Equivalent D=3 Supergravity Amplitudes from Double Copies of Three-Algebra and Two-Algebra Gauge Theories

TL;DR

The paper investigates how three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra Chern-Simons-matter theories or two-algebra Yang–Mills theories when both are arranged to exhibit color-kinematics duality. It formalizes a master equivalence between the two double-copy constructions and verifies it explicitly at four and six points for $\mathcal{N}=16,12,10,8$ supergravities, using a rank-deficient $\Theta$ matrix to relate color-ordered amplitudes and extract gravity results. A key result is that odd-multiplicity amplitudes vanish in 3D descendants due to $D=4$ R-symmetry constraints and dimensional reduction, implying that only helicity-conserving, even-multiplicity amplitudes survive after reduction. The findings unify the three-algebra and two-algebra double-copy pictures in $D=3$, suggest potential loop-level UV improvements from nonlocal duality-satisfying numerators, and connect BLG/ABJM-type theories to four-dimensional $\mathcal{N}=8$ supergravity via dimensional reduction.

Abstract

We show that three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra super-Chern-Simons matter theory or that of two-algebra super-Yang-Mills theory, when either theory is organized to display the color-kinematics duality. We prove that only helicity-conserving four-dimensional gravity amplitudes have nonvanishing descendants when reduced to three dimensions; implying the vanishing of odd-multiplicity S-matrix elements, in agreement with Chern-Simons matter theory. We explicitly verify the double-copy correspondence at four and six points for N=12,10,8 supergravity theories and discuss its validity for all multiplicity.