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We investigate the $SO(5)$ Landau problem in the $SO(4)$ monopole gauge fieldbackground by applying the techniques of the non-linear realization of quantumfield theory. The $SO(4)$ monopole carries two topological invariants, thesecond 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, eachof which holds Landau levels. In the $n$-sector, $N$th Landau level eigenstatesconstitute 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 realizationmatrix. In the $n=0$ sector, the emergent quantum geometry of the lowest Landaulevel is identified as the fuzzy four-sphere with radius being proportional tothe difference between $I_+$ and $I_-$. The Laughlin-like wavefunction isconstructed by imposing the $SO(5)$ lowest Landau level projection to themany-body wavefunction made of the Slater determinant. We also analyze therelativistic version of the $SO(5)$ Landau model to demonstrate theAtiyah-Singer index theorem in the $SO(4)$ gauge field configuration.

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