Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria

A semiclassical AdS effective field theory needs more than a large central charge. It needs a controlled set of light single-trace operators, separation from additional single-particle states, and interaction data compatible with a derivative expansion. The word “gap” is not one number: fixed-spin dimension gaps, higher-spin twist gaps, and the usable bulk cutoff answer different questions. Multi-trace towers built from the light fields are expected and do not by themselves spoil single-trace sparsity.

Required background. Central Charge, Newton Coupling, and the Planck Scale fixes loop suppression; Weakly Coupled Bulk Fields from Connected Correlators fixes interaction scaling; Large-N and Sparse-Spectrum CFT Data develops the boundary spectral data.

Helpful background. Large-Gap Constraints and CFT-Side Locality Tests develops quantitative constraints.

Dimension, twist, and bulk thresholds

Choose a finite light single-trace sector $\mathcal{L}$. Along a family of CFTs, this choice must be held fixed while the proposed gap is taken large. At each fixed spin, define the first omitted single-trace dimension by

$$\Delta_{\mathrm{gap}}^{(J)} =\inf\left\{ \Delta_{\mathcal{O}}: \mathcal{O}\text{ is single trace}, \ J_{\mathcal{O}}=J, \ \mathcal{O}\notin\mathcal{L} \right\}.$$

This asks when a new particle of a specified spin appears relative to the chosen light sector. It is not yet a global higher-spin test. Writing the twist of a primary as $\tau_{\mathcal O}=\Delta_{\mathcal O}-J_{\mathcal O}$, define

$$\Delta_{\mathrm{HS}} = \inf_{\substack{ \mathcal{O}\text{ single trace} \\ J_{\mathcal{O}}>2 }} \Delta_{\mathcal{O}}, \qquad \tau_{\mathrm{HS}} = \inf_{\substack{ \mathcal{O}\text{ single trace} \\ J_{\mathcal{O}}>2 }} \bigl(\Delta_{\mathcal{O}}-J_{\mathcal{O}}\bigr),$$

If $\mathcal{L}$ itself contains a light field of spin above two, $\Delta_{\mathrm{HS}}$ is small and the finite-field Einstein criterion already fails. Across many spins, a dimension gap and a twist gap are not interchangeable: a nearly conserved current can have large $\Delta$ merely because its spin is large while retaining twist near $d-2$. Camanho et al. 2016, § 5.4, pp. 40–42, and § 8, pp. 47–50 formulate their causality scale using the dimension of the lightest spin-above-two single-particle operator in their weakly coupled gravity setting; follow the next article’s causality analysis. Whenever a theorem or bound assumes a twist gap, its declared definition and spin range must be retained rather than replaced by a dimension gap.

The dimension becomes a physical threshold only after applying the spin-dependent AdS dictionary. For a scalar, the relation is exact:

$$m^2L^2=\Delta(\Delta-d), \qquad mL=\Delta-\frac d2+O(\Delta^{-1}) \quad(\Delta\gg d).$$

For a totally symmetric spin-$J$ field with $J\ge1$, in the standard massive-field convention,

$$m_J^2L^2 = (\Delta+J-2)(\Delta-J-d+2).$$

A conserved current has $\Delta=J+d-2$ and therefore gives a massless gauge field; Bonifacio et al. 2019, § 3.2, eq. (3.6), p. 13 give the general symmetric-tensor mass–dimension relation. At fixed $J$ and $\Delta\gg J,d$, the mass again obeys $m_JL=\Delta-d/2+O(\Delta^{-1})$. If the minimizing spin grows with the gap, that fixed-spin approximation cannot be used; the exact relation and the theorem’s dimension-or-twist hypothesis must be kept.

Thus a large dimension at fixed spin gives a large single-particle mass in AdS units. It supplies a candidate heavy-particle threshold, not automatically the full EFT cutoff. Perturbative unitarity, species effects, Kaluza–Klein modes, or another strong-coupling scale may intervene earlier.

Independent criteria and their implications

The independent inputs and their jobs are summarized below. The gap rows are deliberately separate.

