Celestial Holography

• Celestial holography is a duality that recasts Minkowski scattering amplitudes into correlation functions on the celestial sphere using conformal primary wavefunctions.
• It leverages infinite-dimensional symmetries, including BMS, Virasoro, and w₁₊∞ algebras, to relate soft theorems and Ward identities in both gravity and gauge theories.
• Methodologies such as Mellin transforms and conformal block decompositions provide a framework unifying infrared effects, memory phenomena, and operator dynamics in quantum gravity.

Celestial holography is a proposed duality that reformulates scattering amplitudes in asymptotically flat $(d+2)$-dimensional Minkowski spacetimes as correlation functions of a $d$-dimensional conformal field theory (CFT) defined on the celestial sphere at null infinity. This program seeks to generalize the success of AdS/CFT to the physically relevant case of zero cosmological constant, establishing a holographic correspondence in flat spacetimes. The celestial CFT (CCFT) provides an organizing framework unifying infrared dynamics, infinite-dimensional symmetries, and observable structures such as soft theorems, memory effects, and the emergent operator product expansions relevant for quantum gravity.

1. Foundation and Dictionary of Celestial Holography

Celestial holography is fundamentally built on the assertion that the gravitational $\mathcal{S}$-matrix in $(d+2)$-dimensional Minkowski space can be re-expressed as correlators of local operators on the celestial sphere, a $d$-dimensional space at null infinity (Pasterski et al., 2021). The traditional momentum eigenstate basis is traded for a conformal basis, where states are labeled by boost eigenvalues $\Delta$ and celestial coordinates $(z, \bar z)$. The key step is to Mellin transform each external energy variable: $$\mathcal{O}_\Delta(z, \bar{z}) = \int_0^\infty d\omega\, \omega^{\Delta-1} a(\omega, z, \bar z),$$ so amplitude data is recast as CCFT correlators: $$\langle \mathcal{O}_{\Delta_1}(z_1, \bar{z}_1)\cdots \mathcal{O}_{\Delta_n}(z_n, \bar{z}_n) \rangle.$$

The celestial operators transform as primary fields under the Lorentz group $SL(2, \mathbb{C})$, which acts as the global conformal group on the sphere. Asymptotic symmetry analyses—most notably the enhancement of the BMS group (which includes supertranslations and superrotations)—give rise to an infinite-dimensional algebra constraining the CCFT, laying a foundation for correspondence with 2D CFT structures such as the Virasoro and $w_{1+\infty}$ algebras (Raclariu, 2021, Donnay, 2023).

2. Symmetries, Soft Theorems, and Ward Identities

A central insight fueling celestial holography is the observed correspondence between soft theorems and Ward identities for asymptotic symmetries. The leading and subleading soft graviton theorems, for example,

$$\lim_{\omega \to 0} \mathcal{A}_{n+1}(\omega) \sim S_n^{(0)} \mathcal{A}_n + S_n^{(1)} \mathcal{A}_n,$$

translate in the conformal basis to insertions of specific conserved currents and the stress tensor of the CCFT (Raclariu, 2021). For gravity, the subleading soft factor is precisely the Ward identity for the Virasoro symmetry on the celestial sphere: $$\langle T(z) \prod_{i=1}^n \mathcal{O}_i(z_i, \bar{z}_i) \rangle = \sum_{k=1}^n \left[\frac{h_k}{(z-z_k)^2} + \frac{1}{z-z_k} \partial_{z_k} \right] \langle \prod_{i} \mathcal{O}_i(z_i, \bar{z}_i) \rangle.$$ Likewise, the leading soft graviton theorem is associated with the conservation of the supertranslation current, and their modes furnish the extended BMS algebra (Donnay et al., 2021, Donnay, 2023). In the gauge theory setting, leading soft photon and gluon theorems correspond to $U(1)$ and Kac–Moody symmetries, respectively.

These symmetry constraints generate nontrivial recursion relations, operator product expansions, and selection rules for celestial correlators. The structure goes beyond the Virasoro algebra, with the emergence of a $w_{1+\infty}$ algebra organizing an infinite tower of soft charges (Donnay, 2023, Freidel et al., 2022, Crawley et al., 2023, Bu et al., 6 Apr 2024).

3. Construction of Celestial Amplitudes and Operator Bases

Celestial amplitudes are systematically constructed via Mellin transforms of standard momentum-space amplitudes (Pasterski et al., 2021, Pasterski, 2023). The external states—either massless or massive—are mapped to conformal primary wavefunctions on the celestial sphere:

• For massless particles,

$$\varphi_\Delta(\hat{q};x) \propto 1/(\hat{q}\cdot x)^\Delta,$$

with $\hat{q}$ parameterizing the sphere.

• For massive particles (using Milne/hyperbolic slicing), scaling and Bessel function factors give rise to a continuum of scaling dimensions.

Recent developments identify both continuous ("principal series": $\Delta=1+i\lambda$, $\lambda\in\mathbb{R}$) and discrete ($\Delta \in \mathbb{Z}$) complete orthogonal bases. The discrete basis naturally captures gravitational memory and Goldstone modes, making it well-adapted for reconstructing Schwartz-class signals and charge quantization (Freidel et al., 2022).

