Comments on Celestial CFT and $AdS_{3}$ String Theory

TL;DR

This paper extends a celestial CFT model based on the $H_{3}^{+}$-WZNW model, demonstrating its ability to holographically generate tree-level scattering amplitudes and deriving key correlators and equations.

Contribution

It constructs a holographic dictionary linking celestial CFT operators to Euclidean $AdS_{3}$ string theory, offering an alternative to celestial Liouville theory.

Findings

Generated tree-level MHV scattering amplitudes for gluons and gravitons.

Derived celestial stress-energy tensor and computed two- and three-point functions.

Formulated differential equations characterizing celestial amplitudes using KZ equations.

Abstract

In a recent work, Ogawa et al. (2024) proposed a model for celestial conformal field theory (CFT) based on the $H_{3}^{+}$-Wess-Zumino-Novikov-Witten (WZNW) model. In this paper, we extend the model advanced by Ogawa et al. (2024), demonstrating how it can holographically generate tree-level MHV scattering amplitudes for both gluons and gravitons when analytically continued to the ultra-hyperbolic Klein space $\mathbf{R}_{2}^{2}$, thereby offering an alternative to celestial Liouville theory. We construct a holographic dictionary in which vertex operators and conformal primaries in celestial CFT are derived from their worldsheet counterparts in Euclidean $AdS_{3}$ (bosonic) string theory. Within this dictionary, we derive the celestial stress-energy tensor, compute the two- and three-point functions, and determine the celestial operator product expansion (OPE). Additionally, we derive a system of partial differential equations that characterises the celestial amplitudes of our model, utilising the Knizhnik--Zamolodchikov (KZ) equations and worldsheet Ward identities. In the Appendix, we provide a concise introduction to the $H_{3}^{+}$-WZNW model, with emphasis on its connection to Euclidean $AdS_{3}$ string theory.