Isospin particle on $S^{2}$ with arbitrary number of supersymmetries

TL;DR

This paper investigates the supersymmetric quantum mechanics of an isospin particle on a sphere with a spherically symmetric Yang-Mills background, revealing arbitrary supersymmetry levels, nonlinear algebra for many fermions, and spontaneous SUSY breaking for even numbers.

Contribution

It demonstrates that the number of supersymmetries on $S^{2}$ can be arbitrarily large with a specific gauge field choice, and analyzes the resulting algebra and spectra.

Findings

Supersymmetry can be arbitrarily extended on $S^{2}$.

The superalgebra becomes nonlinear for more than two fermions.

Supersymmetry is spontaneously broken when the number of supersymmetries is even.

Abstract

We study the supersymmetric quantum mechanics of an isospin particle in the background of spherically symmetric Yang-Mills gauge field. We show that on $S^{2}$ the number of supersymmetries can be made arbitrarily large for a specific choice of the spherically symmetric SU(2) gauge field. However, the symmetry algebra containing the supercharges becomes nonlinear if the number of fermions is greater than two. We present the exact energy spectra and eigenfunctions, which can be written as the product of monopole harmonics and a certain isospin state. We also find that the supersymmetry is spontaneously broken if the number of supersymmetries is even.