A Chiral $\mathrm{\Lambda }$-${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ Symmetry of AdS${}_4$ Gravity

Generalising the chiral boundary conditions of ${\mathbb{R}}^{1,3}$ gravity for AdS${}_4$ gravity, we derive chiral locally AdS${}_4$ solutions in the Newman-Unti gauge consistent with a variational principle whose asymptotic symmetry algebra we show, to be an infinite-dimensional chiral extension of $\mathfrak{s}\mathfrak{o}(2,3)$. This symmetry algebra coincides with the chiral ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra in the flat space limit. We posit this symmetry algebra as the chiral version of recently discovered $\mathrm{\Lambda }$-${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra. We postulate line integral charges from the bulk AdS${}_4$ gravity corresponding to this chiral symmetry algebra and show that the charges obey the semi-classical limit of a $\mathcal{W}$-algebra that includes a level $\kappa$ Kac-Moody $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ current algebra. Furthermore, using the standard tools of $2d$ CFT, we derive the quantum version of this $\mathcal{W}$-algebra which may be denoted by $\mathcal{W}(2;(3\mathrm{/}2{)}^2,1^3)$.