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

TL;DR

This work analyzes soft theorems in supergravity amplitudes to illuminate duality symmetries. It introduces an anti-symmetrised double-soft extraction to reveal the full duality group when a U(1) factor is present, applying it to tree-level amplitudes in $4\leq \mathcal{N}<8$ in four dimensions and to $\mathcal{N}=16$ in three dimensions. In 3D, the double-soft limits reproduce the nonlinearly realized $E_{8(8)}$ algebra via an on-shell superspace with $\eta^I$, and one-loop amplitudes obey the same soft theorems as tree level when analyzed with 3D generalized unitarity. The duality constraints are then used to assess possible counterterms, ruling out several dimensionally reduced four-dimensional operators (notably $D^8R^4$) as valid 3D counterterms and highlighting the stronger UV-control provided by the enlarged symmetry in 3D. Overall, the paper provides a concrete framework to connect soft theorems, duality groups, and UV consistency in maximal and non-maximal supergravity theories, with implications for higher-loop behavior and dimensional reductions.

Abstract

It is known that for 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 N<8 supergravity theories, complication arises due to the presence of a U(1) in the U(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<=N<8 supergravity in four dimensions, as well as 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^8R^4 as a valid counter term.

Figures (1)

• Figure 1: (a) The BCFW diagram where the two soft legs are attached to a three-point sub amplitude. (b) Due to the soft kinematics, the diagram factorizes into a three-point times an $n{-}1$-point amplitude in the single soft limit.