Independent boundary criteria, their candidate bulk roles, and their remaining failure modes

Boundary datum	Candidate bulk role	What can still fail
CT ≫ 1, after establishing the graviton dictionary	Small gravitational loop parameter	No factorization or no controlled light sector
Factorization and approximate Fock-space organization	Perturbative multiparticle expansion	No locality or no large heavy-state threshold
Finitely many light single traces below a declared threshold	Finitely many candidate fields in that window	A light tower can begin just above the threshold
Large fixed-spin dimension gaps Δgap(J)	Heavy omitted particles at the declared fixed spins	An unexamined spin sector can remain light
Large global higher-spin dimension gap ΔHS	No spin-above-two one-particle global-AdS state below energy ΔHS/L; a rest-mass threshold needs the spin-dependent dictionary	A lower low-spin, Kaluza–Klein, species, or strong-coupling scale
Large higher-spin twist gap τHS, when assumed by the analysis	Control of the lightcone or Regge sector specified by that analysis	It is not universally interchangeable with a dimension gap or a physical cutoff
Polynomial Mellin growth or suitable Regge and dispersive bounds	Locality control for interactions	The result can be restricted to particular correlators, channels, dimensions, or perturbative orders

No row implies the rows beneath it. In particular, large $C_T$ suppresses graviton loops only after a bulk interpretation has been justified; it is not itself a locality theorem.

Partition a toy spectrum

Consider the following synthetic large-$N$ operator list. It is deliberately simple enough to classify by inspection.

A toy spectrum partitioned into one-particle, multiparticle, and omitted heavy sectors

Operator family	Spin	Dimension	Large-N class	Candidate bulk reading
Tμν	2	d	Light single trace	Graviton
𝒪	0	Δ𝒪 = O(1)	Light single trace	Scalar field
[𝒪𝒪]n,J	J = 0, 2, 4, …	2Δ𝒪 + 2n + J + γn,J	Multi-trace	Two-particle state
X	0	Δgap(0) ≫ 1	Omitted single trace	Heavy scalar threshold
HJ	J > 2	J + τJ	Higher-spin single trace	Heavy or light according to the spin-sensitive gap

For scalar primary constituents, at leading order in a factorizing theory,

$$\Delta_{[\mathcal{O}_i\mathcal{O}_j]_{n,J}} =\Delta_i+\Delta_j+2n+J+\gamma_{n,J},$$

with $\gamma_{n,J}$ suppressed in the declared large-$N$ limit for fixed $n$, fixed $J$, fixed operator families and couplings, and away from degeneracies that require diagonalization. This is not a uniform claim at arbitrarily large excitation number or spin. Identical bosonic scalars allow only even $J$; nonidentical scalars can also produce odd-spin families. Spinning constituents require additional tensor structures. These towers are the expected multiparticle spectrum, not extra elementary fields. Mixing can change the convenient operator basis, and finite-$N$ relations can remove some formal composites, so “multi-trace” is a leading-order organizational statement.

Run the controlled inference

Make the toy list into a definite family $\mathcal C_N$ by taking

$$C_T=c_TN^2, \qquad \mathcal L=\{T_{\mu\nu},\mathcal O\}, \qquad \Delta_{\mathcal O}=O(1),$$

with $c_T>0$, matrix-like factorization for normalized correlators, and the bosonic omitted-state gaps

$$G_{\mathrm{low}}(N) = \min_{J=0,1,2}\Delta_{\mathrm{gap}}^{(J)} \longrightarrow\infty, \qquad G_{\mathrm{HS}}(N) = \Delta_{\mathrm{HS}} \longrightarrow\infty.$$

For the symmetric-tensor sector, additionally require the physical mass gap

$$M_{\mathrm{HS}}L := \inf_{\substack{ \mathcal O\text{ single trace} \\ J_{\mathcal O}>2 }} \sqrt{ (\Delta_{\mathcal O}+J_{\mathcal O}-2) (\Delta_{\mathcal O}-J_{\mathcal O}-d+2) } \longrightarrow\infty.$$

