Topological Photonic Sheaves and Bulk Gauge Continuity: A (4 + k)-Dimensional Geometric Extension of Maxwell Electrodynamics

This work contains some solutions for the category errors, degree mismatches, and dif- ferential geometry of the original U(1) toy model that was described in our earlier Maxwell work. We formulate a geometric extension of abelian gauge theory by embedding four-dimensional spacetime X into a smooth, oriented fiber bundle π : Y →X with a compact, boundaryless k-dimensional Riemannian manifold K as fiber (dimR Y = 4 + k) [1]. To maintain categori- cal consistency, we model the bulk space Y as a complex analytic manifold with even fiber dimension k ∈2N, letting EP = OY (L) ∈Coh(Y ) denote the coherent sheaf of holomor- phic sections of a holomorphic line bundle L →Y [1]. The underlying smooth line bundle supports a U(1) gauge connection locally represented by potentials A with globally defined curvature F ∈Ω2 cl(Y ) [1]. Coupling EP to a bulk Higgs sheaf H ∈Coh(Y ), we formulate the interaction via the derived extension class η ∈Extk+3 Y (π∗EP ⊗L H, OY ) [1]. Through a postulated de Rham realization map cη, the class η determines a closed differential form J ∈Ωk+3 cl (Y ) satisfying dY J = 0 [1]. Fiber integration along K induces a closed 3-form current on physical 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, where F ∈Ω2(X) is the physical electromagnetic field strength [1,4]. For any spatial 3-chain Σ3 ⊂X with boundary ∂Σ3, this induces a localized Gauss-law topological charge: Qsoul(Σ3) = ∫︂ Σ3 Js = ∫︂ ∂Σ3 ∗F. (2) We impose the explicit decoupling condition Jk=0 ≡0, guaranteeing that standard classical electrodynamics is recovered identically on uncompactified four-dimensional manifolds (k = 0) [1]. Coupling Js to the gauge potential via the effective action Seff[A] = −1 4 ∫︁ X F ∧∗F + ∫︁ X A∧Js preserves 4D gauge invariance under A ↦→A+dλ as a direct consequence of current closure dXJs = 0 [1]. Under compactification on an isotropic torus of characteristic scale Λ, we formulate a phenomenological scaling hypothesis for multi-path quantum interference visibility:

@article{deceuster2026_83-topological-photo,
  title = {Topological Photonic Sheaves and Bulk Gauge Continuity: A (4 + k)-Dimensional Geometric Extension of Maxwell Electrodynamics},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.22165691},
  url = {https://peterdeceuster.uk/papers/83-topological-photonic-sheaves-and-bulk-gauge-continuity-a},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/maxwellfix.pdf}
}