The Effective Action of Superrotation Modes

TL;DR

This work derives a boundary description for four-dimensional asymptotically flat gravity by reformulating the radiative-vacuum sector as a gauged so(3,1) Chern–Simons theory at null infinity and performing a Hamiltonian reduction to the celestial sphere. The reduction yields a complex Alekseev–Shatashvili action governing superrotation reparametrizations, with holomorphic solutions encoding Virasoro superrotation vacua and generic configurations explicitly breaking Diff$( ext{S}^2)$. The analysis links the gravitational data at null infinity to a two-dimensional boundary theory on the celestial sphere, tying vacuum structure to the Schwarzian and Virasoro coadjoint orbit dynamics. The results provide a concrete bridge between 4D asymptotic symmetries and celestial CFT structures, suggesting avenues toward understanding soft theorems and memory effects in a boundary framework.

Abstract

Starting from an analysis of four-dimensional asymptotically flat gravity in first order formulation, we show that superrotation reparametrization modes are governed by an Alekseev--Shatashvili action on the celestial sphere. This two-dimensional conformal theory describes spontaneous symmetry breaking of Virasoro superrotations together with the explicit symmetry breaking of more general Diff$(\mathcal{S}^2)$ superrotations. We arrive at this result by first reformulating the asymptotic field equations and symmetries of the radiative vacuum sector in terms of a Chern--Simons theory at null infinity, and subsequently performing a Hamiltonian reduction of this theory onto the celestial sphere.