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_{\rm fam} = {\rm index} D_{S^2 \times \mathbb{C}P^2}=\bigl({\rm index} D_{S^2,m}\bigr) \,\bigl({\rm index} D_{\mathbb{C}P^2,n}\bigr)=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 $n\geq 3$. This homogeneous product isolates the family number in an elementary geometric formula. For a 10D Weyl fermion valued in a visible $\mathbf{16}$ of $\mathrm{Spin}(10)_G$, Lorentz branching shows these internal zero modes yield three left-handed 4D $\mathbf{16}_G$ multiplets. The analysis is restricted to the spectral and index-theoretic problem; moduli stabilization and phenomenological dynamics are deferred to a companion paper.