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Three Chiral Families from a Monopole Product over S2×ℂP2

  • Edward J. Shaya
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Abstract

We study the chiral zero-mode spectrum of the compact six-manifold X6=S2× ℂP2 in a ten-dimensional Kaluza-Klein setting. We review the monopole Dirac index on S2 and the Spinc index analysis, showing that ℂP2 admits a minimal unit-index chiral sector. Dolan and Nash established the Spinc index mechanism on complex projective spaces and showed that ℂP2 can supply a one-family chiral building block with Standard-Model-like quantum numbers. The step taken here is to place the ℂP2 sector on the six-dimensional product S2×ℂP2 and use an independently quantized monopole sector on S2 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 Spinc structure on ℂP2 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≥ 3. This homogeneous product isolates the family number in an elementary geometric formula. For a 10D Weyl fermion valued in a visible 16 of Spin(10)G, Lorentz branching shows these internal zero modes yield three left-handed 4D 16G multiplets. The analysis is restricted to the spectral and index-theoretic problem; moduli stabilization and phenomenological dynamics are deferred to a companion paper.

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Cite

BibTeXRIS
BibTeX
@misc{shaya2026three,
  title = {{Three Chiral Families from a Monopole Product over $S^2\times\mathbbCP^2$}},
  author = {Edward J. Shaya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02907},
  eprint = {2609.02907},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02907},
}
RIS
TY  - UNPB
TI  - Three Chiral Families from a Monopole Product over S^2×ℂP^2
AU  - Edward J. Shaya
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.02907
UR  - https://arxiv.org/abs/2609.02907
AB  - We study the chiral zero-mode spectrum of the compact six-manifold X_6=S^2× ℂP^2 in a ten-dimensional Kaluza-Klein setting. We review the monopole Dirac index on S^2 and the Spin^c index analysis, showing that ℂP^2 admits a minimal unit-index chiral sector. Dolan and Nash established the Spin^c index mechanism on complex projective spaces and showed that ℂP^2 can supply a one-family chiral building block with Standard-Model-like quantum numbers. The step taken here is to place the ℂP^2 sector on the six-dimensional product S^2×ℂP^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 Spin^c structure on ℂP^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≥ 3. This homogeneous product isolates the family number in an elementary geometric formula. For a 10D Weyl fermion valued in a visible 16 of 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.
ER  -