Topological Currents in Extended Electrodynamics, Negative Effective Dispersion, and Supersymmetric Consistency Constraints

We investigate a proposed extension of four-dimensional electrodynamics in which aclosed three-form current Js ∈ Ω3(X) is constructed from higher-dimensional geometric datathrough fiber integration over a (4 + n)-dimensional bulk bundle Y π−→ X. We distinguishtwo differential-form constructions: the pushforward of even-degree Chern–Weil forms builtfrom the Chern character and the Dirac ˆA-genus (whose square equals the Todd class ofthe complexified vertical bundle, Td(Tπ ⊗ C) = ˆA(Tπ )2) over odd-dimensional real fibers(n = 2m − 1), and the use of odd Chern-character data in K−1(Y ) over even-dimensionalcomplex fibers (n = 2m). For a globally defined gauge potential A ∈ Ω1(X) with fieldstrength F = dA, the coupling A ∧ Js defines a gauge-invariant action yielding the modifiedMaxwell equation d ∗ F = Js alongside dF = 0. Because ∗F is globally defined, d ∗ F = Jsrequires Js to be globally exact ([Js]dR = 0) on X, supporting non-zero charge on boundedspatial regions while having vanishing periods on closed three-cycles; we also outline hownontrivial bundles can be coupled via F ∧ Bs when Js = dBs, and identify defect andhigher-gerbe extensions as directions requiring further construction if [Js]dR̸ = 0.We then examine how a dimension-six higher-spatial-derivative operator, defined relative toa preferred background direction uμ associated with ∗Js and acting selectively on the spatiallyprojected magnetic two-form F ⊥μν , modifies photon dispersion while avoiding additionalhigher-time-derivative modes at the classical level. Within the sub-threshold momentumregime 0 < q < qstab = 1/√β = Λn/√αs (with β = αs/Λ2n in natural units ℏ = c = 1), thetruncated quadratic model exhibits a negative dispersion curvature mass m∗(q) ≃ −1/(3βq) =−Λ2n/(3αsq) < 0 while maintaining a positive canonical quadratic Hamiltonian.Finally, we discuss formal local electric-magnetic dualization, connections with theKapustin–Witten framework, and supersymmetric operator protection. We formulate thesupersymmetry connection as an open research conjecture regarding whether microscopicN ≥ 1 BPS completions protect the spatially projected operator structure against radiativetemporal-derivative mixing, and we derive conditional fixed-energy interferometric phaseshifts and benchmark constraints in terms of the differential arm-to-arm coupling variance.