From U(1) to E8: soft theorems in supergravity amplitudes

Abstract

It is known that for $$ \mathcal{N}=8 $$ supergravity, the double-soft-scalar limit of a n-point amplitude is given by a sum of local SU(8) rotations acting on a (n−2)-point amplitude. For $$ \mathcal{N}<8 $$ supergravity theories, complication arises due to the presence of a U(1) in the U( $$ \mathcal{N} $$ ) isotropy group, which introduces a soft-graviton singularity that obscures the action of the duality symmetry. In this paper, we introduce an anti-symmetrised extraction procedure that exposes the full duality group. We illustrate this procedure for tree-level amplitudes in 4 ≤ $$ \mathcal{N}<8 $$ supergravity in four dimensions, as well as $$ \mathcal{N}=16 $$ supergravity in three dimensions. In three dimensions, as all bosonic degrees of freedom transform under the E8 duality group, supersymmetry ensures that the amplitude vanishes in the single-soft limit of all particle species, in contrast to its higher dimensional siblings. Using recursive formulas and generalized unitarity cuts in three dimensions, we demonstrate the action of the duality group for any tree-level and one-loop amplitudes. Finally we discuss the implications of the duality symmetry on possible counter terms for this theory. As a preliminary application, we show that the vanishing of single-soft limits of arbitrary component fields in three-dimensional supergravity rules out the direct dimensional reduction of D 8 R 4 as a valid counter term.