M2-branes and $\mathfrak{q}$-Painlev\'e equations

TL;DR

This paper uncovers a new link between M2-brane effective theories, quiver Chern-Simons models, and $rak{q}$-Painlevé equations, revealing their role in describing moduli spaces of topological strings and gauge theories.

Contribution

It proposes that the grand canonical partition function of a four-node quiver Chern-Simons theory solves the $rak{q}$-Painlevé VI equation, extending known relations in string and gauge theories.

Findings

Partition function solves $rak{q}$-Painlevé VI equation.

Provides evidence for Fredholm determinant representation of $ au$-function.

Analyzes decoupling limits to $rak{q}$-Painlevé III$_3$.

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 Painlev\'e 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 $\mathfrak{q}$-Painlev\'e 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, $\mathfrak{q}$-Painlev\'e $\text{III}_3$, 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 $\mathfrak{q}$-Painlev\'e 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 $\mathfrak{q}$-Painlev\'e$\,\,$III${}_3$.