Bulk locality from the celestial amplitude

In this paper, we study the implications of bulk locality on the celestial
amplitude. In the context of the four-point amplitude, the fact that the bulk
S-matrix factorizes locally in poles of Mandelstam variables is reflected in
the imaginary part of the celestial amplitude. In particular, on the real axis
in the complex plane of the boost weight, the imaginary part of the celestial
amplitude can be given as a positive expansion on the Poincar\'e partial waves,
which are nothing but the projection of flat-space spinning polynomials onto
the celestial sphere. Furthermore, we derive the celestial dispersion relation,
which relates the imaginary part to the residue of the celestial amplitude for
negative even integer boost weight. The latter is precisely the projection of
low energy EFT coefficients onto the celestial sphere. We demonstrate these
properties explicitly on the open and closed string celestial amplitudes.
Finally, we give an explicit expansion of the Poincar\'e partial waves in terms
of 2D conformal partial waves.