BMS supertranslations and Weinberg's soft graviton theorem

Recently it was conjectured that a certain infinite-dimensional "diagonal" subgroup of BMS supertranslations acting on past and future null infinity (${\mathscr I}^-$ and ${\mathscr I}^+$) is an exact symmetry of the quantum gravity ${\cal S}$-matrix, and an associated Ward identity was derived. In this paper we show that this supertranslation Ward identity is precisely equivalent to Weinberg's soft graviton theorem. Along the way we construct the canonical generators of supertranslations at ${\mathscr I}^\pm$, including the relevant soft graviton contributions. Boundary conditions at the past and future of ${\mathscr I}^\pm$ and a correspondingly modified Dirac bracket are required. The soft gravitons enter as boundary modes and are manifestly the Goldstone bosons of spontaneously broken supertranslation invariance.