A No-go Theorem for a Gauge Vector as a Space-time Goldstone

Scalars and fermions can arise as Goldstone modes of non-linearly realised extensions of the Poincare group (with important implications for the soft limits of such theories): the Dirac-Born-Infeld scalar realises a higher-dimensional Poincare symmetry, while the Volkov-Akulov fermion corresponds to super-Poincare. In this paper we classify extensions of the Poincare group which give rise to a vector Goldstone mode instead. Our main result is that there are no healthy interacting $U(1)$ gauge theories that non-linearly realise space-time symmetries beyond gauge transformations. This implies that the special soft limits of e.g. the Born-Infeld vector cannot be explained by space-time symmetries.