Approximate Bulk Locality from Spectral and Mellin Data

Bulk locality is not read from one OPE coefficient, one pole, or one successful polynomial fit. At first nontrivial order in a large-$N$ expansion, it is inferred from a package of data: an approximately Fock-like light sector, a parametrically separated heavy scale, factorization, correctly factorizing Mellin poles, a controlled contact expansion, and suitable boundedness in the complex kinematic domains used by the reconstruction. Under those hypotheses, low-degree Mellin polynomials encode finitely many on-shell AdS contact structures. The lowest physical derivative cutoff estimates the first omitted contact term, while any lower fixed-order stop further restricts the usable domain. The conclusion is approximate locality for specified observables below a cutoff, not an exact local bulk at all energies.

Required background. Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria defines the single-trace gap used here; Bulk Interaction Scaling and Effective Cutoffs supplies the bulk power counting; and Mellin-Space CFT Correlators supplies the Mellin representation and crossing conventions.

Helpful background. Large-Gap Constraints and CFT-Side Locality Tests develops the CFT constraints in detail, while The CFT Regge Limit and Boundedness explains why boundedness is independent information.

Mellin conventions and the tree-level dictionary

Let $\mathcal O_\phi$ be an identical scalar single-trace primary of dimension $\Delta_\phi$, with unit-normalized two-point function. The two cross-ratios $U,V$ contain the position dependence left after removing the standard prefactor. The Gamma-function measure $\mu_{\Delta_\phi}$ carries universal kinematic pole families, while $\mathcal M$ is the dynamical Mellin amplitude. The locality question concerns the pole-subtracted analytic part of $\mathcal M$ and its asymptotic behavior, not the measure by itself. In the convention of the prerequisite Mellin chapter,

$$\mathcal G_{\mathrm{conn}}(U,V) = \int_{\mathcal C}\frac{ds\,dt}{(4\pi i)^2} U^{s/2}V^{t/2-\Delta_\phi} \mu_{\Delta_\phi}(s,t)\,\mathcal M(s,t),$$

where the dimensionless third Mellin variable and measure are

$$u_{\mathrm M}=4\Delta_\phi-s-t,$$ $$\begin{aligned} \mu_{\Delta_\phi}(s,t) ={}&\Gamma^2\!\left(\Delta_\phi-\frac{s}{2}\right) \Gamma^2\!\left(\Delta_\phi-\frac{t}{2}\right) \\ &\times \Gamma^2\!\left(\Delta_\phi-\frac{u_{\mathrm M}}{2}\right). \end{aligned}$$

On the initial Euclidean sheet, $\mathcal C$ consists of vertical contours chosen so that $\operatorname{Re}s$, $\operatorname{Re}t$, and $\operatorname{Re}u_{\mathrm M}$ are all less than $2\Delta_\phi$, giving the three Gamma-function arguments positive real parts and keeping the contour away from their pole families. This condition alone does not guarantee convergence: the asymptotic behavior of $\mathcal M$ and separation from dynamical poles also matter. Analytic continuation can require contour deformations and the addition of crossed residues. Stating the contour and sheet is therefore part of a reproducible Mellin claim, not a cosmetic convention.

The separation between the measure and $\mathcal M$ prevents a common misreading. The measure has poles at $s=2\Delta_\phi+2n$ and in the crossed channels; these generate the generalized-free double-trace families. At leading nontrivial order in large-$N$ factorization, exchange of a single-trace primary $\mathcal X$ of dimension $\Delta_{\mathcal X}$ and spin $J_{\mathcal X}$ can give dynamical poles at the possible locations

$$s=\tau_{\mathcal X}+2m, \qquad \tau_{\mathcal X}=\Delta_{\mathcal X}-J_{\mathcal X}, \qquad m=0,1,2,\ldots,$$

with degree-$J_{\mathcal X}$ Mack-polynomial residues in the transverse Mellin variable; special satellite residues can vanish. At bulk loop order, multi-trace operators also produce dynamical singularities in $\mathcal M$; “Mellin poles equal single-trace exchange” is only a tree-level statement.

