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

## TL;DR

This paper explores generalized 4D fuzzy spheres with extra dimensions within matrix models, revealing their role in higher-spin gauge theories and potential implications for particle generations.

## Contribution

It introduces new classes of fuzzy spheres as solutions in matrix models, linking them to higher-spin theories and complex embeddings with self-intersecting extra dimensions.

## Key findings

- Fuzzy spheres are realized as solutions in Yang-Mills matrix models.
- They can be viewed as projections of coadjoint orbits of SO(6).
- Embeddings include self-intersecting extra dimensions suggesting multiple generations.

## Abstract

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 $S^4$. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions $S^4 \times \mathcal{K}$, which is expected to entail 2+1 generations.