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arXiv · 1711.02063

Cluster integrable systems, q-Painleve equations and their quantization

M. Bershtein·P. Gavrylenko·A. Marshakov
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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.

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M. Bershtein, P. Gavrylenko, A. Marshakov. 2017-11-06. Cluster integrable systems, q-Painleve equations and their quantization. https://doi.org/10.1007/jhep02(2018)077

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