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

TL;DR

The paper demonstrates that nonabelian soft gauge algebras in ${\rm M}^4$ are holographically realized as light-transform algebras of conserved currents in the ${\rm CFT}_3$ dual to ${\rm AdS}_4$, via a conformal map through the Einstein cylinder. The leading soft generator $S^{1,a}$ is identified with a light-transformed current, and its entire $S$-algebra is generated by $SO(4,2)$ descendants; on the AdS$_4$ side, AdS boundary conditions yield a diagonal boundary operator $T^{1,a}$ whose descendants form a complete tower of light-ray operators $L^{p,a}_{\bar m,m}$ that reproduce the full algebra. This establishes a concrete 4D-4D connection between holographic symmetry algebras in Minkowski space and AdS$_4$/CFT$_3$ light-ray structures, with potential extensions to gravity and fixed points. The results hinge on conformal mappings among ${\rm M}^4$, ${\rm EC}^4$, and ${\rm AdS}_4$, Mellin-transform representations, and a careful treatment of boundary conditions that project the correct chiral sector at the AdS boundary.

Abstract

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$_4$.