A Geometric Langlands Correspondence for Photonic Gauge Sectors

We propose a photonic adaptation of the Geometric Langlands program: a conjectural equivalence between categories associated to moduli of G-bundles with photonic boundary/coupling data (the “automorphic” side) and a spectral category built from Langlands-dual local systems augmented by photonic insertions (the “spectral” side). We define a moduli stack parameterizing principal G-bundles on a compact Riemann surface (or compactification of a photonic medium) together with prescribed photonic defect data and coupling forms; we define a photonic Hitchin/Local system space encoding dual-group local systems with gerbe-like photonic fluxes. Our main conjecture states a derived equivalence between two specific mathematical categories, compatible with Hecke operators and photonic Hecke modifications. We work out the rank-one case (abelian Geometric Class Field Theory with photonic defects) explicitly and sketch how Fourier–Mukai transforms, twisted D-module techniques and spectral curve methods adapt to the photonic setting. We conclude with proposed tests (spectral measurements in photonic crystals, discrete lattice checks) and relations to stacky-sheaf viewpoints on photons and cohomological dualities.