Celestial Diamonds: Conformal Multiplets in Celestial CFT

TL;DR

The paper develops a wavefunction-based framework to map 4D massless scattering data to 2D celestial CFT, enabling a complete classification of SL(2,$\mathbb{C}$) primary descendants for spins $s=0,\tfrac{1}{2},1,\tfrac{3}{2},2$. It introduces celestial diamonds, organizing radiative and generalized conformal primaries into nested modules with three descendant types (I–III) and explicit shadow relations, thereby clarifying the structure of conformally soft sectors and their relation to asymptotic symmetries. The authors show that radiative primaries at the diamond corners correspond to soft modes and currents, while bottom corners yield nontrivial generalized primaries whose descendants contribute contact terms in celestial amplitudes, and top corners relate to soft dressings. This framework unifies previously known soft theorems, shadow relations, and helicity degeneracies, providing a bulk-picture and algebraic foundation for the celestial CFT description of infrared structure in gauge theory and gravity.

Abstract

We examine the structure of global conformal multiplets in 2D celestial CFT. For a 4D bulk theory containing massless particles of spin $s=\{0,\frac{1}{2},1,\frac{3}{2},2\}$ we classify and construct all SL(2,$\mathbb{C}$) primary descendants which are organized into 'celestial diamonds'. This explicit construction is achieved using a wavefunction-based approach that allows us to map 4D scattering amplitudes to celestial CFT correlators of operators with SL(2,$\mathbb{C}$) conformal dimension $Δ$ and spin $J$. Radiative conformal primary wavefunctions have $J=\pm s$ and give rise to conformally soft theorems for special values of $Δ\in \frac{1}{2}\mathbb{Z}$. They are located either at the top of celestial diamonds, where they descend to trivial null primaries, or at the left and right corners, where they descend both to and from generalized conformal primary wavefunctions which have $|J|\leq s$. Celestial diamonds naturally incorporate degeneracies of opposite helicity particles via the 2D shadow transform relating radiative primaries and account for the global and asymptotic symmetries in gauge theory and gravity.

Figures (10)

• Figure 1: Diamond illustrating the nested submodule structure. The SL(2,$\mathbb{C}$) module of the primary $|\frac{1-k}{2},\frac{1-\bar{k}}{2}\rangle$ contains three submodules associated with primary descendant operators.
• Figure 2: Primary descendants of type I and II. The dots define the integer lattice $\mathbb{Z}^{2}$ in the $J$-$\Delta$ plane. The origin is at $(\Delta,J)= (1,0)$. Dots in the grey regions correspond to primaries that have primary descendants. The latter are obtained by reflecting across the dashed lines.
• Figure 3: Primary descendants of type III. The dots define the integer lattice $\mathbb{Z}^{2}$ in the $J$-$\Delta$ plane. The origin of the lattice is placed at $(\Delta,J)= (1,0)$.
• Figure 4: Example type I module in celestial CFT. For a radiative state () with $\Delta=1-|J|-n$ where $n\in\mathbb{Z_{>}}$, the predicted primary descendants and their descendants are identically zero ($\times$). The radiative primary () and its descendants () thus form an $(n-1)(2|J|+n-1)$ dimensional vector space. The axes intersect at $(\Delta,J)=(1,0)$.
• Figure 5: Goldstone (a) and memory (b) diamonds for the leading soft photon theorem.