Conformal properties of soft-operators - 1 : Use of null-states

Soft-operators, loosely speaking, are operators which create or annihilate zero energy massless particles on the celestial sphere in Minkowski space. The Lorentz group acts on the celestial sphere by conformal transformation and the soft-operators transform as conformal primary operators of various dimension and spin. Working in space-time dimensions $D=4$ and $6$, we study some properties of the conformal representations with the (leading) soft photon and graviton as the highest weight vectors. Typically these representations contain null-vectors. We argue, from the $S$-matrix point of view, that infinite dimensional asymptotic symmetries and conformal invariance require us to set some of these null-vectors to zero. As a result, the corresponding soft-operator satisfies linear PDE on the celestial sphere. Curiously, these PDEs are equations of motion of Euclidean gauge theories on the celestial sphere with scalar gauge-invariance, i.e, the gauge parameter is a scalar field on the sphere. These are probably related to large $U(1)$ and supertranslation transformations at infinity. Now, the PDE satisfied by the soft-operator can be converted into PDE for the $S$-matrix elements with the insertion of the soft-operator. These equations can then be solved subject to appropriate boundary conditions on the celestial sphere, provided by conformal invariance. The solutions determine the soft $S$-matrix elements, for different helicities of the soft-particle, in terms of a single scalar function. This makes the Ward-identity for the asymptotic symmetry almost integrable. The result of the integration, which we are not able to perform completely, should of course be Weinberg's soft-theorem. Finally, we comment on the similarity between the roles played by null-states in the context of asymptotic symmetry and in string theory in relation to space-time gauge symmetry.