Superconformal quantum mechanics and growth of sheaf cohomology

A bstract We give a geometric interpretation for superconformal quantum mechanics defined on hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes the resolved space and their index degeneracies can also be related to Euler characteristic computed in sheaf cohomology. In special case of Hilbert scheme K points ℂ 2 , we obtain rigorous estimate exponential growth as → ∞. This serves toy model our recently proposed duality between seven dimensional black hole mechanics.