On deformations and extensions of Diff(S2)

Martín Enríquez Rojo,Tomáš Procházka,Ivo Sachs

doi:10.1007/jhep10(2021)133

Abstract

We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that hs[λ], the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of mathfrak{vect} (S2) into mathfrak{vect} (ℂ*). For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization.

Highlights

Flat, gbms [16,17,18,19,20,21], and asymptotically decelerating spatially flat Friedmann-LemaîtreRobertson-Walker (FLRW), gbmss [22], spacetimes at future null infinity, and asymptotically de-Sitter [23] in four spacetime dimensions
We discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization
Diff(S2) plays a major role in membrane theory [9,10,11] and fluid-gravity duality [28, 29], our main motivation emerges from recent investigations in asymptotically flat [16, 17, 19,20,21] and asymptotically spatially flat FLRW [22, 49, 50] spacetimes, where the asymptotic symmetry algebras contain as superrotation subalgebra that of vector fields on S2

Summary

Introduction

Flat, gbms [16,17,18,19,20,21], and asymptotically decelerating spatially flat Friedmann-LemaîtreRobertson-Walker (FLRW), gbmss [22], spacetimes at future null infinity, and asymptotically (anti) de-Sitter [23] in four spacetime dimensions. Non-central extensions of this algebra have been discussed in the context of asymptotic symmetries in null hypersurfaces (including event horizons) [24, 25]. We analyze the structure and deformations of the algebra of globally defined vector fields on the sphere, vect(S2) as well as its “chiral” subalgebra generated by holomorphic and anti-holomorphic vector fields respectively. We embed vect(S2) in the algebra of vector fields on the two-punctured sphere, or punctured complex plane vect(C∗), in order to investigate some of its physically relevant non-central extensions and devising simple free field realizations for them.

Objectives

Discussion

Conclusion