Photon Cohomology and Higher Gerbes: Topological Invariants of Photonic Sectors

We introduce Photon Cohomology, a differential cohomology theory tailored to classify photonic bundle data comprising ordinary U(1)-connections, higher gerbe potentials and multi-form couplings that arise in exotic photonic channels and engineered photonic media. Photon Cohomology is defined as the hypercohomology of a truncated Deligne-type complex (the photon complex) which encodes local connection 1-forms, gerbe 2-forms, and higher-form interaction data together with integral quantization. We construct a characteristic class in Photon Cohomology in degree 'n'  whose non-triviality detects obstruction to trivializing photonic transmission channels and correlates with quantized flux and higher-holonomy. We prove existence and uniqueness of the characteristic class up to torsion under mild geometric hypotheses and provide a Cech–de Rham hybrid construction of representatives. Explicit sample calculations on the 3-torus exhibit a non-trivial characteristic class for canonical gerbe curvatures. Finally, we sketch numerical checks via finite-element discretization of curvature invariants and discuss experimental observables in photonic crystals and metamaterials. The theory relates naturally to differential cohomology (Cheeger–Simons/Deligne), higher gerbes and higher categorical Langlands-type correspondences for electromagnetic sectors.