Quantum gravity is the problem of reconciling general relativity with quantum mechanics. The standard approach attempts this in four-dimensional spacetime, quantizing the metric and working perturbatively around flat space. It fails: the perturbative expansion produces infinitely many Feynman diagrams at each loop order, the series is not Borel summable, and the theory is non-renormalizable. This is not a failure of the specific methods used. It is a signal that flat-space quantum gravity is being approached in the wrong language.
Celestial holography proposes the correct language. Four-dimensional quantum gravity in asymptotically flat spacetime is dual to a two-dimensional conformal field theory living on the celestial sphere at null infinity. The translation between them is the Mellin transform: a bulk graviton with momentum pμ = ω qμ(z, z̅) maps to a conformal primary operator of dimension Δ on the celestial sphere 𝒮², with ω playing the role of energy and (z, z̅) the direction.
The Asymptotic Symmetry Group
The asymptotic symmetry group of flat spacetime is the BMS4 group, generated by supertranslations and superrotations. Its Ward identities are not auxiliary equations imposed by hand; they are the boundary expression of bulk conservation laws. Weinberg's soft graviton theorem — the statement that a scattering amplitude acquires a universal factor when one graviton momentum goes to zero — is the Ward identity of supertranslation symmetry. Cachazo–He–Yuan's extension of this theorem to all multiplicity is the Ward identity of superrotation symmetry. Gravitational memory, the permanent displacement of test masses after a burst of gravitational waves, is spontaneous BMS breaking.
The celestial CFT therefore has BMS4 as its symmetry algebra. At tree level. At loop level the algebra extends further, to the w1+∞ algebra of area-preserving diffeomorphisms of the two-sphere, whose Ward identities encode the one-loop soft theorems. These are not new physical inputs; they are consequences of flat-space gravity already present in the Hilbert action, now visible in the boundary language.
Central Charge Zero
The Virasoro central charge of the celestial CFT vanishes: c = 0. This is the flat-space limit of the AdS/CFT result where c → ∞ with the AdS radius, normalized by Newton's constant: the combination c/G goes to zero as the boundary recedes to flat space. Five independent calculations within the framework confirm this: the BRST ghost sector cancels the matter contribution, the conformal anomaly vanishes between positive- and negative-helicity sectors under shadow pairing, the Liouville action on the sphere vanishes at the principal series, crossing symmetry for BMS representations forces c = 0, and the w1+∞ algebra is consistent at c = 0 only.
A CFT with c = 0 is unusual; most familiar examples have positive central charge. The celestial CFT belongs to a class of logarithmic CFTs where c = 0 is not a degenerate limit but the only consistent value. This is not a problem; it is a constraint that makes the theory more rigid and therefore more predictive.
Deriving Einstein's Equations
Perhaps the most striking result of the paper is that the Einstein field equations are not input. They are output. The Penrose–Ward correspondence encodes the anti-self-dual sector of the gravitational field in a holomorphic vector bundle E over projective twistor space. The self-dual sector, which in Penrose's original formulation required the separate dual twistor space and was the source of the googly problem, is encoded in the T-image of this bundle under the shadow symmetry. The Yang–Mills twistor datum (E, Ê, φ) — the bundle, its shadow, and the Knizhnik–Zamolodchikov pairing at the boundary — reconstructs the full nonlinear Einstein equations in the bulk as the integrability condition of this geometric data.
The equations governing spacetime curvature are the consistency conditions of a two-dimensional conformal field theory on the boundary. Gravity is not quantized; it is recognized as already being quantum.
The full paper constructs the graviton S-matrix from the celestial CFT correlators, verifies the Wightman axioms for the resulting theory, and derives the loop structure through the shadow discontinuity mechanism. It is the foundational reference for the celestial quantum gravity programme within the Shadow Framework.