  We consider the infrared factorisation of non-abelian multi-particle
scattering amplitudes, and we study the form of the universal colour operator
responsible for infrared divergences, when expressed in terms of coordinates on
the `celestial sphere' intersecting the future light-cone at asymptotic
distances. We find that colour-dipole contributions to the infrared operator,
to all orders in perturbation theory, have a remarkably simple expression in
these coordinates, with scale and coupling dependence factorised from
kinematics and colour. Generalising earlier suggestions in the abelian theory,
we then show that the infrared operator can be computed as a correlator of
vertex operators in a conformal field theory of Lie-algebra-valued free bosons
on the celestial sphere. We verify by means of the OPE that the theory
correctly predicts the all-order structure of collinear limits, and the
tree-level factorisation of soft real radiation.
