Cluster integrable systems, q-Painlevé equations and their quantization

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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.

Original languageEnglish
Article number77
JournalJournal of High Energy Physics
Issue number2
Publication statusPublished - 1 Feb 2018


  • Conformal and W Symmetry
  • Integrable Hierarchies
  • Supersymmetric Gauge Theory
  • Topological Strings


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