The AdS/CFT correspondence stands as one of the most profound structural insights in theoretical physics—a concrete realization of the holographic principle that maps quantum gravity in anti-de Sitter spacetime to a conformal field theory on its boundary. For over two decades, this duality has served as our most powerful non-perturbative window into quantum gravity, yielding exact results, resolving conceptual paradoxes, and connecting disciplines from condensed matter physics to quantum information theory. Yet there is an uncomfortable truth lodged at the heart of this triumph: our universe is not anti-de Sitter.

Cosmological observations indicate a positive cosmological constant—an accelerating expansion that places us in a de Sitter-like spacetime, not the negatively curved geometry where holography is best understood. Flat spacetime, the arena of scattering experiments and asymptotic particle physics, is equally removed from the AdS setting. The question then becomes whether holography is a peculiar feature of negatively curved spacetimes, or whether it reflects something deeper about quantum gravity itself—a structural principle that should extend to all physically relevant backgrounds.

Extending gauge/gravity duality beyond its AdS cradle is not merely a technical exercise. It strikes at the foundations of what we believe about the quantum nature of spacetime. If holography is universal, then every spacetime—de Sitter, flat, cosmological—should admit a dual description in terms of lower-dimensional degrees of freedom. The search for these generalizations forces us to confront new conceptual challenges: boundaries that lie in the future rather than at spatial infinity, symmetry algebras that replace the conformal group, and observables that resist the neat packaging of correlation functions in a fixed background. What follows is an examination of where this expansion stands and what it demands of our theoretical imagination.

The structural elegance of AdS/CFT rests on a specific geometric fact: anti-de Sitter spacetime possesses a timelike conformal boundary. This boundary is not a physical wall but a mathematical surface at spatial infinity where the geometry approaches a well-defined conformal structure. A conformal field theory—a quantum theory invariant under the conformal group SO(d,2)—lives naturally on this boundary, and Maldacena's conjecture identifies the partition function of quantum gravity in the bulk with the generating functional of correlators in this boundary CFT. The isometry group of AdS_d+1 matches precisely the conformal group of the d-dimensional boundary theory, providing the kinematic skeleton on which the entire correspondence hangs.

This matching of symmetries is not incidental—it is the reason AdS/CFT works so cleanly. The conformal group organizes boundary operators into representations, the radial direction of AdS maps onto the renormalization group scale, and the bulk-to-boundary propagator provides a dictionary translating between gravitational fields and CFT data. Calculations of entanglement entropy, black hole thermodynamics, and even transport coefficients in strongly coupled systems all flow from this architecture. The Ryu-Takayanagi formula, for instance, connects boundary entanglement to bulk minimal surfaces with a precision that has been verified in numerous limits.

But anti-de Sitter spacetime carries a negative cosmological constant, producing a geometry that curves inward—a spacetime that acts like a confining box, where light signals bounce back from infinity in finite time. Our observed universe, by contrast, has a small positive cosmological constant. The expansion of space is accelerating, not decelerating. Spatial infinity in a de Sitter universe is not a timelike boundary but a pair of spacelike surfaces: one in the asymptotic past and one in the asymptotic future. The conformal structure changes fundamentally, and with it, the entire boundary framework.

Flat spacetime presents a different but equally severe departure. Its boundary—null infinity, denoted I⁺ and I⁻—is neither timelike nor spacelike but null. The asymptotic symmetry group is not the conformal group but the infinite-dimensional Bondi-van der Burg-Metzner-Sachs (BMS) group, which has a radically different structure. There is no obvious conformal field theory waiting at null infinity, and the natural observables of flat space physics—S-matrix elements—do not map straightforwardly onto boundary correlators in the AdS sense.

The limitation, then, is both precise and profound. AdS/CFT is not merely a conjecture about one particular spacetime; it is a framework whose internal consistency depends on geometric and algebraic structures that other spacetimes do not share. Extending holography requires either discovering analogous structures in de Sitter and flat backgrounds or fundamentally rethinking what a holographic dual means when the familiar boundary architecture is absent.

