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

Abstract

We study CFT${}_3$ realizations and celestial representations of the ${\mathcal{L}}_{\Lambda }{w}_{1+\infty }$ algebra. Free scalar and Dirac theories yield the same one-particle light-ray action. This action extends to the critical $O(N)$ model at $N=\infty$. At large but finite $N$, we use the light-ray algebra under its closure assumptions and check prescribed iterated limits of the frequency kernel. In the dimension-completed celestial hard-graviton representation, a closed raising subalgebra annihilates $G_3^{+}$ exactly. We construct two nonzero singular descendants in its induced module and prove that both vanish in the celestial realization. They implement the finite-spin condition and massless spin-two shortening of the generated highest-weight sector. The nonzero-$\Lambda$ quadratic Casimir has eigenvalue three throughout the hard family. The null descendants and low-spin Ward identities yield ordered differential--difference equations for celestial correlators.