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(5)-equivariant projections of quantized coadjoint orbits of SO(6). We show that they arise as solutions in Yang-Mills matrix models, which naturally leads to higher-spin gauge theories on S4. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions S ⁴ ×K, which is expected to entail 2 + 1 generations.