On Three-Algebra and Bi-Fundamental Matter Amplitudes and Integrability of Supergravity

TL;DR

This paper investigates the structure of tree-level amplitudes in bi-fundamental matter theories using Lie three-algebras, revealing dimension-dependent relations and connections to supergravity integrability.

Contribution

It derives Kleiss-Kuijf-like relations and explores color-kinematics duality in bi-fundamental theories, highlighting differences across dimensions and linking to supergravity integrability.

Findings

Color-kinematics duality exists in 2D but not in higher dimensions.

Four- and six-point amplitudes are related between theories, explaining previous results.

In 2D, amplitudes satisfy Yang-Baxter equations and vanish as expected from integrability.

Abstract

We explore tree-level amplitude relations for SU(N)xSU(M) bi-fundamental matter theories. Embedding the group-theory structure in a Lie three-algebra, we derive Kleiss-Kuijf-like relations for bi-fundamental matter theories in general dimension. We investigate the three-algebra color-kinematics duality for these theories. Unlike the Yang-Mills two-algebra case, the three-algebra Bern-Carrasco-Johansson relations depend on the spacetime dimension and on the detailed symmetry properties of the structure constants. We find the presence of such relations in three and two dimensions, and absence in D>3. Surprisingly, beyond six point, such relations are absent in the Aharony-Bergman-Jafferis-Maldacena theory for general gauge group, while the Bagger-Lambert-Gustavsson theory, and its supersymmetry truncations, obey the color-kinematics duality like clockwork. At four and six points the relevant partial amplitudes of the two theories are bijectively related, explaining previous results in the literature. In D=2 the color-kinematics duality gives results consistent with integrability of two-dimensional $\mathcal{N}=16$ supergravity: The four-point amplitude satisfies a Yang-Baxter equation; the six- and eight-point amplitudes vanish for certain kinematics away from factorization channels, as expected from integrability.