Soft Algebras in AdS$_4$ from Light Ray Operators in CFT$_3$

Flat Minkowski space (M$^4$) and AdS$_4$ can both be conformally mapped to the Einstein cylinder. The maps may be judiciously chosen so that some null generators of the $\mathcal{I}^+$ boundary of M$^4$ coincide with antipodally-terminating null geodesic segments on the boundary of AdS$_4$. Conformally invariant nonabelian gauge theories in M$^4$ have an asymptotic $S$-algebra generated by a tower of soft gluons given by weighted null line integrals on $\mathcal{I}^+$. We show that, under the conformal map to AdS$_4$, the leading soft gluons are dual to light transforms of the conserved global symmetry currents in the boundary CFT$_3$. The tower of light ray operators obtained from the $SO(3,2)$ descendants of this light transform realize a full set of generators of the $S$-algebra in the boundary CFT$_3$. This provides a direct connection between holographic symmetry algebras in M$^4$ and AdS$_4$.