DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Resurgence, Painlevé equations and conformal blocks

Abstract

We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. Here, the resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross–Witten–Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively.

Authors:
ORCiD logo [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)
OSTI Identifier:
1594816
Grant/Contract Number:  
SC0010339; PHY-1125915
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 52; Journal Issue: 46; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; resurgence; Painlevé systems; conformal blocks

Citation Formats

Dunne, Gerald V. Resurgence, Painlevé equations and conformal blocks. United States: N. p., 2019. Web. doi:10.1088/1751-8121/ab3142.
Dunne, Gerald V. Resurgence, Painlevé equations and conformal blocks. United States. https://doi.org/10.1088/1751-8121/ab3142
Dunne, Gerald V. Mon . "Resurgence, Painlevé equations and conformal blocks". United States. https://doi.org/10.1088/1751-8121/ab3142. https://www.osti.gov/servlets/purl/1594816.
@article{osti_1594816,
title = {Resurgence, Painlevé equations and conformal blocks},
author = {Dunne, Gerald V.},
abstractNote = {We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. Here, the resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross–Witten–Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively.},
doi = {10.1088/1751-8121/ab3142},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 46,
volume = 52,
place = {United States},
year = {2019},
month = {10}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 10 works
Citation information provided by
Web of Science

Figures / Tables:

Table 1 Table 1: Identification between the Painlevé Ⅲ boundary condition parameters in (2.25, 2.26) and the parameters appearing the associated Mathieu spectral problem.

Save / Share:

Works referenced in this record:

Holonomy of the Ising model form factors
journal, December 2006

  • Boukraa, S.; Hassani, S.; Maillard, J-M
  • Journal of Physics A: Mathematical and Theoretical, Vol. 40, Issue 1
  • DOI: 10.1088/1751-8113/40/1/005

On Painlevé/gauge theory correspondence
journal, September 2017

  • Bonelli, Giulio; Lisovyy, Oleg; Maruyoshi, Kazunobu
  • Letters in Mathematical Physics, Vol. 107, Issue 12
  • DOI: 10.1007/s11005-017-0983-6

The Two-Dimensional Ising Model
book, December 1973


Painlevé transcendents and ${\mathcal{P}}{\mathcal{T}}$-symmetric Hamiltonians
journal, October 2015


Operator product expansion convergence in conformal field theory
journal, November 2012


Seiberg-Witten Prepotential from Instanton Counting
journal, January 2003


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


Critical point phenomena - The role of series expansions
journal, June 1974


Critical values of the Yang–Yang functional in the quantum sine-Gordon model
journal, December 2011


Wall-crossing, Hitchin systems, and the WKB approximation
journal, February 2013


Holomorphic anomaly and quantum mechanics
journal, December 2017

  • Codesido, Santiago; Mariño, Marcos
  • Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 5
  • DOI: 10.1088/1751-8121/aa9e77

Monodromy problem and the boundary condition for some Painlevé equations
journal, January 1982

  • Jimbo, Michio
  • Publications of the Research Institute for Mathematical Sciences, Vol. 18, Issue 3
  • DOI: 10.2977/prims/1195183300

Isomonodromic Tau-Functions from Liouville Conformal Blocks
journal, December 2014

  • Iorgov, N.; Lisovyy, O.; Teschner, J.
  • Communications in Mathematical Physics, Vol. 336, Issue 2
  • DOI: 10.1007/s00220-014-2245-0

Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
journal, February 2019

  • Forrester, P. J.; Perk, J. H. H.; Trinh, A. K.
  • Journal of Statistical Mechanics: Theory and Experiment, Vol. 2019, Issue 2
  • DOI: 10.1088/1742-5468/ab00e6

Random Unitary Matrices, Permutations and Painlevé
journal, November 1999

  • Tracy, Craig A.; Widom, Harold
  • Communications in Mathematical Physics, Vol. 207, Issue 3
  • DOI: 10.1007/s002200050741

