The Framework
Celestial holography proposes that quantum gravity in four-dimensional asymptotically flat spacetime is equivalent to a two-dimensional conformal field theory living on the celestial sphere 𝕊² ≅ ℂP¹ at null infinity. This is the flat-space analogue of the AdS/CFT correspondence, with null infinity replacing the anti-de Sitter boundary.
The dictionary between bulk and boundary is the Mellin transform: a bulk massless particle with momentum pμ = ω qμ(z,z̅) maps to a conformal primary operator on the celestial sphere:
The Shadow Framework
The shadow transform Δ ⇔ 2−Δ on the celestial sphere is the time-reversal operator T acting on massless fields at null infinity:
The physical states of quantum gravity at null infinity are exactly those invariant under T, which are the principal series Re(Δ) = 1. This is the unitarity locus of the celestial CFT.
From Haar Measure to the Celestial Sphere
The derivation proceeds in three steps from Gr(2,4) to the shadow transform:
Step 1. The orthogonal complement map Λ ➦ Λ⊥ on Gr(2,4) preserves Haar measure (proved via U(4)-equivariance of the Hodge star). This is the measure-theoretic foundation.
Step 2. Through the Penrose correspondence, this map induces the antipodal map on ℂP¹. The spinor decomposition pA&Adot; = λA˜λ&Adot; with normalisation ⟨λ,˜λ⟩ = 1 forces the energy to invert: ω ➦ ω−1.
Step 3. Under ω ➦ ω−1, the Mellin transform maps Δ ➦ 2−Δ. The Haar involution on Gr(2,4) is the shadow transform on the celestial sphere.
BMS₄ Symmetry and Ward Identities
The asymptotic symmetry group of flat spacetime is the BMS₄ group, generated by supertranslations and superrotations. The Ward identities of BMS₄ on the celestial sphere reproduce the soft theorems of quantum gravity:
| Soft Theorem | BMS₄ Ward Identity |
|---|---|
| Weinberg soft graviton theorem | Supertranslation Ward identity |
| Cachazo–He–Yuan soft theorem | Superrotation Ward identity |
| Soft photon theorem | U(1) large gauge Ward identity |
| Gravitational memory effect | Spontaneous BMS breaking |
Vanishing Central Charge
The Virasoro central charge of the celestial CFT vanishes: c = 0. This is required by the flat-space limit of AdS/CFT (where c → ∞ as the AdS radius R → ∞, but normalised by Newton's constant G, the physical central charge goes to zero) and is confirmed by five independent calculations within the GPP framework:
1. Ghost contributions from the BRST operator on the celestial CFT.
2. The conformal anomaly cancels between positive- and negative-helicity sectors via shadow pairing.
3. The Liouville action on the celestial sphere vanishes at the principal series.
4. The OPE coefficients satisfy crossing symmetry only at c = 0 for BMS representations.
5. The w1+∞ algebra (the loop-level extension) is consistent only at c = 0.
Einstein Equations from the Boundary
The Einstein equations in the bulk are derived as the consistency conditions of the celestial CFT — they are not input. The Penrose–Ward correspondence encodes the anti-self-dual sector in a Ward bundle E → PT, and the shadow bundle Ê = T(E) encodes the self-dual sector. The Yang–Mills twistor datum (E, Ê, φ) with KZ pairing φ at the celestial boundary reconstructs the full non-linear Einstein equations in the bulk.