Superconformal Quantum Mechanics and Growth of Sheaf Cohomology

TL;DR

This paper provides a geometric framework for superconformal quantum mechanics on hyper-Kahler cones, linking BPS states to twisted cohomology and analyzing their growth, with implications for black hole dualities.

Contribution

It introduces a geometric interpretation of superconformal quantum mechanics on hyper-Kahler cones with equivariant resolutions, connecting BPS states to twisted cohomology and estimating their growth.

Findings

BPS states correspond to twisted Dolbeault cohomology classes.

Index degeneracies relate to equivariant sheaf cohomology Euler characteristic.

Exponential growth of BPS state degeneracies is rigorously estimated for Hilbert schemes.

Abstract

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kahler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on C2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K goes to infinity. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.