Centrally extended BMS4 Lie algebroid

TL;DR

The paper addresses the challenge of central extensions in the BMS4 algebra for 4D asymptotically flat spacetimes by constructing a centrally extended Lie algebroid using a field-dependent 2-cocycle. It develops both BRST and vertex-operator-algebra formalisms and provides explicit realizations on the two-punctured Riemann sphere and on the cylinder, showing how zero-mode shifts in the shear and news arise under the sphere-to-cylinder map. The main contributions include explicit central terms K, mode realizations with l_m, bar{l}_m, and P_{k,l}, and a discussion of NS vs R boundary conditions. This framework advances the formal understanding of asymptotic symmetries at null infinity and sets the stage for applications to holography, soft theorems, and potential generalizations to other Riemann surfaces.

Abstract

We explicitly show how the field dependent 2-cocycle that arises in the current algebra of 4 dimensional asymptotically flat spacetimes can be used as a central extension to turn the BMS4 Lie algebra, or more precisely, the BMS4 action Lie algebroid, into a genuine Lie algebroid with field dependent structure functions. Both a BRST formulation, where the extension appears as a ghost number 2 cocyle, and a formulation in terms of vertex operator algebras are introduced. The mapping of the celestial sphere to the cylinder then implies zero mode shifts of the asymptotic part of the shear and of the news tensor.