At fixed excitation numbers, the first two single traces identify candidate graviton and scalar fields, while $[\mathcal O\mathcal O]_{n,J}$ identifies their two-particle states. The omitted low-spin and higher-spin single traces set distinct candidate heavy-particle thresholds, with $M_{\mathrm{HS}}$—not $\Delta_{\mathrm{HS}}/L$ alone—providing the higher-spin rest-mass threshold. If the correlator data also pass the locality tests below, the resulting controlled bosonic symmetric-traceless field content is the graviton plus one scalar. The full field content requires separate gap checks for fermionic and other Lorentz representations. This remains a conditional EFT inference, not a proof of a unique exact bulk dual.

The observable inputs are operator dimensions, spins and Lorentz representations, single- versus multi-trace organization, and normalized connected correlators. The inference applies at fixed low excitation number and at energies below every omitted threshold. Its controls are the $1/C_T$ loop parameter and the inverse physical mass gaps. Other Lorentz representations require their own mass–dimension dictionaries. Mixing, unresolved degeneracies, finite-$N$ relations, anomalous dimensions at large $n$ or $J$, and an unseen lower threshold are explicit uncertainties. This makes the conclusion reproducible without silently turning spectral evidence into a nonperturbative duality theorem.

Run the spin-sensitive adversarial test

Now compare two theories with the same large $C_T$, the same factorization, and the same sparse $J\le2$ spectrum:

A spin-blind sparsity test gives the same verdict for two spectra with different Einstein-regime conclusions

Datum	Spectrum A: higher-spin sector heavy	Spectrum B: low spin-four current
Shared low-spin data	CT ∼ N2; light Tμν and 𝒪; large omitted gaps for J ≤ 2	The same
Spin above two	ΔHS and the relevant τHS both grow in the declared regime	An extra single-trace spin-four primary K4 with Δ4 = d + 2 + γ4 and γ4 = 0 or ≪ 1
Strongest spectral conclusion	Eligible for a finite-field Einstein EFT, conditional on the interaction and lower-cutoff tests	Loop suppression and low-spin sparsity can survive, but a finite-field Einstein EFT does not

For Spectrum B,

$$\Delta_{\mathrm{HS}}\le d+2+\gamma_4=O(1), \qquad \tau_{\mathrm{HS}}\le d-2+\gamma_4=O(1).$$

Thus both candidate higher-spin gap hypotheses fail in this fixed-spin example. The spin-four bulk mass is

$$m_4^2L^2 = \gamma_4(d+4+\gamma_4),$$

so an exactly conserved current is massless and a weakly broken current is light. The strongest surviving bulk claim is therefore a possible weakly interacting higher-spin or collective description, if the rest of the dictionary supports it—not Einstein gravity coupled to finitely many low-spin fields over an improving cutoff window. In three dimensions, under the additional hypotheses of Maldacena and Zhiboedov 2013, § 1, one exactly conserved higher-spin current forces an infinite conserved tower and free-boson- or free-fermion-type current correlators. That rigidity theorem is not being asserted outside its stated scope.

From spectral data to a local EFT

Integrating out a heavy sector can generate a derivative expansion,

$$\delta\mathcal{A} \sim \left(\frac{E}{\Lambda_{\mathrm{heavy}}}\right)^p,$$

where $p>0$ is the leading omitted power in the process being studied. A spectrum alone does not prove that the coefficients are local or controlled. For the $d=2,4$ low-dimension scalar four-point problem, Heemskerk et al. 2009, § 7, pp. 36–39, and § 9, p. 40 match bounded-spin leading $O(N^{-2})$ crossing solutions to local AdS interactions. Their unbounded-spin extension uses stated singularity and convergence assumptions and represents solutions as convergent sums of fixed-spin solutions; excluding nonlocal infinite-derivative sums requires additional coefficient falloff. This is neither an all-orders nor an unconditional locality theorem. Fitzpatrick and Kaplan 2013, §§ 1 and 6 separate perturbativity, approximate Fock space below the gap, and polynomial Mellin boundedness. Caron-Huot et al. 2021, § 1.1.1, pp. 4–5, and § 5, especially pp. 48–50 and § 5.4, pp. 61–63 obtain sharp large-gap bounds on Wilson coefficients in specified weakly coupled light-spectrum settings.

