On symmetries of gravitational on-shell boundary action at null infinity

Abstract: We revisit the gravitational boundary action at null infinity of asymptotically flat spacetimes. We fix the corner ambiguities in the boundary action by using the constraint that (exponential of) the on-shell action leads to tree-level 5-point amplitude in eikonal approximation in a generic supertranslated vacuum. The subleading soft graviton theorem follows naturally from the on-shell action when the BMS group is extended to include superrotations. An infinite tower of Goldstone modes is proposed by `generalizing' the Geroch tensor to incorporate a set of divergence-free symmetric traceless tensors on the sphere. This generalization leads to $\text{sub}^{n}$-leading soft insertions in the tree-level $\mathcal{S}$ matrix, thus paving the way to understanding the infinite tower of tree-level symmetries within this framework.