The gravitational S-matrix from the path integral: asymptotic symmetries and soft theorems

We extend a previously developed formulation of the S-matrix, based on a path integral with asymptotic boundary conditions, to include gravity. The path integral defines a Carrollian boundary partition function whose invariance under asymptotic symmetries implies Ward identities obeyed by the associated boundary correlators, which are simply related to standard S-matrix elements. We develop this in the context of extended BMS transformations at tree level. Modulo well-known subtleties associated with poles in the superrotations and corner terms, this leads to an efficient derivation of the leading and subleading soft graviton theorems from BMS symmetry. Our general arguments are verified by explicit diagrammatic computation of specific terms in the partition function, which are shown to satisfy the Ward identities. We also show how, in our context, the subleading soft theorem is fixed by Poincar\'e Ward identities together with the leading soft theorem.