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

Yu-tin Huang, Arthur E. Lipstein

TL;DR

The paper develops a supertwistor-space framework to constrain and compute tree-level amplitudes in 3D and 6D maximal superconformal theories. By enforcing superconformal invariance and rationality, it shows the 3D case yields a unique four-point amplitude matching BLG, suggesting BLG is the sole Lagrangian theory with classical OSp(8|4) symmetry in three dimensions; ABJM amplitudes are obtainable by truncation of the N=8 results and share related Yangian symmetry. In contrast, the 6D analysis finds that all tree-level amplitudes must vanish if only (2,0) tensor multiplets are present, leading to the conjecture that such a theory cannot be described by a local or nonlocal Lagrangian without additional degrees of freedom. Collectively, the work demonstrates the power of on-shell, supertwistor methods in revealing the underlying algebraic and dynamical structures of higher-dimensional superconformal theories and clarifies the landscape of possible Lagrangian descriptions for M2- and M5-brane related 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.

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

TL;DR

The paper develops a supertwistor-space framework to constrain and compute tree-level amplitudes in 3D and 6D maximal superconformal theories. By enforcing superconformal invariance and rationality, it shows the 3D case yields a unique four-point amplitude matching BLG, suggesting BLG is the sole Lagrangian theory with classical OSp(8|4) symmetry in three dimensions; ABJM amplitudes are obtainable by truncation of the N=8 results and share related Yangian symmetry. In contrast, the 6D analysis finds that all tree-level amplitudes must vanish if only (2,0) tensor multiplets are present, leading to the conjecture that such a theory cannot be described by a local or nonlocal Lagrangian without additional degrees of freedom. Collectively, the work demonstrates the power of on-shell, supertwistor methods in revealing the underlying algebraic and dynamical structures of higher-dimensional superconformal theories and clarifies the landscape of possible Lagrangian descriptions for M2- and M5-brane related 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.

Paper Structure

This paper contains 21 sections, 127 equations, 7 figures.

Table of Contents

  1. Introduction
  2. 3D Supertwistors and Superconformal Generators
  3. 3D $\mathcal{N}=8$ On-shell Superspace
  4. S-matrix for 3D $\mathcal{N}=8$ Superconformal Theories
  5. 3-point S-matrix
  6. 4-point S-matrix
  7. Consistency Conditions on the 4-point Amplitude
  8. Comparison to BLG
  9. ABJM Four-point Amplitude from BLG
  10. 6D Amplitudes in Supertwistor Space
  11. 6D Superconformal Group
  12. Polarization Tensor
  13. 6D $\mathcal{N}=(2,0)$ On-shell Superspace
  14. S-matrix for the 6D $\mathcal{N}=(2,0)$ Theory
  15. 3-point S-matrix
  16. ...and 6 more sections

Figures (7)

  • Figure 1: 3-pt vertex for SO(4) BLG theory
  • Figure 2: Tree-level 4-pt scalar amplitude for SO(4) BLG theory
  • Figure 3: Four-point amplitude in the 6d maximal theory. In the limit $s_{12}\rightarrow 0$, we are probing the s-channel residue.
  • Figure 4: Color-ordered propagators. The gauge field $A_{\mu}$ is represented by a wavy line and the gauge field $\hat{A}_{\mu}$ is represented by a curly line.
  • Figure 5: Color-ordered 3-point vertices. The first diagram corresponds to the first two terms in Eq (B.1) and the second diagram corresponds to the third and fourth terms in Eq (B.1).
  • ...and 2 more figures