Cluster Toda chains and Nekrasov functions

Abstract

In this paper the relation between the cluster integrable systems and $q$-difference equations is extended beyond the Painlevé case. We consider the class of hyperelliptic curves when the Newton polygons contain only four boundary points. The corresponding cluster integrable Toda systems are presented, and their discrete automorphisms are identified with certain reductions of the Hirota difference equation. We also construct non-autonomous versions of these equations and find that their solutions are expressed in terms of 5d Nekrasov functions with the Chern-Simons contributions, while in the autonomous case these equations are solved in terms of the Riemann theta-functions.

arXiv comment32 pages, 13 figures, small corrections, references added

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• Quantum spectral problems and isomonodromic deformations
• Instantons to the people: the power of one-form symmetries
• Combinatorial expressions for the tau functions of $q$-Painlevé V and III equations
• SU(N) q-Toda equations from mass deformed ABJM theory
• M2-branes and $\mathfrak{q}$-Painlevé equations
• Cluster integrable systems and spin chains
• Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
• On $q$-Isomonodromic Deformations and $q$-Nekrasov Functions
• On a 5D UV completion of Argyres-Douglas theories
• Painlevé Kernels and Surface Defects at Strong Coupling
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux
• Blowup relations on ${\mathbb{C}}^2/{\mathbb{Z}}_2$ from Nakajima-Yoshioka blowup relations