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:
shadow_equals_time_reversal) is machine-checked in Lean 4.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 (argued 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 supported by five arguments within the GPP framework. In the formalization the statement c2D = 0 ⇔ c4DWeyl = 0 is proved only conditionally, with the physical input (“Link 6”) carried as an explicit hypothesis:
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 proposal is that the Einstein equations in the bulk arise as consistency conditions of the celestial CFT rather than being 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 is proposed to reconstruct the full non-linear Einstein equations in the bulk.