At the same tree order, polynomial terms have no exchange poles and are Mellin images of local AdS contact diagrams. A scalar contact vertex with $2k$ covariant derivatives produces a polynomial of degree at most $k$; its degree-$k$ part can vanish or mix with lower-degree curvature and equation-of-motion terms. For identical scalars, crossing requires symmetry under permutations of $s,t,u_{\mathrm M}$. Since their sum is fixed, a symmetric linear polynomial is only a constant. The first few independent structures can therefore be represented as

$$\mathcal M_{\mathrm{contact}} =a_0 +a_2\left(s^2+t^2+u_{\mathrm M}^2\right) +a_3\,st u_{\mathrm M}+\cdots.$$

The tree-level contact calculation is explicit in Penedones 2011, § 2.1. After quotienting by integrations by parts, leading equations of motion, and local field redefinitions, each independent polynomial labels an on-shell contact structure, not a unique off-shell vertex. A “contact ambiguity” is an ambiguity only when one reconstructs an amplitude from incomplete pole or discontinuity data; a fully specified correlator itself is not ambiguous.

What the reconstruction results establish

Several results often compressed into “large gap implies locality” have different scopes.

• Heemskerk, Penedones, Polchinski, and Sully 2009, §§ 2.4, 4, and 7–8 formulate a broad locality conjecture and solve the scalar four-point crossing problem at first nontrivial order in $1/N^2$ in the bounded-spin setting, then discuss convergence and inclusion of the stress tensor. This is not a theorem that arbitrary sparse CFT data define a local quantum gravity theory.
• Fitzpatrick and Kaplan 2013, §§ 1–3 formulate perturbative large-$N$ structure, an approximate low-energy Fock space, and polynomially bounded Mellin amplitudes as separate criteria for an AdS EFT. Their argument treats polynomial boundedness as an additional input; the paper gives evidence, not a general proof, that unitarity plus a gap enforces it.
• Caron-Huot, Mazáč, Rastelli, and Simmons-Duffin 2021, § 1.1.1, pp. 5–6; § 2.4, pp. 18–19 and footnote 9; §§ 5.1–5.4, pp. 48–63 (PDF) derive sharp gap-suppressed bounds on quartic Wilson coefficients in a specified leading-order setup with a light scalar and stress tensor below a large higher-spin twist gap. Their derivation also states technical assumptions concerning second-sheet collinear behavior and the truncation used to construct the dispersive functionals. The bounds are much stronger than dimensional analysis, but their hypotheses should not be silently generalized to every spectrum, correlator, or loop order.
• Bhat and Zahed 2023, §§ 1 and 4.1–4.2 obtain analytic two-sided bounds for many scalar AdS Wilson coefficients from a celestial transform and crossing-symmetric Mellin dispersion relation. Their result assumes the stated weakly coupled scalar setup and critical-twist conditions; it is not an assumption-free test for arbitrary correlators.
• Chang, Landau, and Simmons-Duffin 2025, §§ 1 and 4–5 construct the spinning dispersive-sum-rule dictionary relating stress-tensor correlators to graviton dispersion relations. They explicitly leave light-state matching and additional non-shock graviton sum rules for future work, so scalar four-point bounds do not by themselves complete the Einstein-sector locality argument.

Thus, the safe inference on this page is conditional and perturbative: when the light sector, factorization order, subtraction scheme, Mellin domain, boundedness assumptions, and error controls are all stated, the correlator can admit a local AdS EFT description to the tested order.

