Abstract
We develop an in-depth analysis of the SO(4) Landau models on S3 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 S3, 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 S3 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.
Highlights
The Landau models are physical models that manifest the non-commutative geometry in a most obvious way
The gauge fixing enabled us to elaborate the previous works about the SO(4) Landau models and bring the new observations, such as the properties of the SO(4) monopole harmonics and the SO(4) symmetry of the relativistic operators
We took into account the spin connection of three-sphere to construct the relativistic Landau operators
Summary
The Landau models are physical models that manifest the non-commutative geometry in a most obvious way It is well known [1, 2] that the fuzzy two-sphere geometry [3, 4, 5] is realized in the SO(3) Landau model [6, 7] that provides a set-up of the 2D quantum Hall effect [8]. Through a full investigation of the SO(4) Landau model, we learn properties specific to the odd dimensional Landau model and associated non-commutative geometry whose analyses are technically difficult in higher dimensions. We demonstrate that the obtained matrix geometry is identical to that of the fuzzy three-sphere This is the first derivation of the odd dimensional matrix geometry in the context of the Landau model.
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