The Conformal $w$-Algebra

Abstract

Einstein gravity in four asymptotically flat dimensions is governed by a tower of soft theorems obeying an $\mathcal{L}{w}_{1+\infty }$ commutator algebra. This algebra is deformed to ${\mathcal{L}}_{\Lambda }{w}_{1+\infty }$ in the conformally related AdS${}_4$ and dS${}_4$ spacetimes. Here we construct the larger ``conformal $w$-algebra", denoted $\mathcal{C}{w}_{1+\infty }$, which acts on conformal gravity in all of these spacetimes. It includes $\mathcal{L}{w}_{1+\infty }$, ${\mathcal{L}}_{\Lambda }{w}_{1+\infty }$ and the $\mathfrak{so}(4,2)$ 4D conformal algebra as subalgebras. We show that $\mathcal{C}{w}_{1+\infty }$ is the conformal completion of either $\mathcal{L}{w}_{1+\infty }$ or ${\mathcal{L}}_{\Lambda }{w}_{1+\infty }$. A derivation of $\mathcal{C}{w}_{1+\infty }$ is given as the symmetry of the twistor action of self-dual conformal gravity.

16 pages + 1 appendix, 1 figure