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

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.

v2: published version, appendix on numerator identities at 5 points added, 34 pages, many feynmf diagrams

References in corpus (6)

• New Relations for Gauge-Theory Amplitudes
• Double Field Theory
• Minimal Basis for Gauge Theory Amplitudes
• The gauge algebra of double field theory and Courant brackets
• The Momentum Kernel of Gauge and Gravity Theories
• The Kinematic Algebra From the Self-Dual Sector

Cited by in corpus (31)

• Perturbative spacetimes from Yang-Mills theory
• The double copy: Bremsstrahlung and accelerating black holes
• Inelastic Black Hole Scattering from Charged Scalar Amplitudes
• Gravity Amplitudes as Generalized Double Copies
• A new gauge-invariant double copy for heavy-mass effective theory
• Double Copy from Homotopy Algebras
• BRST-Lagrangian Double Copy of Yang-Mills Theory
• On the kinematic algebra for BCJ numerators beyond the MHV sector
• Are all supergravity theories Yang-Mills squared?
• A double copy for asymptotic symmetries in the self-dual sector
• Twin Supergravities from Yang-Mills Squared
• The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order
• Off-Shell Color-Kinematics Duality for Chern-Simons
• Self-Dual Gravity
• New heavenly double copies
• Kinematic Lie Algebras From Twistor Spaces
• $D=6$, $\mathcal{N}=(2,0)$ and $\mathcal{N}=(4,0)$ theories
• Classical Spin Gravitational Compton Scattering
• Berends-Giele currents in Bern-Carrasco-Johansson gauge for $F^3$- and $F^4$-deformed Yang-Mills amplitudes
• The Gauge Structure of Double Field Theory follows from Yang-Mills Theory
• Amplitudes, Hopf algebras and the colour-kinematics duality
• Gauge $\times$ Gauge on Spheres
• Flat Self-dual Gravity
• A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator
• Kinematic Hopf algebra for amplitudes and form factors
• Double copy for Lagrangians at trilinear order
• Homotopy double copy and the Kawai-Lewellen-Tye relations for the non-abelian and tensor Navier-Stokes equations
• A homotopy BV algebra for Yang-Mills and color-kinematics
• BCJ, worldsheet quantum algebra and KZ equations
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Next-to-MHV Yang-Mills kinematic algebra