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

Abstract

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. Dolan and Nash established the ${\mathrm{Spin}}^c$ index mechanism on complex projective spaces and showed that $\mathbb{C}{P}^2$ can supply a one-family chiral building block with Standard-Model-like quantum numbers. The step taken here is to place the $\mathbb{C}{P}^2$ sector on the six-dimensional product $S^2\times \mathbb{C}{P}^2$ and use an independently quantized monopole sector on $S^2$ to control the multiplicity. Unlike Calabi--Yau model-building constructions, where the family number is tied to vector-bundle and quotient data, this homogeneous product gives the three-family count through the factorized Dirac-index formula: $N_{\mathrm{fam}}=\mathrm{index}{D}_{S^2\times \mathbb{C}{P}^2}=\left(\mathrm{index}{D}_{S^2,m}\right)\left(\mathrm{index}{D}_{\mathbb{C}{P}^2,n}\right)=m\frac{n^2-1}{8}$. The canonical complex ${\mathrm{Spin}}^c$ structure on $\mathbb{C}{P}^2$ has $n=3$ and unit index. We show that this index is saturated by exactly one chiral zero mode and no opposite-chirality vectorlike partner. Thus, two flux sectors produce exactly three chiral internal zero modes: $(m,n)=(3,3)$ and $(m,n)=(1,5)$. Saturation is proved for all odd $n\ge 3$. This homogeneous product isolates the family number in an elementary geometric formula. For a 10D Weyl fermion valued in a visible $16$ of $\mathrm{Spin}(10{)}_G$, Lorentz branching shows these internal zero modes yield three left-handed 4D ${16}_G$ multiplets. The analysis is restricted to the spectral and index-theoretic problem; moduli stabilization and phenomenological dynamics are deferred to a companion paper.

10 pages,0 figures