Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory

TL;DR

This work develops a three-dimensional analog of the BCFW recursion for ABJM theory by introducing a nonlinear $SO(2,\mathbb{C})$ momentum shift, and proves its validity via vanishing large-$z$ behavior using background-field methods. It combines a Grassmannian integral formulation on the orthogonal Grassmannian OG$(k,2k)$ with explicit 4-, 6-, and 8-point verifications to reproduce known results and generate new amplitudes. The authors prove that all tree-level ABJM amplitudes possess dual superconformal symmetry, using an inductive argument that the recursion preserves dual inversion, and extend this symmetry to the cut-constructible parts of loop amplitudes via generalized unitarity. The results reveal a deep connection between the ABJM Grassmannian formula and on-shell recursion, and suggest a broader amplitude/Wilson-loop duality picture in $AdS_4/CFT_3$ contexts with potential extensions to other 3d theories. Overall, the paper provides a powerful on-shell framework for ABJM amplitudes, unifying recursion, Grassmannian methods, and dual conformal symmetry at both tree and loop levels.

Abstract

We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams or the Grassmannian integral formula. As an application of the recursion relation, we prove that all tree-level amplitudes of the ABJM theory have dual superconformal symmetry. Using generalized unitarity methods, we extend this symmetry to the cut-constructible parts of the loop amplitudes.

Figures (11)

• Figure 1: Diagrammatic representation of a background field computation.
• Figure 2: Gluon propagator connecting to scalar vertex cancels against contact vertex in the leading order.
• Figure 3: Gluon propagator connecting to fermion vertex.
• Figure 7: Recursion relation applied to the 6-point amplitude.
• Figure 8: Recursion relation applied to the 8-point amplitude.