Quiver Symmetries and Wall-Crossing Invariance

We study the BPS particle spectrum of five-dimensional superconformal field theories on ℝ⁴× S¹ 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 F, and local del Pezzo dP₃ and dP₅ geometries. Remarkably, the BPS spectrum consists of two copies of suitable 4d𝒩= 2 spectra, augmented by Kaluza-Klein towers.