Journal of High Energy Physics

, 2018:77 | Cite as

Cluster integrable systems, q-Painlevé equations and their quantization

Open Access
Regular Article - Theoretical Physics


We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.

We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.


Supersymmetric Gauge Theory Conformal and W Symmetry Integrable Hierarchies Topological Strings 


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This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Center for Advanced Studies, SkoltechMoscowRussia
  3. 3.Department of Mathematics and Laboratory for Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussian Federation
  4. 4.Institute for Information Transmission ProblemsMoscowRussia
  5. 5.Independent University of MoscowMoscowRussia
  6. 6.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  7. 7.Institute for Theoretical and Experimental PhysicsMoscowRussia
  8. 8.Theory Department of Lebedev Physics InstituteMoscowRussia

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