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In this paper we investigate a novel connection between the effective theory of M2-branes on (C 2 /Z 2 ×C 2 /Z 2 )/Z k and the q-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver N = 4 Chern-Simons matter theory solves the q-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local dP 5 and, via geometric engineering, five dimensional N f = 4 SU(2) N = 1 gauge theory on a circle. The results we find extend the known relation between ABJM theory, q-Painlevé III 3 , and topological strings on local P 1 × P 1 . From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the q-Painlevé VI τ -function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to N f = 0, corresponding to q-Painlevé III 3 . * bonelli(at)sissa.it † fgloblek(at)sissa.it ‡ naotaka.kubo(at)yukawa.kyoto-u.ac.jp § nosaka(at)yukawa.kyoto-u.ac.jp ¶ tanzini(at)sissa.it

The partition function of N = 2 super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual Ω-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a non-autonomous Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical solution is obtained just by fixing the perturbative part of the IR prepotential as its asymptotic boundary condition for the RGE. We analyse the explicit solutions of the τ-system for all the classical groups at the diverse levels, extend our analysis to affine twisted Lie algebras and provide conjectural bilinear relations for the τ-functions of linear quiver gauge theory.

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