U(1) Gauge Potentials on de Sitter Spacetime

TL;DR

By leveraging $\mathfrak{so}(1,4)$ representation theory, the paper builds Verma modules of smooth vector fields, 1-forms, and 3-forms on $dS^{4}$ and classifies their finite-dimensional irreducible submodules. It explicitly constructs highest-weight vector fields for two dominant weight families, maps vector fields to 1-forms via $\mathbf{g}_{\flat}$, and analyzes Maxwell and Proca dynamics sector by sector. A key result is that, under a Lorentz gauge and assuming only finite-dimensional sectors, there are no smooth source-free electromagnetic fields on de Sitter space; currents determine the corresponding potentials with a direct sectoral mass term that appears as an imaginary Proca mass due to $dS^{4}$ curvature. The work provides a rigorous group-theoretic framework for electromagnetism on curved spacetimes and highlights foundational questions about light propagation and metric measurements in de Sitter backgrounds.

Abstract

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.