On deformations and extensions of $\text{Diff}(S^2)$

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 $hs[\lambda ]$, 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 $\mathfrak{vect}(S^2)$ into $\mathfrak{vect}({\mathbb{C}}^{\ast })$. 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.

18+5 pages, 1 figure. Attached Mathematica notebook. v3: discussion added on page 11 and reference added