Čech-Dolbeault Realizations, Green-Schwarz Anomaly Consistency, and Heterotic Embeddings of Topological Photonic Sheaves

We formulate a cohomological construction and string-theoretic embedding framework 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]. 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 closed, integral differential 9-form representative J ∈Ω9 cl,Z(Y ) whose leading vertical com- ponent J(3,6) = ω3(X) ∧νK (where νK ∈Ω6 cl,Z(K) is a normalized closed integral repre- sentative of the top cohomology class satisfying ∫︁ 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 string compactifi- cations under consistent adjoint trace conventions Tr ≡Tr248 for E8 and tr ≡trvector for Lorentz curvature. In these conventions, the irreducible gauge and gravitational anoma- lies vanish through exceptional Casimir identities, allowing the ten-dimensional anomaly 12-form to factorize as I12(F, R) = X4 ∧X8. The classical variation of the Green-Schwarz counterterm SGS ∝ ∫︁ Y B2 ∧X8 cancels the resulting factorized one-loop anomaly, providing a consistent ten-dimensional background for the proposed anomaly-inflow interpretation of Js [1]. Furthermore, evaluating the Atiyah-Singer index on the Tian-Yau quotient Calabi-Yau threefold K = ˜︁K/Z3 with Euler characteristic χ(K) = −18/3 = −6 under the standard embedding VF ∼= TK, we derive a net chiral index ind(DK) = n27 −n27 = 1 2|χ(K)| = 3, matching the net three-family generation count required by the Standard Model modulo vector-like pairs and Wilson-line symmetry breaking. Finally, we compute the leading-order visibility suppression in precision optical interferometry 1 −V ≈1 2⟨(∆ϕs)2⟩∼∥ω(η)∥2Λ−12 [1] and quantify correlated loop-level corrections to the effective weak mixing angle sin2 θeff W and the muon anomalous magnetic moment (g −2)µ. 1

@article{deceuster2026_78-ech-dolbeault-rea,
  title = {Čech-Dolbeault Realizations, Green-Schwarz Anomaly Consistency, and Heterotic Embeddings of Topological Photonic Sheaves},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.22165642},
  url = {https://peterdeceuster.uk/papers/78-ech-dolbeault-realizations-green-schwarz-anomaly-consistency-and},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/green1.pdf}
}