M2-branes and $\mathfrak{q}$-Painlevé equations

Abstract

In this paper we investigate a novel connection between the effective theory of M2-branes on $({\mathbb{C}}^2/{\mathbb{Z}}_2\times {\mathbb{C}}^2/{\mathbb{Z}}_2)/{\mathbb{Z}}_k$ and the $\mathfrak{q}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver $\mathcal{N}=4$ Chern-Simons matter theory solves the $\mathfrak{q}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local ${\text{dP}}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text{SU}(2)$ $\mathcal{N}=1$ gauge theory on a circle. The results we find extend the known relation between ABJM theory, $\mathfrak{q}$-Painlevé ${\text{III}}_3$, and topological strings on local ${\mathbb{P}}^1\times {\mathbb{P}}^1$. From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the $\mathfrak{q}$-Painlevé 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 $\mathfrak{q}$-PainlevéIII${}_3$.

63 pages, 7 figures

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