Extended Supersymmetries and the Dirac Operator

TL;DR

This paper explores the relationship between supersymmetry, geometry, and gauge fields in quantum systems, deriving conditions for supercharges and applying results to Kahler spaces like CP^n.

Contribution

It establishes new links between supercharges, geometric restrictions, and gauge configurations, providing a framework to analyze Dirac zero modes on curved spaces.

Findings

Derived relations between supercharges and space geometry.

Identified integrability conditions for supersymmetric systems.

Applied results specifically to Kahler spaces CP^n.

Abstract

We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges that exist and restrictions on the geometry of the underlying spaces as well as the admissible gauge field configurations. From the superalgebra with two or more real supercharges we infer the existence of integrability conditions and obtain a corresponding superpotential. This potential can be used to deform the supercharges and to determine zero modes of the Dirac operator. The general results are applied to the Kahler spaces CP^n.