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, a gap to additional single-trace states—especially higher-spin states—and interaction data compatible with a derivative expansion. Multi-trace towers built from the light fields are expected and do not by themselves spoil 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 owns the boundary spectral data.

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

Separate the spectral conditions

Let $\Delta_{\mathrm{gap}}$ denote the dimension of the first heavy single-trace primary outside a chosen light sector. Three conditions should be recorded separately.

1. Low-spectrum sparsity: only a controlled number of single traces lie below the gap.
2. Higher-spin gap: single traces with spin $J>2$ are parametrically heavy if an Einstein-like regime is claimed.
3. Large central charge: $C_T\gg1$ suppresses bulk loops at the AdS scale.

The AdS mass relation for a scalar,

$$m^2L^2=\Delta(\Delta-d),$$

shows why a dimension gap becomes a mass hierarchy. Schematically,

$$\Lambda_{\mathrm{gap}}L\sim\Delta_{\mathrm{gap}}.$$

The proportionality and its spin dependence belong to the detailed dictionary; the hierarchy is the point needed here.

Partitioning a model spectrum

Given a list of primaries, classify them as:

• light single traces, interpreted as candidate elementary bulk fields;
• multi-traces assembled from the light sector, interpreted as multiparticle states;
• heavy single traces above $\Delta_{\mathrm{gap}}$, which set the EFT threshold.

Integrating out the heavy sector generates higher-derivative interactions. At energy $E$,

$$\delta{\cal A}\sim \left(\frac{E}{\Lambda_{\mathrm{gap}}}\right)^p$$

for the first allowed derivative order $p$, provided the expansion is local and coefficients are controlled.

Sparsity is observable- and threshold-dependent. A spectrum can be sparse below one chosen dimension without possessing a parametrically large gap as $N\to\infty$.

Why spin matters

Suppose the low scalar spectrum is sparse but there is a conserved or nearly conserved current at every even spin. The theory then has an infinite tower of light single-trace fields. Factorization can still hold and bulk loops can still be suppressed, but a truncation to Einstein gravity plus finitely many low-spin fields is unavailable.

This is the decisive counterexample to “sparse without qualification.” The higher-spin gap must be stated separately from scalar sparsity and from the central charge.

Strength of the inference

Large $C_T$, a sparse light single-trace spectrum, a large higher-spin gap, and suitably bounded correlators are strong conditions for a local semiclassical bulk EFT in known holographic classes. The perturbative constructive evidence is given by Heemskerk et al. 2009 and Fitzpatrick and Kaplan 2013. These results are not, in this generality, a theorem guaranteeing a unique nonperturbative bulk or a top-down string construction.

The hierarchy is taken with $N\to\infty$, $\Delta_{\mathrm{gap}}\to\infty$, and $EL/\Delta_{\mathrm{gap}}\to0$ in a declared order. Holding the gap fixed while increasing $N$ suppresses loops but not higher-derivative effects. Volume IX owns measurement and bootstrap bounds on the spectrum. Chapter 8 develops locality diagnostics from Mellin and Regge data.

Evidence cutoff. The status of general sufficiency claims is fixed to 25 July 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

• Fitzpatrick, A. Liam, and Jared Kaplan. 2013. “AdS Field Theory from Conformal Field Theory,” Journal of High Energy Physics 02, 054.
• Heemskerk, Idse, João Penedones, Joseph Polchinski, and James Sully. 2009. “Holography from Conformal Field Theory,” Journal of High Energy Physics 10, 079.
• Maldacena, Juan M., and Alexander Zhiboedov. 2013. “Constraining Conformal Field Theories with a Higher Spin Symmetry,” Journal of Physics A 46, 214011.