Three Chiral Families from a Monopole Product over $S^2\times\mathbb{C}P^2$

We study the chiral zero-mode spectrum of the compact six-manifold $X_6=S^2\times \mathbb{C}P^2$ in a ten-dimensional Kaluza-Klein setting. We review the monopole Dirac index on $S^2$ and the $\mathrm{Spin}^{c}$ index analysis, showing that $\mathbb{C}P^2$ admits a minimal unit-index chiral sector.