Scalar and spinor field actions on fuzzy S 4: fuzzy $\mathbb{C}{{\text{P}}^3}$ as a $S_F^2$ bundle over $S_F^4$

Abstract

We present a manifestly Spin(5) invariant construction of squashed fuzzy $CP^3$ as a fuzzy $S^2$ bundle over fuzzy $S^4$. We develop the necessary projectors and exhibit the squashing in terms of the radii of the $S^2$ and $S^4$. Our analysis allows us give both scalar and spinor fuzzy action functionals whose low lying modes are truncated versions of those of a commutative $S^4$.