Resurgence in sine-Gordon quantum mechanics: exact agreement between multi-instantons and uniform WKB
journal, September 2015

  • Misumi, Tatsuhiro; Nitta, Muneto; Sakai, Norisuke
  • Journal of High Energy Physics, Vol. 2015, Issue 9
  • DOI: 10.1007/JHEP09(2015)157

Form factor expansions in the 2D Ising model and Painlevé VI
journal, October 2010


Liouville Correlation Functions from Four-Dimensional Gauge Theories
journal, January 2010

  • Alday, Luis F.; Gaiotto, Davide; Tachikawa, Yuji
  • Letters in Mathematical Physics, Vol. 91, Issue 2
  • DOI: 10.1007/s11005-010-0369-5

Conformal bootstrap in Liouville field theory
journal, October 1996


Quantum Painlevé-Calogero correspondence
journal, July 2012

  • Zabrodin, A.; Zotov, A.
  • Journal of Mathematical Physics, Vol. 53, Issue 7
  • DOI: 10.1063/1.4732532

Quantum Painlevé-Calogero correspondence for Painlevé VI
journal, July 2012

  • Zabrodin, A.; Zotov, A.
  • Journal of Mathematical Physics, Vol. 53, Issue 7
  • DOI: 10.1063/1.4732534

The anisotropic Ising correlations as elliptic integrals: duality and differential equations
journal, October 2016


An introduction to lattice gauge theory and spin systems
journal, October 1979


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


On information loss in AdS3/CFT2
journal, May 2016

  • Fitzpatrick, A. Liam; Kaplan, Jared; Li, Daliang
  • Journal of High Energy Physics, Vol. 2016, Issue 5
  • DOI: 10.1007/JHEP05(2016)109

TBA equations and resurgent Quantum Mechanics
journal, January 2019

  • Ito, Katsushi; Mariño, Marcos; Shu, Hongfei
  • Journal of High Energy Physics, Vol. 2019, Issue 1
  • DOI: 10.1007/JHEP01(2019)228

Asymptotics of the Instantons of Painlevé I
journal, March 2011

  • Garoufalidis, Stavros; Its, Alexander; Kapaev, Andrei
  • International Mathematics Research Notices, Vol. 2012, Issue 3
  • DOI: 10.1093/imrn/rnr029

Stark effect of hydrogenic ions
journal, February 1982


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


The sixth Painlevé transcendent and uniformization of algebraic curves
journal, February 2016


Painlevé III and 2D polymers
journal, December 1994


A primer on resurgent transseries and their asymptotics
journal, May 2019


Uniform asymptotic and JWKB expansions for anharmonic oscillators
journal, March 2000

  • Álvarez, Gabriel; Casares, Carmen
  • Journal of Physics A: Mathematical and General, Vol. 33, Issue 13
  • DOI: 10.1088/0305-4470/33/13/304

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


Exact WKB analysis of N $$ \mathcal{N} $$ = 2 gauge theories
journal, July 2016

  • Ashok, Sujay K.; Jatkar, Dileep P.; John, Renjan R.
  • Journal of High Energy Physics, Vol. 2016, Issue 7
  • DOI: 10.1007/JHEP07(2016)115

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions
journal, August 2018


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


Erratum: Conformal field theory of Painlevé VI
journal, October 2012

  • Gamayun, O.; Iorgov, N.; Lisovyy, O.
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)183

Monodromy dependence and connection formulae for isomonodromic tau functions
journal, May 2018


Complex saddle points in the Gross-Witten-Wadia matrix model
journal, November 2016


Log-Gases and Random Matrices (LMS-34)
book, December 2010


Darboux coordinates, Yang-Yang functional, and gauge theory
journal, July 2011


Seiberg-Witten Theory and Random Partitions
book, January 2007


On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture
journal, June 2011

  • Alba, Vasyl A.; Fateev, Vladimir A.; Litvinov, Alexey V.
  • Letters in Mathematical Physics, Vol. 98, Issue 1
  • DOI: 10.1007/s11005-011-0503-z

