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

TL;DR

This paper develops a three-algebra framework for SU(N)×SU(M) bi-fundamental theories (ABJM/BLG) to organize color-structure and partial amplitudes, deriving Kleiss-Kuijf–like relations from the fundamental identity. It then analyzes color-kinematics (BCJ) duality in BLG and ABJM theories, showing robust duality and gravity double-copy in BLG (and D≤3) but limited/absent duality in ABJM beyond six points, with dimension playing a crucial role. In D=2, the double copy yields integrable two-dimensional N=16 supergravity amplitudes and reveals Yang-Baxter structures and vanishing higher-point amplitudes away from factorization channels. The work also uncovers bonus relations from large-z behavior and a surprising, potentially bijective, relationship between ABJM and BLG partial amplitudes up to eight points, highlighting rich three-algebra–driven structure in gauge and gravity amplitudes.

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.

Figures (2)

• Figure 1: A diagrammatic representation of the fundamental identity in eq. (\ref{['fundamental']}). The color factors of the diagrams are related through $c_{A}=-c_{B}+c_{C}+c_{D}$.
• Figure 2: A graphical derivation of the identity \ref{['ABJM-KK-6pt']}. The two end points of each chain is identified to form a closed path. Applying the basic four-term identity to the second line gives the third line. The fourth line, $(a)+(b)+(c)$, is the same as the third, except that we turned the arrows to prepare for closing the path in the opposite direction. To obtain the fifth line, we apply \ref{['ABJM-shift2']} to (a) and \ref{['ABJM-shift1']} to (b). We leave (c) as it is, but add and subtract the same term next to it. Now, in addition to two closed paths, there are a total of nine diagrams with open arrows. Using the basic four-term identity, we group them into three closed paths. We colored the diagrams to show which terms are combined. The resulting five distinct closed paths on the right-hand-side and the original term on the left-hand-side together give the desired identity \ref{['ABJM-KK-6pt']}.