Necessary, Sufficient, and Heuristic Bulk Criteria

No single familiar large-$N$ signal is sufficient for a local Einstein bulk. Factorization supports weak interactions, a large stress-tensor coefficient supports a large Planck hierarchy, spectral sparsity limits the light field content, a higher-spin gap removes a central obstruction to Einstein dynamics, and Mellin boundedness supports a derivative expansion. Each conclusion is conditional on a specified sector and precision. The useful result is therefore an implication map with explicit counterexamples, not a checklist promoted to a theorem.

Required background. Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria defines light-state sparsity; Higher-Spin Gaps and Einstein-Regime Obstructions isolates the spin-sensitive condition; and Corrections, Nonuniform Limits, and Failure Modes supplies the finite-$N$, finite-gap, and scaled-limit qualifications.

Helpful background. Falsifiers, Negative Results, and Counterexamples gives the general logic of a decisive failure test, while Large-Gap Constraints and CFT-Side Locality Tests develops the boundary-data constraints.

The target determines what is necessary

A condition is necessary only relative to a target conclusion. Three targets should not be conflated:

1. A weakly interacting bulk field sector: connected correlators of normalized single-trace operators are parametrically suppressed and admit a particle-like organization.
2. A perturbative local AdS effective field theory: the light single-trace sector is sparse, heavy exchanges are separated by a gap, and correlators admit a bounded derivative expansion below that gap.
3. An Einstein regime: in addition, single-trace operators of spin $J>2$ are parametrically heavy enough that higher-spin exchanges do not compete with graviton dynamics in the domain of interest.

A theory can satisfy the first target and fail the other two. Conversely, a bulk with light higher-spin gauge fields is still a bulk, but it is not an Einstein EFT. The word “necessary” without its target silently excludes valid non-Einstein duals.

A counterexample-indexed implication map

The main implications are best stated with their missing converses.

Factorization. If normalized connected $n$-point functions scale as $N^{2-n}$ in a matrix-like sector, then cubic and higher bulk interactions are parametrically weak after canonical normalization. This is necessary for the ordinary classical-field limit of that sector. It is not sufficient for geometry or locality: generalized-free correlators factorize but need not define a standalone local CFT with a stress tensor, and vector models factorize while retaining infinitely many light higher-spin currents.

Large $C_T$. In a theory already known to have an Einstein-like AdS dual,

$$C_T\propto\frac{L^{d-1}}{G_{d+1}}$$

turns $C_T\gg1$ into a Planck hierarchy. Without the dictionary, a large stress-tensor coefficient is boundary data, not proof of a metric bulk. It says nothing by itself about the string scale, higher-spin gap, or number of light species.

Sparse light single-trace spectrum. A finite set of light single-trace primaries supports a bulk description with finitely many elementary fields below a cutoff. The threshold, spin range, and growth with $N$ must be stated. Sparsity alone does not determine the interactions or prove locality.

Large higher-spin gap. Suppressing single-trace fields with $J>2$ is necessary for an Einstein-like regime over a parametrically large energy interval. It is not necessary for a higher-spin bulk. Causality constraints on higher-derivative graviton interactions show why new higher-spin states must appear when certain corrections become large Camanho et al. 2016.

Mellin analyticity and boundedness. Exchange poles with factorizing residues plus polynomially bounded contact ambiguities support perturbative AdS EFT. A finite set of poles and low-energy coefficients is insufficient because an entire, rapidly growing addition can preserve that finite data. The constructive relation between large-$N$, large-gap CFT correlators and local bulk vertices was developed by Heemskerk et al. 2009 and Fitzpatrick and Kaplan 2013; it is a perturbative statement under hypotheses, not a nonperturbative existence theorem.

Top-down dictionary. A fully specified boundary theory, bulk string construction, dictionary, and controlled parameter regime provide an existence example far stronger than correlated indicators. Even then, a supergravity truncation has finite-$N$, string-scale, Kaluza–Klein, and nonperturbative limits. One example does not prove a universal criterion for all CFTs.

Four theories evaluated on the same map

The comparison below uses “yes” only for the stated large-$N$ sector; it does not erase coupling or operator-class qualifications.

Theory class	Factorization	Sparse light single-trace sector	Large $J>2$ gap	Licensed conclusion
Generalized free field data	Yes, at leading order	Can be imposed kinematically	Not enough information	A crossing-consistent mean-field correlator; not by itself a complete local CFT or gravitational dual
Large-$N$ vector model	Yes, with vector scaling	No in the higher-spin sector	No; currents of unbounded spin remain light	A weakly coupled higher-spin organization, not an Einstein regime
Weakly coupled matrix large-$N$ theory	Yes, and often genus-organized	Commonly no because many single-trace operators remain light	Commonly no	String-like topology counting without a local low-energy Einstein truncation
Strong-coupling top-down $AdS_5/CFT_4$ example	Yes	A controlled supergravity sector, with Kaluza–Klein qualifications	A string-scale gap growing with coupling	Perturbative string/supergravity bulk in a declared $N$ and coupling hierarchy

The vector-model row is realized concretely by the critical $O(N)$ model and its higher-spin AdS proposal Klebanov and Polyakov 2002. It defeats the inference

$$\text{factorization}+\text{large central charge} \quad\Longrightarrow\quad \text{Einstein gravity}.$$

The matrix row shows that double-line topology is also insufficient: at weak boundary coupling, long towers of low-dimension single-trace operators obstruct a large string-scale gap. The top-down row refers to the large-$N$, strong ’t Hooft-coupling regime of the $AdS_5\times S^5$ correspondence Maldacena 1998. It is a positive existence example, but the Kaluza–Klein tower means that “pure five-dimensional Einstein gravity with no other light fields” would still be an overstatement.

