$SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background

We investigate the $SO(5)$ Landau problem in the $SO(4)$ monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The $SO(4)$ monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the $SU(2)$ monopole and anti-monopole indices, $I_+$ and $I_-$. The energy levels of the $SO(5)$ Landau problem are grouped into $\text{Min}(I_+, I_-) +1$ sectors, each of which holds Landau levels. In the $n$-sector, $N$th Landau level eigenstates constitute the $SO(5)$ irreducible representation with $(p,q)_5=(N+I_+ + I_--n, N+n)_5$ whose function form is obtained from the $SO(5)$ non-linear realization matrix. In the $n=0$ sector, the emergent quantum geometry of the lowest Landau level is identified as the fuzzy four-sphere with radius being proportional to the difference between $I_+$ and $I_-$. The Laughlin-like wavefunction is constructed by imposing the $SO(5)$ lowest Landau level projection to the many-body wavefunction made of the Slater determinant. We also analyze the relativistic version of the $SO(5)$ Landau model to demonstrate the Atiyah-Singer index theorem in the $SO(4)$ gauge field configuration.