A low-degree polynomial and its first omitted term

First subtract the explicitly retained light exchanges in all channels, using a declared prescription, and call the remainder $\mathcal M_{\mathrm{res}}$. Suppose it is approximated on a compact complex Mellin domain, kept a fixed distance from exchange poles, by a crossing-symmetric polynomial $P_K$ of degree at most $K$. Let $Q$ be the largest local momentum or curvature scale sampled in the bulk, with the boundary-to-local redshift specified as on the power-counting prerequisite.

Two scales must be kept distinct. Let $\Lambda_{\mathrm{der}}$ be the lowest physical omitted-particle, omitted-tower, or genuine UV strong-coupling scale that controls the local derivative expansion, and define $\Delta_{\mathrm{der}}=L\Lambda_{\mathrm{der}}$. Separately, let $\Lambda_{\mathrm{stop}}$ be the first scale at which the particular fixed-order calculation loses control. A lower loop crossing or a large logarithm can set $\Lambda_{\mathrm{stop}}$ and demand resummation or reorganization without becoming a physical derivative cutoff. The physical expansion requires

$$\frac{QL}{\Delta_{\mathrm{der}}} =\frac{Q}{\Lambda_{\mathrm{der}}}\ll1, \qquad Q<\Lambda_{\mathrm{stop}}\leq\Lambda_{\mathrm{der}}.$$

In the fixed-$\Delta_\phi$, massless-external flat-space limit used in Penedones’s Eqs. (9) and (33)–(34), $S_{ij}=2\beta s_{ij}/L^2$ under an auxiliary $\beta$ integral; the inverse transform instead samples $s_{ij}=L^2S_{ij}/(2\alpha)$. Here $S_{ij}$ are dimensionful flat-space Mandelstam invariants, $s_{ij}$ are dimensionless Mellin invariants, and $\alpha,\beta$ are auxiliary inverse- and forward-transform integration variables. If $\Delta_\phi$ scales with $L$ so that the external bulk mass remains finite, the corresponding massive-external prescription and saddle must be stated separately. Thus $s_{ij}=O((QL)^2)$ is transform or saddle power counting after the flat-space limit and complex direction have been specified, not a pointwise pole-free Mellin domain. Independently, the compact reconstruction domain must remain a positive distance from every pole that has not been explicitly subtracted Penedones 2011, Eqs. (9), (33)–(34), and § 2.2 (PDF).

The derivative cutoff is the lowest relevant physical obstruction: a higher-spin or additional-scalar threshold, a Kaluza–Klein or string tower, or genuine UV strong coupling may win. Only when the first omitted heavy single-trace state is the lowest such obstruction may one identify $\Delta_{\mathrm{der}}\simeq\Delta_{\mathrm{gap}}$. Species enhancement can lower $\Lambda_{\mathrm{stop}}$ through loops without by itself deciding that identification; only when the loop failure signals a genuine physical cutoff does it also lower $\Lambda_{\mathrm{der}}$.

Integrating out one heavy field illustrates the expansion and its remainder. Let $\widehat s$ be a dimensionful flat-patch Mandelstam invariant and let $M_H$ be the heavy mass. Then

$$\frac{1}{M_H^2-\widehat s} = \frac{1}{M_H^2} \sum_{n=0}^{K} \left(\frac{\widehat s}{M_H^2}\right)^n +R_{K+1}(\widehat s),$$

with

$$R_{K+1}(\widehat s) = \frac{\widehat s^{K+1}} {M_H^{2K+2}\left(M_H^2-\widehat s\right)}.$$

If $\lvert\widehat s\rvert\leq\eta M_H^2$ with $0<\eta<1$, the geometric series gives the actual bound

$$\left\lvert M_H^2R_{K+1}\right\rvert \leq \frac{\eta^{K+1}}{1-\eta}.$$

An AdS exchange has a complete Mellin pole ladder. After matching its low-local-energy contribution with a declared flat-space or wavepacket prescription and restricting away from unretained singularities, that contribution admits a derivative expansion. Merely being “below the first Mellin pole” is not equivalent to $Q\ll M_H$. If the highest retained contact structure has $2K$ derivatives, compare the first omitted term with a nonzero retained contact contribution carrying $2k_0$ derivatives:

$$\frac{\delta\mathcal M_{K+1}}{\mathcal M_{k_0}} = O\!\left[ \frac{c_{K+1}}{c_{k_0}} \left(\frac{QL}{\Delta_{\mathrm{der}}}\right)^{2(K+1-k_0)} \right].$$