Takeaway

AdS/CFT works because anti-de Sitter spacetime has a timelike conformal boundary whose symmetry group matches that of a conformal field theory—a coincidence of structure that de Sitter and flat spacetimes do not share, making extension far more than a change of sign in the cosmological constant.

The first serious attempt to extend holography to de Sitter spacetime came from Strominger's dS/CFT proposal in 2001. The central idea is deceptively simple: if de Sitter spacetime has a spacelike boundary at future infinity (I⁺), then perhaps a Euclidean conformal field theory lives there, encoding the bulk gravitational dynamics. The isometry group of (d+1)-dimensional de Sitter space is SO(d+1,1), which is precisely the conformal group of d-dimensional Euclidean space. The kinematic matching is present, just as in AdS/CFT—but the physics it implies is radically different.

The most unsettling difference is the nature of the boundary itself. In AdS/CFT, the boundary is timelike, meaning an observer can send and receive signals from it in finite boundary time. It serves as a natural home for a unitary quantum theory with a well-defined Hilbert space and time evolution. The future boundary of de Sitter space, however, is spacelike—it lies in the infinite future. No observer inside de Sitter spacetime can ever reach it or communicate with it. This raises an immediate conceptual crisis: what does it mean for the physics experienced by a local observer if the dual theory lives at a boundary that is causally inaccessible? The dual CFT, if it exists, would be Euclidean rather than Lorentzian, lacking the standard notion of unitary time evolution that underpins quantum mechanics.

Further complications arise from the finite entropy of de Sitter space. The Gibbons-Hawking entropy of the cosmological horizon, S = A/4G, suggests that the Hilbert space of quantum gravity in de Sitter is finite-dimensional. This stands in stark contrast to the infinite-dimensional Hilbert spaces of conformal field theories, which possess operator algebras of unbounded complexity. If the bulk quantum gravity theory has a finite number of degrees of freedom, the boundary dual cannot be a standard CFT in any conventional sense. Proposals involving finite-dimensional matrix models or constrained Hilbert spaces have been explored, but none has achieved the level of precision available in the AdS context.

The problem of observables compounds these difficulties. In AdS/CFT, boundary correlators have a clear operational meaning: they are computed by standard QFT methods on the boundary and correspond to bulk scattering experiments via the extrapolate dictionary. In de Sitter, the natural observables are cosmological correlators—the statistical properties of fluctuations frozen at superhorizon scales, precisely what we measure in the cosmic microwave background. The wavefunction of the universe at I⁺ plays the role of the partition function, and its coefficients encode these cosmological correlators. Recent work on the cosmological bootstrap and the wavefunction approach has made significant progress in computing these objects using consistency conditions—unitarity, locality, and symmetry—without relying on a specific bulk Lagrangian.

Yet for all this progress, dS/CFT remains a conjecture without a concrete example. In AdS/CFT, we have explicit string-theoretic constructions: type IIB strings on AdS₅ × S⁵ dual to N=4 super-Yang-Mills. No analogous top-down construction exists for de Sitter holography. String theory notoriously resists stable de Sitter vacua—the KKLT construction and its descendants remain debated—and without a controlled microscopic realization, dS/CFT lacks the anchor that gives AdS/CFT its computational power. The conceptual framework is suggestive, the symmetry matching is real, but the edifice remains incomplete.

Takeaway

De Sitter holography forces us to contemplate a dual theory that lives at a causally inaccessible boundary in the infinite future, operates with a finite-dimensional Hilbert space, and lacks a single concrete string-theoretic realization—each a fundamental departure from the AdS paradigm.

Flat spacetime might seem like the simplest arena for physics, yet constructing its holographic dual has proven remarkably subtle. The natural observables of flat space quantum gravity are scattering amplitudes—the S-matrix—and these are defined between asymptotic states at past and future null infinity. The boundary of flat spacetime is not a single connected surface but a collection of null and spacelike components: past and future null infinity (I⁻ and I⁺), spatial infinity (i⁰), and the timelike infinities (i^±). This intricate boundary structure admits no straightforward conformal field theory interpretation.

