Quiver symmetries and wall-crossing invariance

TL;DR

This paper explores the BPS spectrum of 5D superconformal theories via quivers derived from Calabi-Yau geometries, revealing symmetries that determine wall-crossing invariants and providing explicit solutions for certain geometries.

Contribution

It introduces a method to compute wall-crossing invariants using quiver symmetries linked to Calabi-Yau geometries, with new exact conjectural formulas for specific local geometries.

Findings

Derived equations for wall-crossing invariants from quiver symmetries.

Provided explicit conjectural formulas for local Hirzebruch and del Pezzo geometries.

Revealed the BPS spectrum structure as two copies of 4D spectra with Kaluza-Klein towers.

Abstract

We study the BPS particle spectrum of five-dimensional superconformal field theories (SCFTs) on $\mathbb{R}^4\times S^1$ with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi-Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi-Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich-Soibelman wall-crossing invariant. We solve these equations iteratively: the pattern arising from the solution is naturally extended to an exact conjectural expression, that we provide for the local Hirzebruch $\mathbb{F}_0$, and local del Pezzo $dP_3$ and $dP_5$ geometries. Remarkably, the BPS spectrum consists of two copies of suitable $4d$ $\mathcal{N}=2$ spectra, augmented by Kaluza-Klein towers.