M2-branes and q-Painlevé equations

In this paper we investigate a novel connection between the effective theory of M2-branes on (C-2/Z(2)xC(2)/Z(2))/Z(k) and the q-deformed Painleve 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-Painleve 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-Painleve III3, and topological strings on local P-1 x P-1. From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the q-Painleve VI tau-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-Painleve III3.