Superconformal Quantum Mechanics on K\"ahler Cones

TL;DR

This paper studies supersymmetric quantum mechanics on Kähler cones, revealing superconformal symmetry and computing a superconformal index across various examples, advancing understanding of quantum systems with geometric singularities.

Contribution

It introduces a superconformal quantum mechanics framework on Kähler cones, establishing the existence of a superconformal algebra and calculating the associated index for multiple cases.

Findings

Unresolved Kähler cones exhibit $rak{u}(1,1|2)$ superconformal symmetry.

The proposed quantum mechanics has a discrete spectrum of unitary, lowest weight representations.

Superconformal index computed for diverse Kähler cone examples.

Abstract

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.