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Quantum Hall Effects (QHEs) on the complex Grassmann manifolds$\mathbf{Gr}_2(\mathbb{C}^N)$ are formulated. We set up the Landau problem in$\mathbf{Gr}_2(\mathbb{C}^N)$ and solve it using group theoretical techniquesand provide the energy spectrum and the eigenstates in terms of the $SU(N)$Wigner ${\cal D}$-functions for charged particles on$\mathbf{Gr}_2(\mathbb{C}^N)$ under the influence of abelian and non-abelianbackground magnetic monopoles or a combination of these thereof. In particular,for the simplest case of $\mathbf{Gr}_2(\mathbb{C}^4)$ we explicitly write downthe $U(1)$ background gauge field as well as the single and many-particleeigenstates by introducing the Pl\"{u}cker coordinates and show by calculatingthe two-point correlation function that the Lowest Landau Level (LLL) atfilling factor $\nu =1$ forms an incompressible fluid. Our results are inagreement with the previous results in the literature for QHE on ${\mathbbC}P^N$ and generalize them to all $\mathbf{Gr}_2(\mathbb{C}^N)$ in a suitablemanner. Finally, we heuristically identify a relation between the $U(1)$ Halleffect on $\mathbf{Gr}_2(\mathbb{C}^4)$ and the Hall effect on the odd sphere$S^5$, which is yet to be investigated in detail, by appealing to the alreadyknown analogous relations between the Hall effects on ${\mathbb C}P^3$ and${\mathbb C}P^7$ and those on the spheres $S^4$ and $S^8$, respectively.

Quantum Hall Effect on the Grassmannians $\mathbf{Gr}_2(\mathbb{C}^N)$