The familiar $2K+2$ power applies only when $k_0=0$. If the leading amplitude is exchange-dominated, its momentum dependence must instead be retained explicitly. The gap alone neither fixes $c_{K+1}/c_{k_0}$ nor proves the estimate is a rigorous bound. One needs matching information, positivity or dispersive bounds, or an explicit natural-coefficient assumption.

For example, consider

$$P_2(s,t,u_{\mathrm M}) =a_0+a_2\left(s^2+t^2+u_{\mathrm M}^2\right).$$

It contains two on-shell contact structures: a zero-derivative term and one independent four-derivative term, modulo redundancies. There is no new symmetric two-derivative structure because $s+t+u_{\mathrm M}$ is fixed. If $QL\leq\eta\Delta_{\mathrm{der}}$, $Q<\Lambda_{\mathrm{stop}}$, and Wilson coefficients are naturally normalized relative to a nonzero $a_0$, the first generic six-derivative term is of order $\eta^6$. A symmetry, a tuning, or $a_0=0$ changes that comparison; finite-$N$ loops are a separate expansion.

What the calculation licenses

The observable is the pole-subtracted connected scalar four-point Mellin amplitude at the first nontrivial large-$N$ order, restricted to the declared compact complex domain. The retained data are the two structures in $P_2$; the controls are $\eta=QL/\Delta_{\mathrm{der}}<1$, $Q<\Lambda_{\mathrm{stop}}$, and the assumed natural size of the first nonzero six-derivative coefficient. Under those conditions the first omitted contact contribution is $O[(c_3/c_0)\eta^6]$ relative to a nonzero zero-derivative term, reducing to $O(\eta^6)$ when the coefficient ratio is naturally of order one. This is an error estimate for one correlator and one order—not a rigorous gap-only bound, an all-correlator locality theorem, or a nonperturbative completion.

Poles, boundedness, and what each hypothesis contributes

A locality inference combines logically distinct observations:

• Sparse light data limit the number of bulk fields that must remain explicit below the cutoff.
• The lowest physical heavy or strong-coupling threshold sets the domain in which an expansion in $Q/\Lambda_{\mathrm{der}}$ is possible.
• Large-$N$ factorization suppresses connected correlators and organizes bulk loops.
• Exchange-pole residues test whether the same OPE coefficients consistently determine different channels.
• Polynomial boundedness in declared complex directions excludes analytic additions whose asymptotics cannot be represented at any fixed EFT order.
• Crossing and unitarity constrain, but do not by themselves prove, an approximately local bulk description.

At a fixed EFT order, one useful schematic condition is that, after the required pole subtractions and at fixed $t$ in a named strip,

$$\lvert\mathcal M_{\mathrm{sub}}(s,t)\rvert \leq C(t)\left(1+\lvert s\rvert\right)^p$$

as $\lvert s\rvert\to\infty$ in the specified complex directions and away from poles, with crossed-channel analogues. The subtraction order, strip, direction, and uniformity in $t$ are part of the assertion. A low-degree fit on a compact real patch does not test this asymptotic condition. Nonperturbatively, even defining the Mellin transform can require subtractions; the canonical Mellin Regge bound is first established along $\lvert\operatorname{Im}s\rvert\to\infty$ at fixed $t$ in a holomorphy strip, while extending it to other directions requires additional assumptions Penedones, Silva, and Zhiboedov 2020, §§ 2.5 and 4.3–4.4.

Mellin boundedness and the Lorentzian Regge limit are related but are not interchangeable tests. The latter is taken after continuation to the appropriate second sheet and constrains the full correlator or suitably resummed amplitude, not an isolated exchange diagram. In the tree-level eikonal and shockwave regime analyzed by Camanho, Edelstein, Maldacena, and Zhiboedov, the offending non-Einstein on-shell graviton three-point structures cannot generically be repaired by finitely many spin-$J\leq2$ exchanges. An infinite higher-spin tower is required in four bulk dimensions and whenever their $\alpha_4$ structure is nonzero; in bulk dimension $D>4$ with $\alpha_4=0$ and $\alpha_2\neq0$, they leave open repair by finitely many mixed-symmetry states Camanho et al. 2016, § 3.2, pp. 18–19; § 5.4, p. 41; §§ 8–8.1, pp. 47–50 (PDF). This is a scoped obstruction, not a proof that every large-gap CFT has an Einstein bulk; see Higher-Spin Gaps and Einstein-Regime Obstructions for the detailed claim boundary.

Adversarial amplitude with acceptable low-energy data

Let $\mathcal M_{\mathrm{poles}}$ be a crossing-symmetric baseline with acceptable factorizing poles and polynomially bounded asymptotics, and let $P_K$ reproduce the Taylor data being tested through total polynomial degree $K$. Center the Mellin variables at the crossing-symmetric point,

$$\begin{aligned} \sigma&=s-\frac{4\Delta_\phi}{3}, &\tau&=t-\frac{4\Delta_\phi}{3},\\ \rho&=u_{\mathrm M}-\frac{4\Delta_\phi}{3}, &\sigma+\tau+\rho&=0. \end{aligned}$$

For a dimensionless Mellin scale $S_\star>1/\pi$ and a nonzero coefficient $\varepsilon$, set $m=\lfloor K/2\rfloor$ and define the even Taylor remainder

$$\begin{aligned} C_m(z) &=\cos z-\sum_{n=0}^{m}\frac{(-1)^n z^{2n}}{(2n)!},\\ \delta\mathcal M &=\varepsilon\sum_{x\in\{\sigma,\tau,\rho\}} C_m\!\left(\frac{x}{S_\star}\right). \end{aligned}$$

This function is entire and crossing symmetric, introduces no poles, and begins at degree $2m+2>K$ about the symmetric point. Therefore it leaves every Taylor coefficient of total degree at most $K$ unchanged. A finite test cannot distinguish

$$\begin{aligned} \mathcal M_{\mathrm{good}} &=\mathcal M_{\mathrm{poles}}+P_K,\\ \mathcal M_{\mathrm{bad}} &=\mathcal M_{\mathrm{good}}+\delta\mathcal M \end{aligned}$$

to that accuracy. Now take the fixed-$t$ imaginary Mellin Regge ray

$$\sigma=iR, \qquad \tau=0, \qquad \rho=-iR, \qquad R\to+\infty.$$

The $\tau$ term vanishes, while the other two terms give

$$\begin{aligned} \delta\mathcal M &=2\varepsilon\Bigg[ \cosh\!\left(\frac{R}{S_\star}\right) \\ &\qquad-\sum_{n=0}^{m}\frac{(R/S_\star)^{2n}}{(2n)!} \Bigg],\\ \delta\mathcal M &\sim \varepsilon e^{R/S_\star}. \end{aligned}$$

Whenever this fixed-$t$ ray lies in the asserted holomorphy and Mellin Regge strip, the deformation violates every polynomial bound there. On this ray the Gamma measure falls as $e^{-\pi R}$ up to powers, so $S_\star>1/\pi$ leaves the standard initial-sheet vertical-contour integrand exponentially damped even though the amplitude is not polynomially bounded. If the ray does not lie in the asserted strip, that mismatch must itself be reported rather than silently calling it a Regge test.

This is an analytic Mellin-Regge counterfixture, not a claimed Mellin amplitude of a unitary CFT: crossing symmetry, pole data, and finitely many Taylor coefficients do not ensure reflection positivity, OPE positivity, or acceptable second-sheet behavior of a full correlator. Its precise lesson is limited but decisive: the tested finite data alone do not imply the missing boundedness hypothesis.

The strongest surviving statement is narrower: the tested correlator admits a finite-order low-energy contact parametrization in the sampled domain. Calling that parametrization a local bulk EFT additionally requires a gap, a uniform error bound, acceptable Regge behavior, and consistency across correlators.

Orders of limits and evidence ceiling

The useful hierarchy is

$$N\to\infty,\qquad \Delta_{\mathrm{der}}\to\infty,\qquad \frac{QL}{\Delta_{\mathrm{der}}}\to0,$$

but these operations answer different questions. At fixed normalized observables and fixed $QL$, the declared gravitational loop is suppressed only if $\lvert b_D\rvert N_{\mathrm{eff}}(Q)r_G(Q)/C_T\to0$, where $b_D$ is the dimension-dependent loop factor defined in the power-counting prerequisite; every nongravitational weighted coupling sum must be controlled separately. Large $N$ does not by itself suppress $\alpha'$ or string-scale, Kaluza–Klein, or other heavy-threshold corrections, and genus suppression depends on the model-specific $g_s$–$N$ dictionary. Taking $\Delta_{\mathrm{der}}$ large at fixed $Q$ suppresses a specified higher-derivative term only if its matched dimensionless coefficient remains bounded in that family; allowing $Q$ to scale with the derivative cutoff removes that suppression. A Regge limit can leave every fixed compact low-energy domain, so the derivative expansion must not be assumed uniform there. Finite-$N$, finite-derivative-cutoff, fixed-order-stop, pole-subtraction, and domain-truncation errors must be reported separately.

Common pitfalls

Calling every Mellin pole single-trace exchange. The Gamma measure already carries generalized-free double-trace poles, and loop-level Mellin amplitudes develop multi-trace dynamical singularities. State both the Mellin convention and the order in $1/N$.

Using the largest visible gap as the cutoff. The derivative expansion is controlled by the lowest relevant new scale. A high higher-spin gap does not remove a lower scalar or compactification threshold, a genuine strong-coupling cutoff, or a species-enhanced fixed-order stop.

Identifying a real-axis fit with a Regge bound. A compact fit supplies low-energy coefficients. Polynomial boundedness is a complex-asymptotic statement, while the CFT Regge limit is a second-sheet statement about the full correlator.

Exercises

1. Contact data versus a Lagrangian

A residual Mellin amplitude is $P=a+b(s^2+t^2+u_{\mathrm M}^2)$. Explain why determining $a$ and $b$ does not identify a unique off-shell bulk Lagrangian.

Solution — on-shell contact data

Correlators determine on-shell contact structures. Integrations by parts, the leading equations of motion, and local field redefinitions move coefficients among off-shell vertices while leaving the correlator invariant. The two coefficients identify two independent on-shell polynomial structures in the chosen normalization, not a unique Lagrangian basis.

2. First omitted interaction

Suppose the physical derivative cutoff is $\Delta_{\mathrm{der}}=20$, the fixed-order stop obeys $L\Lambda_{\mathrm{stop}}>4$, the sampled scale is $QL=4$, the retained polynomial includes four derivatives, and a nonzero zero-derivative coefficient sets the natural normalization. Estimate the first generic six-derivative contribution.

Solution — six-derivative estimate

The expansion ratio is $QL/\Delta_{\mathrm{der}}=0.2$. A six-derivative term is naturally of order

$$0.2^6=6.4\times10^{-5}$$

relative to a zero-derivative coefficient of comparable natural size. This is not a rigorous bound unless matching, positivity, or dispersion controls the next dimensionless Wilson coefficient.

3. Check the Mellin-Regge counterfixture

For $K=3$, determine $m$, identify the first nonzero polynomial degree of $\delta\mathcal M$, and show its growth on the fixed-$t$ ray $\sigma=iR$, $\tau=0$, $\rho=-iR$.

Solution — exponential imaginary-ray growth

Here $m=\lfloor3/2\rfloor=1$. Subtracting the $n=0$ and $n=1$ terms removes degrees zero and two, so

$$C_1(z)=\cos z-1+\frac{z^2}{2} =\frac{z^4}{24}+O(z^6).$$

The deformation therefore begins at degree four, beyond all data through degree three. On the stated ray, its two nonzero terms equal $C_1(\pm iR/S_\star)=\cosh(R/S_\star)-1-(R/S_\star)^2/2$, so the sum grows as $\varepsilon e^{R/S_\star}$ and cannot obey a polynomial bound there. This verifies only the analytic counterfixture; it does not establish unitary-CFT positivity or furnish a physical second-sheet correlator.

Evidence cutoff: 28 August 2026. The structural and dispersive results cited here support conditional perturbative reconstruction and, in specified scalar setups, gap-suppressed bounds on Wilson coefficients. The spinning stress-tensor dictionary is a major advance, but the cited 2025 analysis still leaves parts of the pure-gravity locality argument open. These results do not prove that an arbitrary CFT satisfying a few spectral tests has an exact local bulk, determine a nonperturbative completion, or convert a polynomial fit into evidence for quantum gravity in nature. Continue with Corrections, Nonuniform Limits, and Failure Modes for the finite-parameter error budget and with CFT Criteria for Approximate Bulk Locality for the later multi-correlator and dispersive analysis. Operator-level locality, reconstruction, and gravitational dressing continue in Bulk Reconstruction and Gravitational Dressing. Mutable evidence assessments belong in the Holography and Quantum Gravity field guide, with finite-$N$ and code-subspace questions tracked in Bulk Reconstruction Beyond Semiclassical Code Subspaces.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bhat, Faizan, and Ahmadullah Zahed. 2023. “A Celestial Route to AdS Bulk Locality,” Journal of High Energy Physics 08, 112. Open PDF.
• Camanho, Xian O.; Edelstein, José D.; Maldacena, Juan; and Zhiboedov, Alexander. “Causality Constraints on Corrections to the Graviton Three-Point Coupling.” Journal of High Energy Physics 2016, 020 (2016). doi:10.1007/JHEP02(2016)020. Open PDF.
• Caron-Huot, Simon; Mazáč, Dalimil; Rastelli, Leonardo; and Simmons-Duffin, David. “AdS Bulk Locality from Sharp CFT Bounds.” Journal of High Energy Physics 2021, 164 (2021). doi:10.1007/JHEP11(2021)164. Open PDF.
• Chang, Cyuan-Han, Yakov Landau, and David Simmons-Duffin. 2025. “Spinning Dispersive CFT Sum Rules and Bulk Scattering,” Journal of High Energy Physics 04, 016. Open PDF.
• Fitzpatrick, A. Liam, and Kaplan, Jared. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 2013, 054 (2013). doi:10.1007/JHEP02(2013)054. Open PDF.
• Heemskerk, Idse; Penedones, João; Polchinski, Joseph; and Sully, James. “Holography from Conformal Field Theory.” Journal of High Energy Physics 2009, 079 (2009). doi:10.1088/1126-6708/2009/10/079. Open PDF.
• Penedones, João. “Writing CFT Correlation Functions as AdS Scattering Amplitudes.” Journal of High Energy Physics 2011, 025 (2011). doi:10.1007/JHEP03(2011)025. Open PDF.
• Penedones, João; Silva, João A.; and Zhiboedov, Alexander. “Nonperturbative Mellin Amplitudes: Existence, Properties, Applications.” Journal of High Energy Physics 2020, 031 (2020). doi:10.1007/JHEP08(2020)031. Open PDF.