Superconformal Quantum Mechanics on K\"ahler Cones

We consider supersymmetric quantum mechanics on a K"{a}hler cone, regulated via a suitable resolution of the conical singularity. The unresolved space has a $\mathfrak{u}(1,1\mathrm{\mid }2)$ superconformal symmetry and we propose the existence of an associated quantum mechanical theory with a discrete spectrum consisting of unitary, lowest weight representations of this algebra. We define a corresponding superconformal index and compute it for a wide range of examples.