Asymptotics and Borel Summability
book, January 2008


Asymptotically free = 2 theories and irregular conformal blocks
journal, December 2013


Quantum geometry of resurgent perturbative/nonperturbative relations
journal, May 2017

  • Basar, Gökçe; Dunne, Gerald V.; Ünsal, Mithat
  • Journal of High Energy Physics, Vol. 2017, Issue 5
  • DOI: 10.1007/JHEP05(2017)087

Possible third-order phase transition in the large- N lattice gauge theory
journal, January 1980


Anharmonic oscillator discontinuity formulae up to second-exponentially-small order
journal, April 2002

  • $Aacute$lvarez, Gabriel; Howls, Christopher J.; Silverstone, Harris J.
  • Journal of Physics A: Mathematical and General, Vol. 35, Issue 18
  • DOI: 10.1088/0305-4470/35/18/302

Hyperasymptotics for Integrals with Saddles
journal, September 1991

  • Berry, M. V.; Howls, C. J.
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 434, Issue 1892
  • DOI: 10.1098/rspa.1991.0119

Uniform WKB, multi-instantons, and resurgent trans-series
journal, May 2014


Exponential asymptotics, transseries, and generalized Borel summation for analytic, nonlinear, rank-one systems of ordinary differential equations
journal, January 1995


A Review of the Sixth Painlevé Equation
journal, July 2014


On determinant representation and integrability of Nekrasov functions
journal, October 2017


Random Matrix Theory and Entanglement in Quantum Spin Chains
journal, October 2004


Studies on holonomic quantum fields, XVII
journal, January 1980

  • Jimbo, Michio; Miwa, Tetsuji
  • Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 56, Issue 9
  • DOI: 10.3792/pjaa.56.405

Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region
journal, January 1976


Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator
journal, July 2000

  • Álvarez, Gabriel; Casares, Carmen
  • Journal of Physics A: Mathematical and General, Vol. 33, Issue 29
  • DOI: 10.1088/0305-4470/33/29/302

Polynomial Hamiltonians associated with Painlevé equations, II. Differential equations satisfied by polynomial Hamiltonians
journal, January 1980

  • Okamoto, Kazuo
  • Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 56, Issue 8
  • DOI: 10.3792/pjaa.56.367

Generating nonperturbative physics from perturbation theory
journal, February 2014


How instanton combinatorics solves Painlevé VI, V and IIIs
journal, July 2013


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


Discrete Painlevé equations for a class of P VI τ-functions given as U(N) averages
journal, June 2005


Studies on holonomic quantum fields, XVI. Density matrix of impenetrable bose gas
journal, January 1979

  • Jimbo, Michio; Miwa, Tetsuji; Mōri, Yasuko
  • Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 55, Issue 9
  • DOI: 10.3792/pjaa.55.317

Poles of integrále tritronquée and anharmonic oscillators. A WKB approach
journal, February 2010


Relation of conformal field theory and deformation theory for the Ising model
journal, January 1987


Infinite conformal symmetry in two-dimensional quantum field theory
journal, July 1984


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

  • Álvarez, Gabriel
  • Journal of Mathematical Physics, Vol. 45, Issue 8
  • DOI: 10.1063/1.1767988

Electron counting statistics and coherent states of electric current
journal, October 1996

  • Levitov, Leonid S.; Lee, Hyunwoo; Lesovik, Gordey B.
  • Journal of Mathematical Physics, Vol. 37, Issue 10
  • DOI: 10.1063/1.531672

Transmutation of a trans-series: the Gross-Witten-Wadia phase transition
journal, November 2017


Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results
journal, June 2013

  • Deift, Percy; Its, Alexander; Krasovsky, Igor
  • Communications on Pure and Applied Mathematics, Vol. 66, Issue 9
  • DOI: 10.1002/cpa.21467

Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory
text, January 1976


Darboux coordinates, Yang-Yang functional, and gauge theory
journal, October 2014

