Quiver Symmetries and Wall-Crossing Invariance

We study the BPS particle spectrum of five-dimensional superconformal field theories on R-4 x 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 Hirze-bruch F-0, and local del Pezzo d P-3 and d P-5 geometries. Remarkably, the BPS spectrum consists of two copies of suitable 4d N = 2 spectra, augmented by Kaluza-Klein towers.