The Kinematic Algebra From the Self-Dual Sector

TL;DR

The work reveals a kinematic Lie algebra, arising from area-preserving diffeomorphisms in the self-dual sector, that governs the kinematic numerators in Yang–Mills theory and, via BCJ squaring, determines gravity amplitudes. By analyzing self-dual Yang–Mills and gravity, the authors show that cubic diagrams with numerators built from kinematic structure constants $F^{p_1p_2p_3}$ satisfy Jacobi identities mirroring the colour algebra, providing a natural realization of colour–kinematics duality. In the full YM theory, the MHV sector can be computed from off-shell cubic diagrams using the Chalmers–Siegel formulation, with numerators $n_s,n_t,n_u$ built from off-shell spinor-like invariants $X(p_i,p_j)$, and the same kinematic Jacobi relations as in the self-dual sector. For gravity, the BCJ squaring prescription holds for MHV amplitudes, with gravity numerators given by the square of YM numerators, and the analysis points toward a cubic, gauge-theory–like formulation underlying all graviton amplitudes; this opens avenues for supersymmetric extensions and deeper connections to string-inspired constructions.

Abstract

We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.

Figures (4)

• Figure 1: The first order correction $\phi^{(1)}(k)$ consists of a single interaction between two particles sourced by $J$.
• Figure 2: The second order correction $\phi^{(2)}(k)$ consists of an interaction between two particles, creating a disturbance which propagates before scattering against a third particle.
• Figure 3: The third order correction $\Phi^{(3)}(k)$, in \ref{['Phi3']}, consists of two different terms. The first one corresponds to the Feynman diagram on the left, and the second one to the diagram on the right.
• Figure 4: Example of a diagram contributing to the 6-point MHV amplitude. The thick line represents the reference leg; the vertex attached to it is of the type $--+$, while the other three vertices are of the type $++-$.