BPS invariants count stable objects in supersymmetric theories — states that saturate a mass bound and are protected against quantum corrections. Computing them requires wall-crossing formulas that track how the invariants change as parameters vary. The Kontsevich-Soibelman wall-crossing formula provides the general framework, but explicit calculations require knowing the quiver — the directed graph encoding the interactions between BPS states.
Aspman, Cirafici, Del Zotto, and Longhi derive a covering formula that expresses rational BPS invariants as sums over covering quivers. An orbifold of the original theory produces a new theory whose BPS spectrum is related to the original by a Galois covering: each BPS state in the orbifold theory corresponds to multiple states in the cover, labeled by the elements of the covering group.
The formula unifies two operations that were previously treated separately: orbifolding (dividing by a discrete symmetry) and wall-crossing (tracking invariants across parameter boundaries). In the covering framework, orbifolding IS a specific wall-crossing transformation — the one associated to the Galois group of the cover. The wall-crossing algebra and the orbifold algebra are the same algebra, acting on different generators.
The practical consequence: computing BPS invariants for orbifold theories reduces to computing them for the covering theory and then applying the covering formula. This is a simplification when the covering theory is better understood — which it often is, because the covering theory has more symmetry.
Two algebraic structures for BPS invariants — one from geometry (orbifolds), one from physics (wall-crossing) — revealed as the same structure in different notation.