# Quiver symmetries and wall-crossing invariance

Fabrizio Del Monte, Pietro Longhi

Fabrizio Del Monte, Pietro Longhi

**2/8/21**Published in : arXiv:2107.14255

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.