One of the main entries in the AdS/CFT dictionary is a relation between the bulk on-shell partition function with specified boundary conditions and the generating function of correlation functions of primary operators in the boundary CFT. In this talk, I will show how to construct a similar relation for gravity in 4d asymptotically flat spacetimes. For simplicity, we will restrict to the leading infrared sector, where a careful treatment of soft modes and their canonical partners leads to a non-vanishing on-shell action. I will show that this action localizes to a codimension-2 surface and coincides with the generating function of 2d CFT correlators involving insertions of Kac-Moody currents. The latter were previously shown, using effective field theory methods, to reproduce the leading soft graviton theorems in 4d. I will conclude with comments on the implications of these results for the computation of soft charge fluctuations in the vacuum.