Narrow Results

We study the supersymmetric quantum mechanics of an isospin particle in the background of spherically symmetric Yang–Mills gauge field. We show that on S² 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.

Noncommutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac–Landau operator eigenstates is obtained based on algebraic methods. We establish $SU(2)$ “gauge” transformation between the relativistic Landau model and the Pauli–Schrödinger nonrelativistic quantum mechanics. After the $SU(2)$ transformation, the Dirac operator and the angular momentum operators are found to satisfy the $SO(3,1)$ algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac–Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres.