Flat Space, de Sitter, and Open Problems

The main idea

AdS/CFT is the sharpest known realization of holography because several favorable structures line up at once:

$$\text{timelike conformal boundary} \quad + \quad \text{well-defined boundary time} \quad + \quad \text{reflecting boundary conditions} \quad + \quad \text{ordinary CFT Hilbert space}.$$

The previous pages explored many examples beyond the canonical $\mathrm{AdS}_5/\mathrm{CFT}_4$ pair, but most of them were still AdS examples. The final question in this module is more radical:

$$\boxed{ \text{What remains of holography when the bulk is not asymptotically AdS?} }$$

Two cases are especially important.

First, asymptotically flat space is the natural arena for the gravitational $\mathcal S$-matrix. Its boundary is not a timelike cylinder. It is null infinity, $\mathscr I^+\cup\mathscr I^-$. The most robust observables are scattering amplitudes, soft charges, memory effects, and asymptotic symmetry data. A modern program, celestial holography, rewrites the flat-space $\mathcal S$-matrix as correlation functions on the celestial sphere.

Second, de Sitter space is the simplest model of a universe with positive cosmological constant. Its asymptotic boundaries are spacelike, and a single observer has a cosmological horizon with finite entropy. This makes its holographic interpretation structurally different from AdS. The natural objects are not boundary time-ordered correlators on a Lorentzian cylinder, but late-time wavefunction coefficients, static-patch observables, horizon degrees of freedom, and possibly Euclidean or finite-dimensional dual descriptions.

Three asymptotic situations. In AdS, the boundary supports an ordinary Lorentzian CFT and the basic dictionary is $Z_{\rm CFT}[J]=Z_{\rm bulk}[\phi\to J]$. In flat space, the natural data live at null infinity and include the gravitational $\mathcal S$-matrix, BMS charges, and celestial amplitudes. In de Sitter, the natural data may be late-time wavefunction coefficients or static-patch/horizon degrees of freedom. The central difficulty beyond AdS is identifying the correct nonperturbative observables and Hilbert-space interpretation.

The punchline is not that flat-space or de Sitter holography is hopeless. It is that the AdS dictionary should not be copied mechanically. The correct lesson from AdS/CFT is not “put a CFT on every boundary.” The correct lesson is:

$$\boxed{ \text{Find the gauge-invariant gravitational observables, identify their asymptotic data, and ask what lower-dimensional structure computes them.} }$$

That sentence is a much better guide to research than the slogan “gravity equals field theory.”

Why AdS is unusually sharp

Global AdS$_{d+1}$ has a timelike conformal boundary with topology

$$\mathbb R_t\times S^{d-1}.$$

After imposing boundary conditions, the bulk problem behaves like a gravitational theory in a box. Signals can reach the boundary and return in finite global time. The boundary has a natural Lorentzian time coordinate, and the dual CFT has a Hamiltonian $H$ generating time translations on the cylinder.

The basic AdS/CFT observables are ordinary CFT observables:

$$Z_{\rm CFT}[J] = \left\langle \exp\!\left(\int J\mathcal O\right) \right\rangle \simeq \exp\!\left(-S_{\rm bulk}^{\rm ren}[\phi_{\rm cl}\to J]\right),$$

with the Lorentzian version giving real-time correlators after choosing the correct contour and interior boundary conditions. States, density matrices, thermal ensembles, modular Hamiltonians, and entanglement entropies all have standard boundary definitions.

This is not merely a convenience. It is why AdS/CFT can provide a nonperturbative definition of quantum gravity with AdS boundary conditions. The CFT Hilbert space exists independently of the bulk approximation. Classical gravity, string perturbation theory, black holes, Witten diagrams, hydrodynamics, and RT/HRT surfaces are calculational regimes inside that exact quantum system.

For non-AdS asymptotics, some of these ingredients fail or change form:

bulk asymptotics	boundary type	natural data	immediate difficulty
AdS	timelike conformal boundary	CFT correlators and states	technically hard, but conceptually sharp
flat	null infinity $\mathscr I^\pm$	$\mathcal S$-matrix, soft charges, memory, radiative data	no ordinary timelike boundary QFT
de Sitter	spacelike $\mathcal I^\pm$ or observer horizon	wavefunction coefficients, static-patch observables	no external boundary time; finite horizon entropy

The problem is not “find the AdS/CFT dictionary again.” The problem is to understand which part of the AdS logic survives after the asymptotic structure changes.

The flat-space limit of AdS/CFT

There is a conservative way to discuss flat-space quantum gravity using AdS/CFT: start with an AdS duality and take the AdS radius large in Planck or string units.

Locally, any large AdS space looks approximately flat. A bulk excitation of energy $E_{\rm flat}$ corresponds to a CFT state with large dimension

$$\Delta \sim L_{\rm AdS} E_{\rm flat},$$

so the flat-space limit is a high-energy, large-dimension limit of CFT data. In this regime, CFT correlators can contain the flat-space bulk $\mathcal S$-matrix.

A useful language is Mellin space. For a CFT four-point function, the Mellin amplitude $\mathcal M(\delta_{ij})$ behaves in many ways like a scattering amplitude: it has factorization poles, residues controlled by exchanged states, and a flat-space limit in which Mellin variables scale like bulk Mandelstam invariants. Schematically,

$$\mathcal M(\delta_{ij}) \quad \xrightarrow[L_{\rm AdS}\to\infty]{} \quad \mathcal T_{\rm flat}(s_{ij}),$$

where $\mathcal T_{\rm flat}$ is a flat-space transition amplitude and $s_{ij}$ are flat-space kinematic invariants. The exact normalization and transform depend on dimension, external masses, and conventions, but the conceptual point is robust: flat-space scattering can be extracted from a suitable scaling limit of AdS/CFT correlators.

This is powerful, but it is not the same as an intrinsic flat-space holographic dual. The flat-space limit of AdS/CFT answers a question of the form:

$$\text{Given a family of AdS/CFT dualities, how do we recover approximately flat bulk scattering?}$$

Intrinsic flat-space holography asks a sharper question:

$$\text{What lower-dimensional theory directly defines asymptotically flat quantum gravity?}$$

These questions are related, but not identical. The first uses AdS as a regulator. The second tries to formulate holography natively at null infinity.

Asymptotically flat space and null infinity

In four-dimensional asymptotically flat gravity, future null infinity $\mathscr I^+$ is reached by taking $r\to\infty$ at fixed retarded time $u=t-r$ and angle $(z,\bar z)$ on the celestial sphere. Near $\mathscr I^+$, the metric has the Bondi form

$$ds^2 = -du^2-2du\,dr +r^2\gamma_{z\bar z}dz\,d\bar z +\cdots,$$

where $\gamma_{z\bar z}$ is the round metric on $S^2$, and the omitted terms encode radiative data, mass aspect, angular momentum aspect, and subleading fields.

Unlike the AdS boundary, $\mathscr I^+$ is null. It is not a spacetime on which an ordinary relativistic QFT lives. It is a null hypersurface whose natural coordinates are

$$(u,z,\bar z).$$

The gravitational $\mathcal S$-matrix maps incoming data on $\mathscr I^-$ to outgoing data on $\mathscr I^+$:

$$\mathcal S: \mathcal H_{\mathscr I^-} \longrightarrow \mathcal H_{\mathscr I^+}.$$

This is the flat-space analog of the AdS question “what does the boundary theory compute?” But the answer is different: the most direct observable is the scattering map, not a finite-volume thermal partition function or local Hamiltonian evolution on a timelike boundary.

BMS symmetry

The asymptotic symmetry group of four-dimensional asymptotically flat gravity is larger than the Poincaré group. The Bondi-Metzner-Sachs group contains ordinary Lorentz transformations together with angle-dependent translations called supertranslations:

$$\delta_f u=f(z,\bar z),\qquad \delta_f z=\delta_f\bar z=0.$$

Here $f(z,\bar z)$ is an arbitrary function on the celestial sphere. In extended versions, the Lorentz transformations may also be enlarged to local conformal transformations or more general diffeomorphisms of the sphere, with corresponding subtleties about regularity and boundary conditions.

A deep modern lesson is the triangular relation

$$\boxed{ \text{soft theorem} \quad \Longleftrightarrow \quad \text{asymptotic-symmetry Ward identity} \quad \Longleftrightarrow \quad \text{memory effect}. }$$

This is one of the cleanest pieces of flat-space holography. It says that the infrared structure of gravitational scattering is controlled by symmetries living at null infinity. The Weinberg soft graviton theorem becomes a Ward identity for supertranslation symmetry; subleading soft theorems are related to angular-momentum-like extensions; and memory effects are the observable imprint of transitions between asymptotic vacua.

This is already holographic in spirit: a bulk scattering process is constrained by charges at infinity. But it is not yet a full microscopic dual in the same sense as AdS/CFT. It controls universal infrared data, not by itself the complete nonperturbative quantum gravity Hilbert space.

Celestial holography

Celestial holography takes a beautiful next step. Instead of writing massless scattering amplitudes in an energy-momentum basis, one writes them in a conformal basis adapted to the celestial sphere.

A null momentum in four dimensions can be parameterized as

$$p^\mu=\omega q^\mu(z,\bar z), \qquad q^2=0, \qquad \omega>0,$$

where $(z,\bar z)$ labels a point on the sphere. One convenient choice is

$$q^\mu(z,\bar z) = \frac{1}{1+z\bar z} \left( 1+z\bar z, z+\bar z, -i(z-\bar z), 1-z\bar z \right).$$

The celestial amplitude is a Mellin transform over the external energies:

$$\widetilde{\mathcal A}(\Delta_i,J_i;z_i,\bar z_i) = \prod_i \int_0^\infty d\omega_i\,\omega_i^{\Delta_i-1} \mathcal A(\omega_i,z_i,\bar z_i;J_i).$$

Here $J_i$ is the four-dimensional helicity, while the two-dimensional conformal weights are

$$h_i=\frac{\Delta_i+J_i}{2}, \qquad \bar h_i=\frac{\Delta_i-J_i}{2}.$$

Lorentz transformations act as global conformal transformations on the celestial sphere:

$$SO(1,3)\simeq PSL(2,\mathbb C).$$

Thus celestial amplitudes transform like two-dimensional conformal correlators. Collinear limits of scattering amplitudes become OPE-like limits on the celestial sphere. Soft gravitons and photons become special low-dimension operators or currents. Infrared symmetries become current-algebra-like structures.

This is a remarkable reorganization of flat-space scattering. It is not, however, an ordinary unitary Euclidean two-dimensional CFT. Several features are unusual:

issue	why it matters
Lorentzian scattering origin	celestial correlators are Mellin transforms of $\mathcal S$-matrix elements, not ordinary Euclidean path-integral correlators
energy conservation	momentum-conserving delta functions become distributional structures in celestial variables
principal series	normalizable massless wavefunctions often use $\Delta=1+i\nu$ rather than real positive dimensions
infrared divergences	gravitational and gauge-theory amplitudes require careful soft dressing or inclusive definitions
massive particles	massive asymptotic states naturally live at timelike infinity, not only on $\mathscr I^\pm$
nonperturbative completion	perturbative amplitudes do not automatically define a complete Hilbert space

A useful attitude is therefore:

$$\boxed{ \text{Celestial holography is a conformal-basis formulation of flat-space scattering, and perhaps a route to a full dual.} }$$

The word “perhaps” is doing real work. The program has a large amount of structural evidence, but its final nonperturbative formulation is still being clarified.

Carrollian holography and the null-boundary viewpoint

Celestial holography is a codimension-two perspective: the celestial sphere is the space of null directions. There is also a codimension-one perspective in which the putative dual theory lives directly on null infinity $\mathscr I^+$, with coordinates $(u,z,\bar z)$.

A null boundary has a degenerate metric structure. The corresponding symmetry is not an ordinary relativistic conformal symmetry but a Carrollian/BMS-like symmetry. Roughly, Carrollian structures arise when the speed of light is taken to zero in the boundary theory, so that time and space scale in a way adapted to null hypersurfaces.

The contrast is useful:

approach	boundary arena	natural variables	strength
celestial	celestial sphere $S^2$	$(\Delta,J,z,\bar z)$	makes Lorentz symmetry look like 2d conformal symmetry
Carrollian/null	null infinity $\mathscr I$	$(u,z,\bar z)$	keeps retarded time and radiative phase space manifest
AdS flat limit	large-radius AdS boundary	large $\Delta$, Mellin variables	derives flat amplitudes from known AdS/CFT structures

These are not necessarily competing pictures. They may be different transforms or regimes of the same underlying flat-space holographic structure.

de Sitter as a holographic problem

de Sitter space is qualitatively different from both AdS and flat space. In global coordinates, $(d+1)$-dimensional de Sitter has metric

$$ds^2 = -dt^2 +\ell_{\rm dS}^2\cosh^2\!\left(\frac{t}{\ell_{\rm dS}}\right)d\Omega_d^2.$$

The future and past boundaries, $\mathcal I^+$ and $\mathcal I^-$, are spacelike. In planar coordinates,

$$ds^2 = \frac{\ell_{\rm dS}^2}{\eta^2} \left( -d\eta^2+d\vec x^{\,2} \right), \qquad \eta<0,$$

future infinity is at $\eta\to0^-$. This looks formally similar to Euclidean AdS after analytic continuation, but the physical interpretation is different: $\eta$ is time, not a radial coordinate.

A single de Sitter observer sees only a static patch,

$$ds^2 = -\left(1-\frac{r^2}{\ell_{\rm dS}^2}\right)dt^2 + \frac{dr^2}{1-r^2/\ell_{\rm dS}^2} +r^2d\Omega_{d-1}^2.$$

There is a cosmological horizon at $r=\ell_{\rm dS}$ with temperature and entropy

$$T_{\rm dS}=\frac{1}{2\pi\ell_{\rm dS}}, \qquad S_{\rm dS} = \frac{A_{\rm hor}}{4G_{d+1}} = \frac{\Omega_{d-1}\ell_{\rm dS}^{d-1}}{4G_{d+1}}.$$

This entropy is one of the deepest clues. In AdS, the boundary CFT has an ordinary Hilbert space whose high-energy states account for large AdS black holes. In a single de Sitter static patch, the horizon entropy suggests a finite number of accessible states of order

$$\dim\mathcal H_{\rm patch} \sim \exp(S_{\rm dS}),$$

at least in a coarse semiclassical sense. How to make this precise is one of the central questions in de Sitter quantum gravity.

The dS/CFT wavefunction idea

One influential proposal relates quantum gravity in de Sitter to a Euclidean CFT living at future infinity. The most common schematic formula is

$$\boxed{ \Psi_{\rm dS}[\varphi_0] \sim Z_{\rm CFT}[\varphi_0]. }$$

Here $\Psi_{\rm dS}$ is a late-time wavefunction of the universe, and $\varphi_0$ denotes boundary data for bulk fields near $\mathcal I^+$. This formula resembles the AdS/CFT relation, but the interpretation is different. In AdS, the boundary value is a source in a Lorentzian or Euclidean CFT partition function. In de Sitter, the object is a cosmological wavefunction, and probabilities are obtained from something like

$$P[\varphi_0]\sim |\Psi_{\rm dS}[\varphi_0]|^2,$$

not directly from a single Euclidean generating functional.

A scalar field illustrates the difference. In planar de Sitter, the late-time homogeneous scalar equation gives

$$\eta^2\phi''-(d-1)\eta\phi' +m^2\ell_{\rm dS}^2\phi=0.$$

With $\phi\sim(-\eta)^\Delta$, one finds

$$\Delta(\Delta-d)+m^2\ell_{\rm dS}^2=0,$$

so

$$\Delta_\pm = \frac d2 \pm \sqrt{\frac{d^2}{4}-m^2\ell_{\rm dS}^2}.$$

For sufficiently heavy stable fields, the square root becomes imaginary. This is one reason de Sitter holography is not expected to be a straightforward unitary Euclidean CFT in the usual sense.

There is nevertheless a productive analytic-continuation relation between Euclidean AdS and de Sitter wavefunction coefficients. Very schematically,

$$z=-i\eta, \qquad L_{\rm AdS}=i\ell_{\rm dS},$$

maps certain Euclidean AdS calculations to de Sitter late-time data. This is useful for perturbative computations of cosmological correlators. But analytic continuation is not by itself a nonperturbative definition of quantum gravity in de Sitter.

Static-patch holography

A different de Sitter perspective focuses on the causal patch of a single observer. This is closer in spirit to black-hole thermodynamics. The observer has a horizon, a temperature, and a finite entropy. Instead of asking for a theory living at $\mathcal I^+$, one asks for a microscopic description of the static patch.

The static patch viewpoint emphasizes:

ingredient	de Sitter static-patch version
horizon	cosmological horizon at $r=\ell_{\rm dS}$
temperature	$T=1/(2\pi\ell_{\rm dS})$
entropy	$A/(4G)$
observables	worldline correlators, horizon response, static-patch density matrix
puzzle	finite entropy versus exact de Sitter symmetry

A useful analogy is the AdS black hole. An exterior observer outside an AdS black hole sees a horizon and thermal physics, but the full nonperturbative definition comes from the boundary CFT. In de Sitter, there is no analogous external timelike boundary. The horizon is not hiding behind an AdS boundary theory; it is part of the observer’s universe.

This makes the Hilbert-space question sharper. Is the static patch described by a finite-dimensional Hilbert space? If so, how is exact de Sitter invariance realized, given that continuous noncompact symmetries act poorly on finite-dimensional unitary Hilbert spaces? Or is the finite entropy only an effective thermodynamic statement about a larger structure? Different proposals answer these questions differently.

Comparing the three cases

Here is a compact comparison that is useful when reading the literature.

feature	AdS	flat space	de Sitter
cosmological constant	$\Lambda<0$	$\Lambda=0$	$\Lambda>0$
asymptotic boundary	timelike	null	spacelike, or observer horizon
natural observable	boundary correlators	$\mathcal S$-matrix and asymptotic charges	wavefunction coefficients or static-patch observables
symmetry	$SO(2,d)$	Poincaré/BMS	$SO(1,d+1)$
lower-dimensional structure	Lorentzian CFT	celestial CFT or Carrollian/BMS theory	Euclidean CFT-like object or finite/horizon system
entropy clue	black holes have $S\sim A/4G$	black holes and soft sectors	cosmological horizon has $S_{\rm dS}=A/4G$
sharpest achievement	nonperturbative AdS quantum gravity	infrared symmetry/soft theorem/memory structure; amplitude reformulations	perturbative wavefunction technology; horizon thermodynamics; partial models
central difficulty	finite $N$, finite $\lambda$ dynamics	full nonperturbative dual of the $\mathcal S$-matrix	Hilbert space, observables, unitarity, finite entropy

This table should make one thing clear: AdS is not the generic case. AdS is the case where the boundary structure happens to match the standard machinery of quantum field theory extraordinarily well.

Open problems as precise research questions

The phrase “open problem” is often too vague. For this topic, the useful open problems are concrete.

1. What are the exact observables?

In AdS, the answer is CFT observables. In flat space, the perturbative answer is the $\mathcal S$-matrix, but quantum gravity complicates local asymptotic states through soft radiation, memory, infrared dressing, and black-hole production. In de Sitter, neither an $\mathcal S$-matrix nor boundary time evolution is available in the usual way.

A precise research question is:

$$\text{What replaces } Z_{\rm CFT}[J]\text{ as the exact object that defines the theory?}$$

For flat space, candidates include dressed $\mathcal S$-matrix data, celestial correlators, or null-boundary phase-space observables. For de Sitter, candidates include wavefunction coefficients, static-patch density matrices, horizon algebras, and observer-centered quantum systems.

2. What is the Hilbert space?

In AdS/CFT, the CFT Hilbert space is the Hilbert space of the theory. In flat space, scattering states are defined at asymptotic infinity, but massless gauge theories and gravity require careful infrared dressing. In de Sitter, a finite horizon entropy suggests a finite number of static-patch states, but global de Sitter symmetry suggests a tension with finite-dimensional unitary representations.

The question is not merely philosophical. The Hilbert space determines which questions are well-posed.

3. How does bulk locality emerge?

In AdS, bulk locality is tied to large $N$, a sparse spectrum, HKLL reconstruction, entanglement wedges, and quantum error correction. In flat space, one must explain how local bulk scattering and causal propagation emerge from celestial or null-boundary data. In de Sitter, one must explain local physics inside an observer patch without an external boundary system of the AdS type.

4. How are horizons encoded?

Flat space has black-hole horizons. De Sitter has a cosmological horizon even in the vacuum. Holography should explain horizon entropy microscopically, not merely reproduce $A/(4G)$ semiclassically.

For AdS black holes, the dual CFT gives a precise statistical framework. For de Sitter, the analogous microscopic counting remains much less settled.

5. What is the role of string theory?

Many AdS examples come from controlled brane constructions and compactifications. Flat-space string theory has a perturbative $\mathcal S$-matrix, but an intrinsic holographic dual is not as explicit as AdS/CFT. de Sitter compactifications and metastable positive-vacuum constructions are technically subtle and conceptually debated. A complete holographic formulation should explain which backgrounds are allowed and which semiclassical effective descriptions are actually realized in quantum gravity.

Common mistakes

Mistake 1: “Every gravitational boundary has a CFT.”

The boundary geometry matters. A timelike AdS boundary naturally supports an ordinary Lorentzian CFT. Null infinity and spacelike de Sitter infinity require different structures.

Mistake 2: “Celestial amplitudes are just ordinary 2d CFT correlators.”

They transform like conformal correlators under the Lorentz group, but they come from Mellin transforms of scattering amplitudes. Distributional support, infrared divergences, principal-series dimensions, and unitarity are different from textbook compact Euclidean CFT.

Mistake 3: “The flat-space limit of AdS/CFT already solves flat-space holography.”

It gives a way to extract flat-space scattering from a family of AdS dualities. It does not automatically identify a native lower-dimensional theory that nonperturbatively defines asymptotically flat quantum gravity.

Mistake 4: “dS/CFT is AdS/CFT with $L\to iL$.”

Analytic continuation is useful for perturbative wavefunction calculations, but it does not settle the Hilbert-space interpretation, probabilities, static-patch physics, horizon entropy, or unitarity.

Mistake 5: “Finite de Sitter entropy means we already know the microscopic Hilbert space.”

The entropy is a profound clue. Turning it into a complete microscopic theory is a much harder task.

A practical reading strategy

For readers trained in AdS/CFT, the best way to enter this literature is not to start with the most speculative claims. Start from the most controlled structures:

1. Flat-space limit of AdS/CFT: learn how bulk scattering is encoded in high-energy limits of CFT correlators.
2. BMS/soft/memory triangle: learn the exact infrared constraints on gravitational scattering.
3. Celestial amplitudes: learn the Mellin transform from momentum-space amplitudes to celestial correlators.
4. de Sitter wavefunction: learn perturbative late-time cosmological correlators and their analytic continuation from Euclidean AdS.
5. Static patch and entropy: learn why de Sitter horizons make the Hilbert-space problem unavoidable.

This route keeps the speculative parts connected to calculational facts.

Exercises

Exercise 1: Boundary type and observable

For each spacetime, identify the boundary type and the most natural class of observables: global AdS, asymptotically flat Minkowski space, and global de Sitter.

Solution

Global AdS has a timelike conformal boundary, usually $\mathbb R_t\times S^{d-1}$, and the natural observables are CFT correlation functions, states, partition functions, and density matrices.

Asymptotically flat Minkowski space has null infinity $\mathscr I^\pm$. The natural perturbative observable is the $\mathcal S$-matrix mapping incoming data on $\mathscr I^-$ to outgoing data on $\mathscr I^+$. Soft charges, memory effects, and BMS Ward identities are also natural asymptotic observables.

Global de Sitter has spacelike future and past infinity, $\mathcal I^+$ and $\mathcal I^-$. Natural objects include late-time wavefunction coefficients and, from an observer-centered viewpoint, static-patch observables associated with the cosmological horizon.

Exercise 2: A null vector on the celestial sphere

Show that

$$q^\mu(z,\bar z) = \frac{1}{1+z\bar z} \left( 1+z\bar z, z+\bar z, -i(z-\bar z), 1-z\bar z \right)$$

is null in mostly-plus signature $(-,+,+,+)$.

Solution

Let $z=x+iy$, so $z\bar z=x^2+y^2$. Then

$$q^0=1, \qquad q^1=\frac{2x}{1+x^2+y^2}, \qquad q^2=\frac{2y}{1+x^2+y^2}, \qquad q^3=\frac{1-x^2-y^2}{1+x^2+y^2}.$$

The spatial norm is

$$(q^1)^2+(q^2)^2+(q^3)^2 = \frac{4x^2+4y^2+(1-x^2-y^2)^2}{(1+x^2+y^2)^2}.$$

Writing $\rho^2=x^2+y^2$, the numerator is

$$4\rho^2+(1-\rho^2)^2 =1+2\rho^2+\rho^4 =(1+\rho^2)^2.$$

Thus the spatial norm is $1$, so

$$q^2=-(q^0)^2+|\vec q|^2=-1+1=0.$$

Exercise 3: de Sitter late-time dimensions

For a massive scalar in planar de Sitter,

$$ds^2 =\frac{\ell^2}{\eta^2} \left(-d\eta^2+d\vec x^{\,2}\right),$$

the late-time homogeneous equation is

$$\eta^2\phi''-(d-1)\eta\phi'+m^2\ell^2\phi=0.$$

Assume $\phi\sim(-\eta)^\Delta$ and derive the two exponents $\Delta_\pm$.

Solution

Substituting $\phi\sim(-\eta)^\Delta$ gives

$$\eta^2\phi''-(d-1)\eta\phi' = \left[\Delta(\Delta-1)-(d-1)\Delta\right]\phi = \Delta(\Delta-d)\phi.$$

The equation becomes

$$\Delta(\Delta-d)+m^2\ell^2=0.$$

Therefore

$$\Delta_\pm = \frac d2 \pm \sqrt{\frac{d^2}{4}-m^2\ell^2}.$$

For $m^2\ell^2>d^2/4$, the exponents are complex. This is one reason the dS/CFT interpretation differs from an ordinary unitary Euclidean CFT with real scaling dimensions.

Exercise 4: Flat limit versus intrinsic flat holography

Explain the difference between extracting a flat-space $\mathcal S$-matrix from the large-radius limit of AdS/CFT and constructing an intrinsic holographic dual of asymptotically flat quantum gravity.

Solution

In the large-radius limit of AdS/CFT, one begins with a well-defined AdS/CFT duality. The CFT lives on the AdS boundary, and suitable high-energy or large-dimension limits of CFT correlators reproduce approximately flat bulk scattering. AdS is being used as a regulator and as a nonperturbative definition.

An intrinsic flat-space holographic dual would instead define asymptotically flat quantum gravity directly in terms of lower-dimensional data at null infinity, without embedding the problem into a family of AdS spacetimes. Candidate languages include celestial correlators, Carrollian theories on $\mathscr I$, BMS charge algebras, and dressed $\mathcal S$-matrix data. The two ideas are related, but the second is conceptually stronger.

Exercise 5: The soft theorem triangle

Give a conceptual explanation of the relation

$$\text{soft theorem} \Longleftrightarrow \text{asymptotic symmetry Ward identity} \Longleftrightarrow \text{memory effect}.$$

Solution

A soft theorem describes the universal behavior of a scattering amplitude when an emitted gauge boson or graviton has vanishing energy. In gravity, the leading soft graviton theorem controls the insertion of a very low-energy graviton.

An asymptotic symmetry has an associated charge at null infinity. The Ward identity for that charge constrains the $\mathcal S$-matrix. For BMS supertranslations, the Ward identity is equivalent to the leading soft graviton theorem.

A memory effect is a classical observable: after radiation passes through null infinity, detectors can be permanently displaced or clocks shifted. This permanent change is the classical imprint of the same asymptotic charge transition. Thus the same structure appears as a quantum soft theorem, a symmetry Ward identity, and a measurable classical memory effect.

Further reading

• Andrew Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory.
• Sabrina Pasterski, Lectures on Celestial Amplitudes.
• Ana-Maria Raclariu, Lectures on Celestial Holography.
• Laura Donnay, Celestial Holography: An Asymptotic Symmetry Perspective.
• A. Liam Fitzpatrick, Jared Kaplan, João Penedones, Suvrat Raju, and Balt van Rees, A Natural Language for AdS/CFT Correlators.
• Takuya Okuda and João Penedones, String Scattering in Flat Space and a Scaling Limit of Yang-Mills Correlators.
• Andrew Strominger, The dS/CFT Correspondence.
• Dionysios Anninos, De Sitter Musings.
• Arjun Bagchi, Shankhadeep Banerjee, Rudranil Basu, and Sudipta Dutta, Holography in Flat Spacetimes: The Case for Carroll.