The mapping to CCFT operators incorporates additional structure from asymptotic symmetries, such as soft currents and higher-spin Goldstone towers, which combine into nontrivial algebras and enable the systematic construction of generalized dressed states (Freidel et al., 2022, Crawley et al., 2023). The presence of shockwave backgrounds or curved asymptotically flat spacetimes enriches the celestial two-point functions, making them true conformal correlators and lifting kinematic delta-function constraints present in Minkowski space (2207.13719).

4. Mathematical Structures, Feynman Rules, and Block Decompositions

The Mellin transform methodology and conformal basis make available a suite of mathematical techniques from CFT and harmonic analysis. These include:

• Split representations: Internal lines in Feynman diagrams are "cut" into pairs of boundary-to-bulk propagators, allowing celestial amplitudes to be decomposed into conformal partial waves and blocks. This leads to systematic conformal block expansions of four-point functions, sometimes involving non-standard intermediate exchanges like staggered modules (Chang et al., 2023).
• Extrapolate dictionaries: Analogs of the AdS/CFT "extrapolate" prescription are formulated for Minkowski space, leading to celestial correlators that can be recast as Witten diagrams in Euclidean AdS for perturbative sectors (Sleight et al., 2023, Iacobacci et al., 29 Jan 2024). This framework leverages analytic continuations from de Sitter or AdS spaces, and spectral representations such as the Källén-Lehmann decomposition for non-perturbative effects.
• Watson–Sommerfeld transforms: The analytic structure of celestial correlators, especially as functions of intermediate scaling dimensions, aligns with Watson–Sommerfeld techniques in S-matrix theory (Iacobacci et al., 29 Jan 2024).
• Chiral and twisted holographies: Relations to 2D chiral algebras, Koszul duality, and twisted holography methods provide efficient means for computing form factors, chiral correlators, and modular structures on the boundary (Costello et al., 2022, Pasterski, 2023, Krishna et al., 2023).

These mathematical ingredients have led to the identification of the CCFT with specific models, such as the $H_3^+$ WZW model, which naturally carries the correct spectrum and correlation functions (Ogawa et al., 18 Apr 2024).

5. Relation to Infrared Physics, Black Holes, and dS/CFT Correspondence

Celestial holography unifies the infrared sector of gravity—including soft theorems, memory effects, and vacuum degeneracies—with conformal symmetry. Memory observables and the construction of soft hair on black holes find natural descriptions as celestial operators and states, complete with representations of $w_{1+\infty}$ and associated infinite-dimensional charge towers (Crawley et al., 2023). Dressed states constructed from coherent superpositions of Goldstone modes resolve infrared divergences and yield finite observables in both Minkowski and curved backgrounds (2207.13719, Freidel et al., 2022).

The connection between celestial correlators and cosmological correlators in de Sitter space (dS/CFT) is further illuminated. Explicit constructions relate celestial operators (defined via Mellin transforms) to the asymptotic (late/early time) data of dS fields. Correlators are mapped via Weyl rescaling, Fourier transformation, and analytic continuation, giving a unified approach to both scattering in flat space and encoding cosmological information, often requiring linear combinations of primary and shadow operators (Furugori et al., 23 Jul 2025).

6. Entanglement Structure and Geometric Duals

Celestial holography naturally accommodates the computation of entanglement Rényi entropies (EREs) and entanglement entropy (EE) in the CCFT, utilizing replica trick methods (Capone et al., 12 Dec 2024). Twist operators in the CCFT are identified as being dual to cosmic branes in the bulk spacetime, and their correlators correspond to partition functions in the presence of these branes. For a spherical subregion, the universal contribution to the entropy is dictated by conformal symmetry, arising from the sphere partition function (for odd $d$) or the Weyl anomaly (even $d$), with the vanishing of the universal term for $d=4$ mod $4$. These results offer geometric interpretations of quantum information observables within a flat space holographic framework.

7. Open Issues and Research Directions

Key outstanding challenges include:

• Refining the holographic dictionary, especially for massive states, higher-point functions, and in physically relevant dimensions (Pasterski et al., 2021, Pasterski, 2023).
• Systematic treatment of loop corrections and their duals as marginal deformations in the CCFT, encoded by double-current operators and topological gauging methods; these are directly tied to the appearance of universal infrared divergences in field theory (He et al., 2023).
• Deeper understanding of operator spectra—including the physical significance of staggered modules, integer-valued bases, and shadow sectors.
• Bridging Carrollian, twisted, and string-theoretic perspectives, incorporating Carrollian CFTs at null infinity, and elucidating the role of chiral string models (Donnay et al., 2022, Krishna et al., 2023).
• Practical tools for computing observables, generalizing to nontrivial backgrounds, and exploring information-theoretic aspects such as EE and EREs (Capone et al., 12 Dec 2024).

Celestial holography continues to serve as an organizing principle for synthesizing infinite-dimensional symmetry, conformal field theory, and the quantum dynamics of gravity in flat and cosmological spacetimes. Its dictionary encompasses soft physics, quantum information, algebraic structures, and links to both mathematical and phenomenological quantum gravity research.