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 $\mathfrak{ u}(1,1|2)$ 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.
Categories: hep-th