  • Nekrasov, N. A.; Rosly, A. A.; Shatashvili, S. L.
  • Theoretical and Mathematical Physics, Vol. 181, Issue 1
  • DOI: 10.1007/s11232-014-0209-3

Operator product expansions on the vacuum in conformal quantum field theory in two spacetime dimensions
text, January 1975


Reductions of Self-Dual Yang-Mills Fields and Classical Systems
journal, October 1990


Liouville Correlation Functions from Four-dimensional Gauge Theories
text, January 2009


Wall-crossing, Hitchin Systems, and the WKB Approximation
preprint, January 2009


On combinatorial expansion of the conformal blocks arising from AGT conjecture
text, January 2010


A Review on The Sixth Painleve' Equation
text, January 2012


The toroidal block and the genus expansion
text, January 2012


Uniform WKB, Multi-instantons, and Resurgent Trans-Series
text, January 2014


Isomonodromic tau-functions from Liouville conformal blocks
text, January 2014


Non-Perturbative Quantum Geometry II
text, January 2014


Classical limit of irregular blocks and Mathieu functions
text, January 2015


On Information Loss in AdS$_3$/CFT$_2$
preprint, January 2016


Holomorphic Anomaly and Quantum Mechanics
text, January 2016


Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
text, January 2017


Bands and gaps in Nekrasov partition function
text, January 2017


A Primer on Resurgent Transseries and Their Asymptotics
text, January 2018


TBA equations and resurgent Quantum Mechanics
text, January 2018


The large-N expansion of unitary-matrix models
text, January 1996


Classical conformal blocks and accessory parameters from isomonodromic deformations
journal, April 2018


Conformal field theory of Painlevé VI
journal, October 2012

  • Gamayun, O.; Iorgov, N.; Lisovyy, O.
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/jhep10(2012)038

Isomonodromic Tau-Functions from Liouville Conformal Blocks
journal, December 2014

  • Iorgov, N.; Lisovyy, O.; Teschner, J.
  • Communications in Mathematical Physics, Vol. 336, Issue 2
  • DOI: 10.1007/s00220-014-2245-0

On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture
journal, June 2011

  • Alba, Vasyl A.; Fateev, Vladimir A.; Litvinov, Alexey V.
  • Letters in Mathematical Physics, Vol. 98, Issue 1
  • DOI: 10.1007/s11005-011-0503-z

On Painlevé/gauge theory correspondence
journal, September 2017

  • Bonelli, Giulio; Lisovyy, Oleg; Maruyoshi, Kazunobu
  • Letters in Mathematical Physics, Vol. 107, Issue 12
  • DOI: 10.1007/s11005-017-0983-6

The sixth Painlevé transcendent and uniformization of algebraic curves
journal, February 2016


Form factor expansions in the 2D Ising model and Painlevé VI
journal, October 2010


Darboux coordinates, Yang-Yang functional, and gauge theory
journal, July 2011


How instanton combinatorics solves Painlevé VI, V and IIIs
journal, July 2013


Reductions of self-dual Yang-Mills fields and classical systems
journal, August 1990


Unitary Matrix Models and PainlevÉ iii
journal, December 1996


Monodromy dependence and connection formulae for isomonodromic tau functions
journal, May 2018


Monodromy problem and the boundary condition for some Painlevé equations
journal, January 1982

  • Jimbo, Michio
  • Publications of the Research Institute for Mathematical Sciences, Vol. 18, Issue 3
  • DOI: 10.2977/prims/1195183300

OPE Convergence in Conformal Field Theory
text, January 2012


Classical irregular block, N=2 pure gauge theory and Mathieu equation
text, January 2014


Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB
text, January 2015


Bands and gaps in Nekrasov partition function
text, January 2017


Painleve III and 2D Polymers
text, January 1994


Random Matrix Theory and Entanglement in Quantum Spin Chains
text, January 2004


Figures / Tables found in this record:

    Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.