Removing one hypothesis at a time

An adversarial sufficiency test starts from a proposed package and deletes one assumption.

Delete factorization. Connected correlators remain order one, so canonical bulk interactions are not parametrically weak. The classical saddle conclusion fails directly.

Delete light-spectrum sparsity. An increasing number of elementary bulk fields lies below the proposed cutoff. A finite-field EFT conclusion fails, although a more complicated bulk description may survive.

Delete the higher-spin gap. Vector models supply an explicit counterexample to an Einstein inference. The surviving claim is a possible higher-spin bulk.

Delete polynomial boundedness. Add a crossing-symmetric entire function with exponential Regge growth that begins beyond all measured low-degree terms. Finite low-energy matching survives; controlled local-EFT behavior does not.

Delete finite-$N$ control. Two exact functions can share every coefficient of the genus expansion and differ by $e^{-N}$. Perturbative bulk agreement survives; uniqueness of the exact completion does not.

Delete a fixed-theory specification. An ensemble-averaged model and a single boundary theory can agree on selected averaged observables. The data then do not decide which interpretation is intended.

If no counterexample is known after deleting a hypothesis, the correct entry is unresolved, not sufficient. Absence of a counterexample in a surveyed class is evidence about that survey, not a theorem.

A conditional sufficiency package

For a unitary $d$-dimensional CFT with a unique stress tensor, the following package is strong evidence for a perturbative local AdS sector:

$$\begin{gathered} C_T\gg1,\qquad \langle\mathcal O_1\cdots\mathcal O_n\rangle_{\mathrm{conn}} \ \text{factorizes at large }N,\\ \text{a sparse light single-trace spectrum},\qquad \Delta_{\mathrm{gap}}(J>2)\gg1,\\ \text{crossing-consistent Mellin amplitudes with controlled Regge growth}. \end{gathered}$$

With a declared energy window $EL\ll\Delta_{\mathrm{gap}}$, this package supports reconstruction order by order in $1/N$ and the derivative expansion. Calling it “sufficient” requires specifying the sense: it can be sufficient for a perturbative correlator representation by local AdS fields to a stated order, given the analytic assumptions. It is not currently a general theorem guaranteeing a unique exact quantum-gravity Hilbert space, all topologies, a fixed-theory interpretation, or a nonperturbative completion. For a broad pedagogical account of these distinctions, see Harlow 2016, §§2–4.

Evidence ceiling and research handoff

Evidence cutoff: 25 July 2026. Known top-down examples and perturbative reconstruction results make the combined criteria scientifically powerful. Known vector, generalized-free, weak-coupling, and rapidly growing Mellin examples establish that several tempting converses are false. No general classification known at this cutoff turns the indicator package into a unique nonperturbative bulk theorem for arbitrary CFTs. New examples, improved bounds, and claimed sufficiency results require a dated comparison against every counterexample class above; rigorous existence or reconstruction theorems belong to mathematical QFT.

Exercises

1. A CFT family has $C_T\sim N^2$ and factorization, but a conserved current at every even spin. What is the strongest bulk statement supported by these facts?

   Solution

   They support a weakly interacting large-$N$ bulk organization with light higher-spin fields. They do not support an Einstein EFT with a parametrically separated higher-spin scale. Additional dictionary and locality evidence is still required even for the higher-spin interpretation.

2. Why is a top-down dual pair stronger evidence than satisfying the five boundary indicators, yet still insufficient for a universal criterion?

   Solution

   A top-down construction specifies both theories, the dictionary, charges, and tunable regimes, so it establishes a concrete existence example rather than an inference from correlated data. It remains one class of examples with its own compactification and corrections; it does not prove that every CFT satisfying similar indicators has the same kind of bulk or a unique completion.

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

• 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.
• 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.
• Harlow, Daniel. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88, 015002 (2016). doi:10.1103/RevModPhys.88.015002.
• 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.
• Klebanov, Igor R., and Polyakov, Alexander M. “AdS Dual of the Critical $O(N)$ Vector Model.” Physics Letters B 550, 213–219 (2002). doi:10.1016/S0370-2693(02)02980-5.
• Maldacena, Juan M. “The Large $N$ Limit of Superconformal Field Theories and Supergravity.” Advances in Theoretical and Mathematical Physics 2, 231–252 (1998). doi:10.1023/A:1026654312961.