These are powerful perturbative locality results, not a general theorem that every theory satisfying a short spectral checklist has a unique nonperturbative bulk or a top-down string construction. Let $\Lambda_{\mathrm{HS}}$ be the physical mass of the first omitted spin-above-two particle and $\Lambda_{\mathrm{low\text{-}spin}}$ the first omitted one-particle threshold with $J\le2$, including scalars, fermions, vectors, or additional spin-2 fields as applicable. Let $\Lambda_{\mathrm{KK}}$ be the first unretained compactification mode, $\Lambda_{\mathrm{species}}$ the scale at which the accumulated light species invalidate the gravitational expansion, and $\Lambda_{\mathrm{strong}}$ the earliest additional interaction or perturbative-unitarity scale not already represented by the other entries. The usable cutoff is the first of these scales,

$$\Lambda_{\mathrm{EFT}} = \min\bigl( \Lambda_{\mathrm{HS}}, \Lambda_{\mathrm{low\text{-}spin}}, \Lambda_{\mathrm{KK}}, \Lambda_{\mathrm{species}}, \Lambda_{\mathrm{strong}} \bigr).$$

A claimed window must then control at least two independent errors:

$$\epsilon_{\mathrm{grav\,loop}} \sim \frac{N_{\mathrm{light}}(E)}{C_T} (EL)^{d-1}, \qquad \epsilon_{\mathrm{deriv}} \sim \left(\frac{E}{\Lambda_{\mathrm{EFT}}}\right)^q.$$

Here $N_{\mathrm{light}}(E)$ is a coupling-weighted count of species active at energy $E$, and $q>0$ is the first power not retained in the EFT expansion. The species scale is determined parametrically by

$$\frac{N_{\mathrm{light}}(\Lambda_{\mathrm{species}})}{C_T} (\Lambda_{\mathrm{species}}L)^{d-1} \sim1.$$

If this effective count is approximately constant, $N_{\mathrm{light}}\simeq N_s$, then

$$\Lambda_{\mathrm{species}}L \sim \left(\frac{C_T}{N_s}\right)^{1/(d-1)}.$$

For $N_s=O(1)$ this coincides parametrically with the Planck or gravitational strong-coupling scale, up to convention-dependent factors; if the number of active species grows with energy, the defining relation is implicit. For a fixed light sector, fixed order-one couplings, and $EL=O(1)$, the species factor is $O(1)$ and the loop estimate reduces to the familiar $C_T^{-1}$ scaling. Additional matter self-interactions can carry their own loop parameters. The derivative formula is likewise schematic and theory-dependent. If the higher-spin state is the first cutoff, its spin stays bounded, and its dimension is large compared with that spin and $d$, then $\Lambda_{\mathrm{EFT}}L\sim\Delta_{\mathrm{HS}}$ at leading order. Otherwise use the exact mass–dimension relation and the actual lowest physical scale. Increasing $N$ at fixed thresholds and fixed light content improves the gravitational loop expansion but not the derivative expansion. In familiar matrix examples, these controls are often governed by $N$ and the ’t Hooft coupling respectively. The limits must be uniform over the energy range claimed.

Common pitfalls

Counting multi-traces as new elementary fields. Their leading dimensions and OPE structure identify the expected multiparticle tower. Sparsity is a statement about single traces after mixing is handled.

Calling any large number “the gap.” State the spin sector, whether dimension or twist is used, the light sector being excluded, and the parameter in which the separation grows.

Equating a heavy-particle threshold with the EFT cutoff. The smallest of the heavy-state, species, perturbative-unitarity, compactification, and strong-coupling scales controls the usable window.

Exercises

1. Double traces and sparsity

Let one light bosonic scalar single trace have dimension $\Delta$. Explain why the infinite even-spin tower $[\mathcal{O}\mathcal{O}]_{n,J}$ does not contradict a finite number of light single-trace fields.

