We find new spontaneously generated fuzzy extra dimensions emerging from a certain deformation of $N=4$ supersymmetric Yang-Mills (SYM) theory with cubic soft supersymmetry breaking and mass deformation terms. First, we determine a particular four di
mensional fuzzy vacuum that may be expressed in terms of a direct sum of product of two fuzzy spheres, and denote it in short as $S_F^{2, Int}times S_F^{2, Int}$. The direct sum structure of the vacuum is revealed by a suitable splitting of the scalar fields in the model in a manner that generalizes our approach in cite{Seckinson}. Fluctuations around this vacuum have the structure of gauge fields over $S_F^{2, Int}times S_F^{2, Int}$, and this enables us to conjecture the spontaneous broken model as an effective $U(n)$ $(n < {cal N})$ gauge theory on the product manifold $M^4 times S_F^{2, Int} times S_F^{2, Int}$. We support this interpretation by examining the $U(4)$ theory and determining all of the $SU(2)times SU(2)$ equivariant fields in the model, characterizing its low energy degrees of freedom. Monopole sectors with winding numbers $(pm 1,0),,(0,pm1),,(pm1,pm 1)$ are accessed from $S_F^{2, Int}times S_F^{2, Int}$ after suitable projections and subsequently equivariant fields in these sectors are obtained. We indicate how Abelian Higgs type models with vortex solutions emerge after dimensionally reducing over the fuzzy monopole sectors as well. A family of fuzzy vacua is determined by giving a systematic treatment for the splitting of the scalar fields and it is made manifest that suitable projections of these vacuum solutions yield all higher winding number fuzzy monopole sectors. We observe that the vacuum configuration $S_F^{2, Int}times S_F^{2, Int}$ identifies with the bosonic part of the product of two fuzzy superspheres with $OSP(2,2)times OSP(2,2)$ supersymmetry and elaborate on this feature.
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 techniques and 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-abelian background magnetic monopoles or a combination of these thereof. In particular, for the simplest case of $mathbf{Gr}_2(mathbb{C}^4)$ we explicitly write down the $U(1)$ background gauge field as well as the single and many-particle eigenstates by introducing the Pl{u}cker coordinates and show by calculating the two-point correlation function that the Lowest Landau Level (LLL) at filling factor $ u =1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for QHE on ${mathbb C}P^N$ and generalize them to all $mathbf{Gr}_2(mathbb{C}^N)$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect 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 already known 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.
