Quantum Hall effect on odd spheres

We solve the Landau problem for charged particles on odd dimensional spheres $S^{2k-1}$ in the background of constant $\mathrm{SO}(2k-1)$ gauge fields carrying the irreducible representation ($\frac{I}{2},\frac{I}{2},\dots ,\frac{I}{2}$). We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner $\mathcal{D}$-functions, and for odd values of $I$, the explicit local form of the wave functions in the lowest Landau level (LLL). The spectrum of the Dirac operator on $S^{2k-1}$ in the same gauge field background together with its degeneracies is also determined, and in particular, its number of zero modes is found. We show how the essential differential geometric structure of the Landau problem on the equatorial $S^{2k-2}$ is captured by constructing the relevant projective modules. For the Landau problem on $S^5$, we demonstrate an exact correspondence between the union of Hilbert spaces of LLLs, with $I$ ranging from 0 to $I_{\max }=2K$ or $I_{\max }=2K+1$ to the Hilbert spaces of the fuzzy ${\mathbb{C}P}^3$ or that of winding number $\pm 1$ line bundles over ${\mathbb{C}P}^3$ at level $K$, respectively.