Solution — multi-traces and particle number

At leading large $N$ and fixed $n,J$, the tower has dimensions $2\Delta+2n+J+\gamma_{n,J}$ for $J=0,2,4,\ldots$ and is generated by products of the same light operator. Its members represent two-particle states with radial excitation $n$ and angular momentum $J$. Single-trace sparsity limits independent one-particle species, not the multiparticle Hilbert space built from them; the approximation need not remain uniform when $n$ or $J$ grows with $N$ or with the gap.

2. Two parameters, two errors

Suppose $C_T=N^2$, the number and couplings of light species stay fixed, and the physical higher-spin cutoff is the lowest cutoff with $\Lambda_{\mathrm{HS}}L=20$ independent of $N$. What happens to the loop and derivative errors as $N\to\infty$ at fixed $EL$?

Solution — independent loop and gap limits

Because the light species factor is fixed, the gravitational loop error scales as $N^{-2}$ and vanishes. The derivative error remains $O[(EL/20)^q]$, independent of $N$, and is small only if $EL\ll20$. There is a classical limit, but no parametrically improving derivative expansion.

3. A spin-blind false positive

Theories A and B both have $C_T\sim N^2$, factorization, one light scalar, and no other $J\le2$ single trace below a gap that grows with $N$. Theory A has both $\Delta_{\mathrm{HS}}\to\infty$ and the higher-spin twist gap required by its locality analysis. Theory B instead has a conserved spin-four single trace with $\Delta_4=d+2$. What is the strongest bulk conclusion licensed for each theory, and which hypotheses fail in B?

Solution — the spin-four obstruction

Theory A passes the stated spectral necessary conditions for a graviton-plus-scalar Einstein EFT, but locality of its interactions and the absence of a lower cutoff still need independent tests. Theory B can retain large-$C_T$ loop suppression, factorization, and a sparse low-spin sector. However, its gaps satisfy $\Delta_{\mathrm{HS}}\le d+2=O(1)$ and $\tau_{\mathrm{HS}}\le d-2=O(1)$, while the spin-four field is massless because $m_4^2L^2=(d+4)\times0=0$. The exact failed hypotheses are the parametrically large higher-spin dimension gap and the declared twist-gap condition. The remaining data may be compatible with a classical higher-spin or collective bulk, but not with a finite-field Einstein EFT over an improving cutoff window.

Continue to Higher-Spin Gaps and Einstein-Regime Obstructions for the decisive spin-sensitive test and CFT Criteria for Approximate Bulk Locality for the full correlator criteria.

Evidence cutoff. Sufficiency claims and cited literature were reviewed through 28 August 2026.

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

• Bonifacio, James, Kurt Hinterbichler, Austin Joyce, and Rachel A. Rosen. 2019. “Shift Symmetries in (Anti) de Sitter Space,” Journal of High Energy Physics 02, 178. Open PDF.
• Camanho, Xian O., Jose D. Edelstein, Juan Maldacena, and Alexander Zhiboedov. 2016. “Causality Constraints on Corrections to the Graviton Three-Point Coupling,” Journal of High Energy Physics 02, 020. Open PDF.
• Caron-Huot, Simon, Dalimil Mazáč, Leonardo Rastelli, and David Simmons-Duffin. 2021. “AdS Bulk Locality from Sharp CFT Bounds,” Journal of High Energy Physics 11, 164. Open PDF.
• Fitzpatrick, A. Liam, and Jared Kaplan. 2013. “AdS Field Theory from Conformal Field Theory,” Journal of High Energy Physics 02, 054. Open PDF.
• Heemskerk, Idse, João Penedones, Joseph Polchinski, and James Sully. 2009. “Holography from Conformal Field Theory,” Journal of High Energy Physics 10, 079. Open PDF.
• Maldacena, Juan M., and Alexander Zhiboedov. 2013. “Constraining Conformal Field Theories with a Higher Spin Symmetry,” Journal of Physics A: Mathematical and Theoretical 46, 214011. Open PDF.