A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an ${\alpha }^{\mathrm{\prime }}$-weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the ${\alpha }^{\mathrm{\prime }}$ deformed colour structure observed in open string and semi-abelian $Z$-theory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra.