Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms

TL;DR

This paper reveals a Lie algebraic structure behind Yang-Mills theory's colour-kinematics duality, identifying it as the Drinfeld double of the Lie algebra of vector fields, and explains the duality's limitations at higher points.

Contribution

It demonstrates that the kinematic numerators satisfy a Jacobi identity linked to the Drinfeld double, providing a new algebraic understanding of colour-kinematics duality in Yang-Mills theory.

Findings

Identifies the Drinfeld double as the underlying Lie algebraic structure.

Shows the Jacobi identity for kinematic numerators is satisfied off-shell.

Explains the limitations of the duality at higher points and suggests modifications.

Abstract

Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.