U(1) Gauge Potentials on de Sitter Spacetime

The smooth 1-form Verma module of $\mathfrak{so}(1,4)$ is acquired, which can be regarded as the U(1) gauge potential on de Sitter spacetime. It is shown that electromagnetic fields could not be source free on de Sitter background.

💡 Research Summary

The paper investigates the structure of U(1) gauge potentials on four‑dimensional de Sitter (dS₄) spacetime by exploiting the representation theory of the Lie algebra so(1,4). Starting from an explicit embedding of dS₄ in a five‑dimensional Minkowski space, the authors introduce global coordinates (χ, ζ, θ, φ) and write down the Killing vectors that generate the so(1,4) symmetry. They then seek vector fields v_λ that serve as highest‑weight vectors of an irreducible so(1,4) module, imposing the standard Cartan‑Chevalley conditions: eigenvalue equations with respect to the Cartan generators h_{α₁}, h_{α₂} and annihilation by the raising operators e_{α}. Solving these constraints yields two families of solutions distinguished by the second weight N₂: the trivial case N₂=0 and the non‑trivial case N₂=±2. In the N₂=0 sector the solution reduces to a scalar function ϕ_{Nλ₁}=(e^{-iθ}coshχ cosζ)^N, and the associated vector field is simply the gradient of this scalar, v_{Nλ₁}=l dϕ_{Nλ₁}. In the N₂=2 sector a more intricate vector field appears, built from the same scalar multiplied by a fixed Lie‑algebra element e_{α₁+2α₂}.

Having identified the highest‑weight vectors, the authors construct the full Verma modules for vector fields (X(dS₄)λ), 1‑forms (Ω¹(dS₄)λ), and 3‑forms (Ω³(dS₄)λ) by repeatedly applying the lowering operators L{f{α₁}+α₂}, L{f_{α₁+2α₂}}, etc. The musical isomorphism g♭ maps vector fields to 1‑forms, showing that the 1‑form modules are isomorphic to the vector‑field modules. Crucially, every 1‑form in the λ=Nλ₁ sector is exact: Ω¹(dS₄)_{Nλ₁}=d