New Fuzzy Extra Dimensions from $SU(\mathcal{N})$ Gauge Theories

We start with an $SU(\mathcal{N})$ Yang-Mills theory on a manifold $\mathcal{M}$, suitably coupled to two distinct set of scalar fields in the adjoint representation of $SU(\mathcal{N})$, which are forming a doublet and a triplet, respectively under a global $SU(2)$ symmetry. We show that a direct sum of fuzzy spheres $S_F^{2 , Int} := S_F^2(\ell) \oplus S_F^2 (\ell) \oplus S_F^2 \left ( \ell + \frac{1}{2} \right ) \oplus S_F^2 \left ( \ell - \frac{1}{2} \right )$ emerges as the vacuum solution after the spontaneous breaking of the gauge symmetry and lay the way for us to interpret the spontaneously broken model as a $U(n)$ gauge theory over $\mathcal{M}\times S_F^{2Int}$. Focusing on a $U(2)$ gauge theory we present complete parameterizations of the $SU(2)$-equivariant, scalar, spinor and vector fields characterizing the effective low energy features of this model. Next, we direct our attention to the monopole bundles $S_F^{2 , \pm} := S_F^2 (\ell) \oplus S_F^2 \left ( \ell \pm \frac{1}{2} \right )$over$S_F^2 (\ell)$withwindingnumbers$\pm 1$, which naturally come forth through certain projections of $S_F^{2Int}$, and discuss the low energy behaviour of the $U(2)$ gauge theory over ${\cal M} \times S_F^{2 , \pm}$\mathrm{.}Westudymodelswith$k$-component multiplet of the global $SU(2)$, give their vacuum solutions and obtain a class of winding number $\pm (k-1)$ monopole bundles $S_F^{2,\pm (k-1)}$ as certain projections of these vacuum solutions. We make the observation that $S_F^{2 , Int}$isindeedthebosonicpartofthe$N=2$fuzzysuperspherewith$OSP(2,2)$supersymmetryandconstructthegeneratorsofthe$osp(2,2)$ Lie superalgebra in two of its irreducible representations using the matrix content of the vacuum solution $S_F^{2Int}$. Finally, we show that our vacuum solutions are stable by demonstrating that they form mixed states with non-zero von Neumann entropy.