Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
Chih-Hao Fu, Kirill Krasnov
TL;DR
This work identifies a Lie-algebraic origin for color–kinematics duality in Yang–Mills theory by revealing the Drinfeld double of the Lie algebra of vector fields as the underlying structure. The YM cubic vertex defines a YM bracket on vector fields, which alone fails Jacobi, but the quartic vertex cancels this failure at four points, a mechanism interpretable as the Jacobi identity of a twisted Drinfeld double. A symmetric-tensor twist (and gauge-fixing) within the double accounts for the observed off-shell Jacobi-like identities and provides a framework to organize higher-point numerators, with partial success at five points requiring additional higher-point interactions to restore full duality. The analysis connects off-shell YM dynamics to Drinfeld double twists and suggests concrete routes to modify Feynman rules (e.g., a five-point vertex or dimension-specific terms) to extend color–kinematics duality beyond four points, with implications for gravity via the double-copy construction.
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.