5D BPS quivers and KK towers

We explore BPS quivers for D = 5 theories, compactified on a circle and geometrically engineered over local Calabi-Yau 3-folds, for which many of known machineries leading to (refined) indices fail due to the fine-tuning of the superpotential. For Abelian quivers, the counting reduces to a geometric one, but the technically challenging L-2 cohomology proved to be essential for sensible BPS spectra. We offer a mathematical theorem to remedy the difficulty, but for non-Abelian quivers, the cohomology approach itself fails because the relevant wavefunctions are inherently gauge-theoretical. For the Cartan part of gauge multiplets, which suffers no wall-crossing, we resort to the D0 picture and reconstruct entire KK towers. We also perform numerical checks using a multi-center Coulombic routine, with a simple hypothesis on the quiver invariants, and extend this to electric BPS states in the weak coupling chamber. We close with a comment on known Donaldson-Thomas invariants and on how L-2 index might be read off from these.

Automated summary

This study focuses on BPS quivers for D = 5 theories, compactified on a circle and engineered over local Calabi-Yau 3-folds. It discusses the challenges faced in known machineries and proposes mathematical theorems to address the difficulties. The study finds that for Abelian quivers, the counting can be reduced to a geometric one, while for non-Abelian quivers, the cohomology approach fails.