State-Operator Correspondence in Celestial Conformal Field Theory

TL;DR

This work develops a concrete bulk-to-boundary dictionary for celestial holography by introducing 2D boundary states on oriented circles on the celestial sphere. It shows how to construct a BPZ inner product for CCFT boundary states and demonstrates that the corresponding adjoint operation uses a shadow transform, aligning the boundary structure with the bulk shadow product. By linking conformal primary wavefunctions, their shadows, and the Mellin-transformed S-matrix, the authors connect CCFT correlators to 4D scattering data while clarifying how boundary states encode the same information as bulk states, though in a different organization. Overall, the paper provides a principled CCFT framework for analyzing celestial amplitudes and clarifies the interplay between bulk and boundary scattering in flat space holography.

Abstract

The bulk-to-boundary dictionary for 4D celestial holography is given a new entry defining 2D boundary states living on oriented circles on the celestial sphere. The states are constructed using the 2D CFT state-operator correspondence from operator insertions corresponding to either incoming or outgoing particles which cross the celestial sphere inside the circle. The BPZ construction is applied to give an inner product on such states whose associated bulk adjoints are shown to involve a shadow transform. Scattering amplitudes are then given by BPZ inner products between states living on the same circle but with opposite orientations. 2D boundary states are found to encode the same information as their 4D bulk counterparts, but organized in a radically different manner.

Figures (2)

• Figure 1: "Northern" states enter in the southern hemisphere on $\mathcal{I}^-$ and exit in the northern one on $\mathcal{I}^+$. "Southern" states enter in the northern hemisphere on $\mathcal{I}^-$ and exit in the southern one on $\mathcal{I}^+$. Due to the antipodal identification between the angular coordinates at $\mathcal{I}^+$ and $\mathcal{I}^-$, a free massless particle will enter and exit at the same point on the celestial sphere.
• Figure 2: On the left we depict the 4D scattering problem. Incoming states are in the Hilbert space $\mathcal{H}_{\rm{in}}$ at $\mathcal{I}^-$ and outgoing states are in the Hilbert space $\mathcal{H}_{\rm{out}}$ at $\mathcal{I}^+$. The bulk scattering problem is to relate these two Hilbert spaces. On the right we depict the 2D scattering problem. In the northern (southern) hemisphere we have northern (southern) states, those originating at the south (north pole) of $\mathcal{I}^-$, and they form the Hilbert space $\mathcal{H}_N$ ($\mathcal{H}_S$). The dots represent operators that annihilate the northern (southern) states on $\mathcal{I}^+$ and the crosses represent antipodally mapped operators that create the northern (southern) states on $\mathcal{I}^-$. The boundary scattering problem is to find the overlap between the states in these two Hilbert spaces.