Topological Electroweak Unification: Higher-Dimensional Photonic Sheaves, Chiral Index Invariants, and the Extended Standard Model

Rejecting the Standard Model (SM) is easily done as we have showed -to great personal satisfaction -in our previous works; however, a constructive theoretical paradigm must fun- damentally improve, complete, and geometrically ground it [1]. In this work, we formulate an extended geometric framework for the Standard Model by asserting that physical four- dimensional spacetime X represents the base manifold of a smooth, oriented fiber bundle π : Y →X with a compact, boundaryless k-dimensional Riemannian spin manifold K as fiber (dimR Y = 4+k, k ∈2N) [1]. We model the electroweak gauge sector as a holomorphic principal GC EW -bundle (GC EW = SL(2, C)×C∗) with associated adjoint bundle ad(PEW ) and coherent adjoint sheaf EEW = OY (ad(PEW )) ∈Coh(Y ) [1]. Following spontaneous symme- try breaking ⟨H⟩= vB ̸= 0, the unbroken photonic reality is formalized as the holomorphic line subbundle Lad γ ⊂ad(PEW ) stabilized by the electric charge generator Q = T 3 + YW , with associated coherent sheaf Eγ = OY (Lad γ ) [1]. To ensure dimensional consistency on the smooth (4 + k)-manifold Y , the bulk photon- Higgs topological coupling is formulated in differential cohomology as an integral charac- ter ˆ︁η ∈ˆ︁Hk+3(Y ) with closed curvature representative J = curv(ˆ︁η) ∈Ωk+3 cl,Z (Y ) satisfying dY J = 0, constructed from the invariant characteristic forms of the gauge and Higgs con- nections [1,2]. Fiber integration along K induces a closed conserved 3-form soul current on spacetime: Js := π∗J = ∫︂ K J ∈Ω3(X), dXJs = π∗(dY J ) = 0. (1) The degree-consistent extended Maxwell equations are formulated as dF = 0 and d ∗F = Js +eJem, yielding a localized Gauss-law topological charge Qsoul(Σ3) = ∫︁ Σ3 Js = ∫︁ ∂Σ3 ∗F − e ∫︁ Σ3 Jem for any spatial 3-chain Σ3 ⊂X with boundary ∂Σ3 [1, 4]. Under the boundary condition Jk=0 ≡0, the framework identically recovers standard four-dimensional electro- dynamics [1]. Beyond abelian electrodynamics, we demonstrate that: (1) on a compact spin mani- fold K, the internal Dirac operator DK yields a net chiral generation index ind(DK) = ∫︁ K ˆ︁A(TK) ∧ch(VF ) = 3 when c1(VF ) = 0 and 1 2 ∫︁ K c3(VF ) = 3, providing a topological realization of a net chiral index of three; (2) coupling Js into the electroweak neutral sec- tor induces scale-dependent corrections to the effective weak mixing angle sin2 θeff W (q2); (3) compactification scale boundary conditions Λ serve as an effective field theory threshold reg- ularizing the Higgs mass against unconstrained quadratic ultraviolet divergences; and (4) topological holonomies across spatial 3-chains predict a leading-order visibility suppression in precision quantum interferometry 1 −V ≈1 2⟨(∆ϕs)2⟩∼∥ω(η)∥2Λ−2k [1]. We establish the differential-cohomological lifting conditions in Cheeger-Simons cohomology [2], formu- late bulk Green-Schwarz anomaly-consistency requirements, and outline experimental tests spanning precision optical interferometry, (g −2)µ, and atomic parity violation. 1

@article{deceuster2026_89-topological-elect,
  title = {Topological Electroweak Unification: Higher-Dimensional Photonic Sheaves, Chiral Index Invariants, and the Extended Standard Model},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.22165671},
  url = {https://peterdeceuster.uk/papers/89-topological-electroweak-unification-higher-dimensional-photonic-sheaves-chiral},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/stmodelnew.pdf}
}