$SO(4)$ Landau Models and Matrix Geometry

TL;DR

This paper analyzes $SO(4)$ Landau models on $S^3$ with $SU(2)$ monopoles, develops matrix geometry representations, and demonstrates fuzzy three-sphere structures emerging in Landau levels, including relativistic and supersymmetric variants.

Contribution

It introduces explicit coordinate representations, constructs $SO(4)$ invariant operators, and derives fuzzy three-sphere geometries from Landau models, advancing understanding of matrix geometries in these systems.

Findings

Fuzzy three-sphere geometry appears in Landau levels.

Dirac-Landau operator contains two fuzzy three-spheres per level.

Mass terms induce interactions between fuzzy spheres.

Abstract

We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the $SU(2)$ monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete coordinate representation of $SO(4)$ operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the $SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we construct an $SO(4)$ invariant Weyl-Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac-Landau and supersymmetric Landau models, are investigated too. With the developed $SO(4)$ technologies, we derive the three-dimensional matrix geometry in the Landau models. By applying the level projection method to the Landau models, we identify the matrix elements of the $S^3$ coordinates as the fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy three-sphere geometry emerges in each of the Landau levels and only in the degenerate lowest energy sub-bands. We also point out that Dirac-Landau operator accommodates two fuzzy three-spheres in each Landau level and the mass term induces interaction between them.