The breakthrough insight came from recognizing that the asymptotic symmetry group of flat spacetime is far richer than the Poincaré group. In the 1960s, Bondi, van der Burg, and Metzner, together with Sachs, showed that the symmetry group at null infinity is the BMS group—an infinite-dimensional extension of the Poincaré group that includes an infinite set of angle-dependent translations called supertranslations. More recently, Strominger and collaborators demonstrated that these BMS symmetries are not merely mathematical curiosities but are physically realized as Ward identities governing soft graviton theorems—the universal behavior of scattering amplitudes when one graviton carries vanishingly small energy. This connection between asymptotic symmetries, soft theorems, and gravitational memory effects forms a triangle of equivalences that has reshaped our understanding of infrared physics in gravity.

Celestial holography builds on this foundation by proposing that the holographic dual of quantum gravity in four-dimensional flat spacetime is a two-dimensional conformal field theory living on the celestial sphere—the sphere of directions at null infinity. Scattering amplitudes, when re-expressed in a basis of boost eigenstates (conformal primary wavefunctions) rather than the usual momentum eigenstates, transform as correlation functions of a 2D CFT. The Lorentz group SL(2,ℂ) acts as the global conformal group on this celestial sphere, and the extended BMS symmetries, together with additional infinite-dimensional enhancements like superrotations, play the role of the larger symmetry algebra of the celestial CFT.

The results have been striking. Celestial operator product expansions (OPEs) have been constructed, conformally soft operators identified, and connections to w_1+∞ algebras—infinite-dimensional symmetry algebras familiar from integrable systems and the quantum Hall effect—have been uncovered. These structures suggest that the celestial CFT is not an ordinary 2D conformal field theory but something more exotic, possibly related to twisted holomorphic theories or chiral algebras. The collinear limits of scattering amplitudes, which control the behavior when external particles become parallel, map directly onto OPE coefficients in the celestial theory, providing a concrete computational dictionary.

Yet flat space holography faces its own open questions. The celestial CFT does not have a standard stress tensor or a conventional notion of a local energy-momentum tensor on the celestial sphere. The mapping between bulk states and boundary operators is less complete than in AdS/CFT, and loop-level amplitudes introduce infrared divergences that must be carefully treated in the celestial basis. Perhaps most fundamentally, celestial holography currently applies to four-dimensional asymptotically flat spacetimes—extending it to other dimensions, or to spacetimes that are only approximately flat, remains a challenge. Nevertheless, the emergence of infinite-dimensional symmetry, the repackaging of the S-matrix as celestial correlators, and the unexpected connections to mathematical structures in 2D CFT represent a genuine expansion of holography's reach into the spacetime that particle physicists actually use.

Takeaway

Flat space holography replaces the conformal boundary of AdS with the celestial sphere at null infinity, trading the conformal group for the infinite-dimensional BMS algebra and recasting scattering amplitudes as correlators in a novel two-dimensional theory—suggesting that holographic structure may be woven into the asymptotic fabric of any spacetime.

The effort to extend holography beyond anti-de Sitter spacetime is not a peripheral research program—it is a test of whether the holographic principle deserves to be called a principle at all. If gauge/gravity duality is merely a peculiarity of negatively curved spacetimes, its implications for quantum gravity are significant but bounded. If it extends to de Sitter and flat backgrounds, it tells us something universal about the quantum structure of spacetime itself.

What emerges from these explorations is that holography does not transplant unchanged into new settings. Each spacetime demands its own conceptual vocabulary: future boundaries and finite Hilbert spaces for de Sitter, celestial spheres and BMS symmetry for flat space. The unifying thread is not a single formula but an architectural conviction—that gravitational degrees of freedom are always encodable on lower-dimensional surfaces.

We are, in a precise sense, learning to read the holographic dictionary in new languages. The grammar is unfamiliar, the vocabulary incomplete, and the pronunciation uncertain. But the conviction that such a dictionary exists—that gravity is always, everywhere, a hologram—continues to drive some of the deepest work in theoretical physics.