  Starting from gravity in asymptotically flat spacetime, the BMS momentum
fluxes are constructed. These are non-local expressions of the solution space
living on the celestial Riemann surface. They transform in the coadjoint
representation of the extended BMS group and correspond to Virasoro primaries
under the action of bulk superrotations. The relation between the BMS momentum
fluxes and celestial CFT operators is then established: the supermomentum flux
is related to the supertranslation operator and the super angular momentum flux
is linked to the stress-energy tensor of the celestial CFT. The transformation
under the action of asymptotic symmetries and the OPEs of the celestial CFT
currents are deduced from the BMS flux algebra.
