Asymptotic Higher Spin Symmetries I: Covariant Wedge Algebra in Gravity

In this paper, we study gravitational symmetry algebras that live on 2-dimensional cuts $S$ of asymptotic infinity. We define a notion of wedge algebra $\mathcal{W}(S)$ which depends on the topology of $S$. For the cylinder $S=\mathbb{C}^*$ we recover the celebrated $Lw_{1+\infty}$ algebra. For the 2-sphere $S^2$, the wedge algebra reduces to a central extension of the anti-self-dual projection of the Poincar\'e algebra. We then extend $\mathcal{W}(S)$ outside of the wedge space and build a new Lie algebra $\mathcal{W}_\sigma(S)$, which can be viewed as a deformation of the wedge algebra by a spin two field $\sigma$ playing the role of the shear at a cut of $\mathscr{I}$. This algebra represents the gravitational symmetry algebra in the presence of a non trivial shear and is characterized by a covariantized version of the wedge condition. Finally, we construct a dressing map that provides a Lie algebra isomorphism between the covariant and regular wedge algebras.