# Centrally extended BMS4 Lie algebroid

## TL;DR

This paper demonstrates how a field-dependent 2-cocycle can extend the BMS4 Lie algebroid into a genuine Lie algebroid, incorporating field-dependent structure functions, with formulations via BRST and vertex operator algebras.

## Contribution

It introduces a method to incorporate a field-dependent 2-cocycle as a central extension of the BMS4 Lie algebroid, enhancing its algebraic structure.

## Key findings

- Field-dependent 2-cocycle enables a genuine Lie algebroid structure.
- BRST and vertex operator algebra formulations are developed.
- Zero mode shifts relate to celestial sphere mapping.

## 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.