We solve the Landau problem for charged particles on odd-dimensional spheres S 2k−1 in the background of constant SO(2k − 1) gauge fields carrying the irreducible representation I 2 , I 2 , · · · , I 2 . We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner D-functions, and for odd values of I the explicit local form of the wave functions in the lowest Landau level (LLL). 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 the 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 LLL's with I ranging from 0 to I max = 2K or I max = 2K + 1 to the Hilbert spaces of the fuzzy CP 3 or that of winding number ±1 line bundles over CP 3 at level K, respectively.

We examine the 5d Yang-Mills matrix model in 0 + 1-dimensions with U (4N ) gauge symmetry and a mass deformation term. We determine the explicit SU (4) ≈ SO(6) equivariant parametrizations of the gauge field and the fluctuations about the classical four concentric fuzzy four sphere configuration and obtain the low energy reduced actions(LEAs) by tracing over the S 4 F s for the first five lowest matrix levels. The LEAs so obtained have potentials bounded from below indicating that the equivariant fluctuations about the S 4 F do not lead to any instabilities. These reduced systems exhibit chaos, which we reveal by computing their Lyapunov exponents. Using our numerical results, we explore various aspects of chaotic dynamics emerging from the LEAs. In particular, we model how the largest Lyapunov exponents change as a function of the energy. We also show that, in the Euclidean signature, the LEAs support the usual kink type soliton solutions, i.e. instantons in 1 + 0-dimensions, which may be seen as the imprints of the topological fluxes penetrating the concentric S 4 F s due to the equivariance conditions, and preventing them to shrink to zero radius. Relaxing the Gauss law constraint in the LEAs in the manner recently discussed by Maldacena and Milekhin leads to Goldstone bosons.