CFT$_3$ Realizations and Celestial Representation Theory of the $\mathcal{L}_Λw_{1+\infty}$ Algebra

Abstract

We study CFT$_3$ realizations and celestial representations of the $\mathcal{L}_Λw_{1+\infty}$ algebra. Free scalar and Dirac theories yield the same one-particle light-ray action. This action extends to the critical $O(N)$ model at $N=\infty$, while regulated moments reproduce the pairwise algebra in the large-but-finite-$N$ correlators studied here. In the celestial hard-graviton module, the projected null conditions select $Δ=3$ uniquely, and a closed raising subalgebra annihilates $G^+_3$ exactly. The nonzero-$Λ$ quadratic Casimir has eigenvalue three throughout the hard module, and the exact annihilators and low-spin Ward identities give compatible differential-difference equations for celestial correlators.