Cluster integrable systems, q-Painleve equations and their quantization

TL;DR

This paper explores the connection between cluster integrable systems and q-Painleve equations, introducing their quantum versions and solutions, and proposing a generalization of the isomonodromy/CFT correspondence.

Contribution

It establishes a link between cluster integrable systems and q-Painleve equations, defines their quantum versions, and offers formal solutions using q-deformed conformal blocks.

Findings

Identification of 16 convex Newton polygons with a single interior point.

Formal solutions for quantum q-Painleve systems using Nekrasov functions.

Proposal of a quantum cluster structure generalizing isomonodromy/CFT correspondence.

Abstract

We discuss the relation between the cluster integrable systems and $q$-difference Painlev\'e equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlev\'e 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 $q$-Painlev\'e 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 Painlev\'e system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.