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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]
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  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.
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Dunne, Gerald V. Resurgence, Painlevé equations and conformal blocks. United States. https://doi.org/10.1088/1751-8121/ab3142
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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.
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@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 = {Mon Oct 21 00:00:00 EDT 2019},
month = {Mon Oct 21 00:00:00 EDT 2019}
}
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https://doi.org/10.1088/1751-8121/ab3142
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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.
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Conformal field theory of Painlevé VI
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Isomonodromic Tau-Functions from Liouville Conformal Blocks
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On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture
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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
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Monodromy dependence and connection formulae for isomonodromic tau functions
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Monodromy problem and the boundary condition for some Painlevé equations
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  • Jimbo, Michio
  • Publications of the Research Institute for Mathematical Sciences, Vol. 18, Issue 3
  • DOI: 10.2977/prims/1195183300

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Classical irregular block, N=2 pure gauge theory and Mathieu equation
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Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB
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Bands and gaps in Nekrasov partition function
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Painleve III and 2D Polymers
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Random Matrix Theory and Entanglement in Quantum Spin Chains
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