Covariant 4-dimensional fuzzy spheres, matrix models and higher spin

We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as $SO\left(5\right)$-equivariant projections of quantized coadjoint orbits of $SO\left(6\right)$. We show that they arise as solutions in Yang–Mills matrix models, which naturally leads to higher-spin gauge theories on S ⁴. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions $S^4\times K$, which is expected to entail 2  +  1 generations. (paper)