BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlev\'e equations

TL;DR

This paper explores the symmetry-induced discrete flows in BPS quivers for 5D SCFTs, linking them to q-Painlevé equations and topological string partition functions, with explicit examples in SU(2) theories.

Contribution

It establishes a novel connection between BPS quiver symmetries, q-Painlevé equations, and topological string theory in five-dimensional superconformal field theories.

Findings

Discrete flows generate BPS spectra and q-difference equations.

Solutions are expressed as topological string partition functions.

Explicit examples for SU(2) theories demonstrate the framework.

Abstract

We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi-Yau geometries describing five dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlev\'e equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $\tau$-functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five dimensional $SU(2)$ pure Super Yang-Mills and $N_f = 2$ on a circle.