Explicit Calabi-Yau Cohomology, Chiral Spectra, and SO(10) ⊂ E8 × E8 Heterotic Embeddings of Topological Photonic Sheaves

We formulate a Grand Unified embedding framework and cohomological representation for higher-dimensional photonic continuity over a ten-dimensional spacetime product man- ifold Y = X × K, where X is four-dimensional physical spacetime and K is a compact Calabi-Yau threefold (k = 6, dimR Y = 10) [1]. The standard embedding of the spin con- nection in an SU(3) factor of the visible E8 breaks the gauge symmetry to E6; subsequent symmetry breaking to SO(10) and the Standard Model group SU(3)C × SU(2)L × U(1)Y is modeled via discrete Wilson lines and bulk Higgsing [1]. The unbroken electromagnetic direction is identified with an adjoint line subbundle Lad γ ⊂ad(PE8) stabilized by the electric charge generator Q = T 3 + YW [1]. Addressing the vanishing of harmonic 1-forms on strict Calabi-Yau manifolds (b1(K) = 0), we utilize the canonical horizontal-vertical graded decomposition Ω9(Y ) = ⨁︁ p+q=9 Ωp(X)⊗ Ωq(K) [1]. We formulate a postulated closed, integral differential 9-form representative J ∈Ω9 cl,Z(Y ) whose leading vertical component J(3,6) = ω3(X) ∧νK (where νK ∈Ω6 cl,Z(K) is a normalized closed integral representative of the fundamental cohomology class of K with ∫︁ K νK = 1, and ω3(X) ∈Ω3 cl,Z(X) is a closed spacetime 3-form) guarantees an exact, non- trivial fiber pushforward soul current Js = π∗J = ∫︁ K J(3,6) = ω3(X) ∈Ω3(X) satisfying dXJs = 0 [1]. We examine the embedding of this structure into E8 × E8 heterotic compactifications under consistent adjoint trace conventions (Tr ≡Tr248 for E8 and tr ≡trvector for Lorentz curvature), where the ten-dimensional anomaly 12-form factorizes as I12(F, R) = X4 ∧X8. The classical variation of the Green-Schwarz counterterm SGS ∝ ∫︁ Y B2 ∧X8 cancels the factorized one-loop anomaly, providing a consistent ten-dimensional background for the proposed anomaly-inflow interpretation of Js [1]. Furthermore, on the Tian-Yau complete intersection Calabi-Yau threefold ˜︁K ⊂P3 × P3 of bidegrees (3, 0), (0, 3), (1, 1) with Euler characteristic χ( ˜︁K) = −18 and freely acting sym- metry G ∼= Z3, the smooth quotient K = ˜︁K/Z3 has χ(K) = −6. Under the standard embedding VF ∼= TK, the Atiyah-Singer index theorem yields a net chiral generation index ind(DK) = n27 −n27 = 1 2|χ(K)| = 3, supplying the topological family count of the Standard Model modulo non-chiral vector-like pairs. Finally, we evaluate the phenomenological signa- tures across precision optical interferometry (1−V ≈1 2⟨(∆ϕs)2⟩∼∥ω(η)∥2Λ−12), loop-level shifts in the effective weak mixing angle sin2 θeff W (q2), and the muon anomalous magnetic moment (g −2)µ [1]. 1

@article{deceuster2026_70-explicit-calabi-y,
  title = {Explicit Calabi-Yau Cohomology, Chiral Spectra, and SO(10) ⊂ E8 × E8 Heterotic Embeddings of Topological Photonic Sheaves},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.22165615},
  url = {https://peterdeceuster.uk/papers/70-explicit-calabi-yau-cohomology-chiral-spectra-and-so10},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/cohom.pdf}
}