models and PainlevÉ iii
journal, December 1996

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations
journal, October 2004

Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions
journal, September 2004

N = ∞ phase transition in a class of exactly soluble model lattice gauge theories
journal, June 1980

Random Unitary Matrices, Permutations and Painlevé
journal, November 1999

Hyperasymptotics for Integrals with Saddles
journal, September 1991

Non-perturbative quantum geometry III
journal, August 2016

Loop equations and 1N expansion
journal, December 1983

Deconstructing zero: resurgence, supersymmetry and complex saddles
journal, December 2016

A boundary value problem associated with the second painlev� transcendent and the Korteweg-de Vries equation
journal, January 1980

Non-intersecting Brownian walkers and Yang–Mills theory on the sphere
journal, March 2011

Level spacing distributions and the Bessel kernel
journal, March 1994

Discrete gap probabilities and discrete Painlev� equations
journal, April 2003

Application of the ?-function theory of Painlev� equations to random matrices: PV, PIII, the LUE, JUE, and CUE
journal, March 2002

Chiral models: Their implication for gauge theories and large N
journal, July 1981

Borel Summability of Perturbative Series in 4D $N = 2$ and 5D $N = 1$ Supersymmetric Theories
journal, May 2016

Unitary matrix models and phase transition
journal, January 1998

Nonperturbative Ambiguities and the Reality of Resurgent Transseries
journal, October 2014

Theta dependence, sign problems, and topological interference
journal, November 2012

How to resum perturbative series in 3d $N = 2$ Chern-Simons matter theories
journal, July 2016

Instantons as a bridgehead at N = ∞
journal, July 1980

The resurgence of instantons in string theory
journal, January 2012

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
journal, December 2008

Two dimensional gauge theories revisited
journal, October 1992

Two-dimensional Yang–Mills theory, Painlevé equations and the six-vertex model
journal, February 2012

Level-spacing distributions and the Airy kernel
journal, January 1994

Tracy-Widom distribution as instanton sum of 2D IIA superstrings
journal, September 2014

On the exact evaluation of 〈det Up〉 in a lattice gauge model
journal, November 1982

Nekrasov functions and exact Bohr-Sommerfeld integrals
journal, April 2010

Resurgent analysis of localizable observables in supersymmetric gauge theories
journal, March 2015

Wilson loops in unitary matrix models at finite N
journal, July 2017

Exact Scaling Functions for One-Dimensional Stationary KPZ Growth
journal, April 2004

On quantum gauge theories in two dimensions
journal, October 1991

Topological Strings from Quantum Mechanics
journal, April 2016

Some properties of large-N two-dimensional Yang-Mills theory
journal, March 1995

Non-perturbative contributions in models with a non-analytic behavior at infinite N
journal, March 1981

The longest increasing subsequence in a random permutation and a unitary random matrix model
journal, January 1998

A Computational Exploration of the Second Painlevé Equation
journal, May 2013

Non-perturbative quantum geometry II
journal, December 2014

Langer–Cherry derivation of the multi-instanton expansion for the symmetric double well
journal, August 2004

2D gravity and random matrices
journal, March 1995

Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems
journal, February 2015

Resummations and non-perturbative corrections
journal, September 2015

Universal randomness
journal, March 2011

Top eigenvalue of a random matrix: large deviations and third order phase transition
journal, January 2014

Quantization of Integrable Systems and four Dimensional Gauge Theories
conference, January 2012

Phase diagram of q-deformed Yang-Mills theory on S2 at non-zero θ-angle
journal, April 2018

Resurgence, Painlevé equations and conformal blocks
journal, October 2019

The grin of Cheshire cat resurgence from supersymmetric localization
journal, January 2018

Eigenvalue instantons in the spectral form factor of random matrix model
journal, March 2019

Schwinger pair production from Padé-Borel reconstruction
journal, January 2020