  We discuss the properties of recently constructed "single-valued" celestial
four-gluon amplitudes. We show that the amplitude factorizes into the "current"
part and the "scalar" part. The current factor is given by the group-dependent
part of the Wess-Zumino-Witten correlator of four holomorphic currents with a
non-vanishing level of Ka\v{c}-Moody algebra. The scalar factor can be
expressed in terms of a complex integral of the Koba-Nielsen form, similar to
the integrals describing four-point correlators in Coulomb gas models and, more
generally, in the infinite central charge limit of Liouville theory. The scalar
part can be also obtained by a dimensional reduction of a single D=4 conformal
block and the shadow block from Minkowski space to the celestial sphere.
