  We consider two-dimensional chiral, first-order conformal field theories
governing maps from the Riemann sphere to the projective light cone inside
Minkowski space -- the natural setting for describing conformal field theories
in two fewer dimensions. These theories have a SL(2) algebra of local bosonic
constraints which can be supplemented by additional fermionic constraints
depending on the matter content of the theory. By computing the BRST charge
associated with gauge fixing these constraints, we find anomalies which vanish
for specific target space dimensions. These critical dimensions coincide
precisely with those for which (biadjoint) cubic scalar theory, gauge theory
and gravity are classically conformally invariant. Furthermore, the BRST
cohomology of each theory contains vertex operators for the full conformal
multiplets of single field insertions in each of these space-time CFTs. We give
a prescription for the computation of three-point functions, and compare our
formalism with the scattering equations approach to on-shell amplitudes.
