Amplitudes of 3D and 6D Maximal Superconformal Theories in Supertwistor Space

TL;DR

This paper constructs superconformal scattering amplitudes in 3D and 6D using supertwistor space, revealing unique solutions in 3D consistent with BLG and ABJM theories, and showing vanishing amplitudes in 6D, suggesting no simple Lagrangian exists.

Contribution

It provides a supertwistor space framework for superconformal amplitudes in 3D and 6D, identifying unique 3D solutions and constraints on 6D amplitudes, linking to known theories.

Findings

3D four-point amplitude matches BLG theory.

ABJM amplitude derived from N=8 results via supersymmetry reduction.

All tree-level amplitudes vanish in 6D, indicating no simple Lagrangian for such theories.

Abstract

We use supertwistor space to construct scattering amplitudes of maximal superconformal theories in three and six dimensions. In both cases, the constraints of superconformal invariance and rationality imply that the three-point amplitude vanishes on-shell, which constrains the four-point amplitude to have vanishing residues in all channels. In three dimensions, we find a unique solution for the four-point amplitude and demonstrate that it agrees with the component result in the BLG theory. This suggests that BLG is the unique three-dimensional theory with classical OSp(8|4) symmetry that admits a Lagrangian description. We also show that one can derive the four-point amplitude of the ABJM theory from our N=8 result by reducing the supersymmetry, which implies that the tree-level Yangian symmetry recently found in ABJM is also present in BLG. In six dimensions, we find that the consistency conditions imply that all tree-level amplitudes vanish. This leads us to conjecture that an interacting six-dimensional theory with classical OSp(8|4) symmetry does not have a Lagrangian description, local or nonlocal, unless the (2,0) tensor multiplets are supplemented by additional degrees of freedom.