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Higher-Spin Gaps and Einstein-Regime Obstructions

Large NN can suppress bulk loops while an infinite tower of light higher-spin fields remains. Such a theory may be holographic and semiclassical, but it is not described over an AdS-scale window by local Einstein gravity coupled to finitely many fields. The missing condition is a parametrically large gap above spin two.

Required background. Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria separates central-charge, sparsity, and spin-gap requirements.

Helpful background. Weakly Broken Higher-Spin Symmetry supplies the CFT data. Unitarity Bounds and Null States explains why exactly conserved currents sit at a unitarity bound.

Higher-spin currents and bulk fields

Section titled “Higher-spin currents and bulk fields”

In a unitary dd-dimensional CFT, a symmetric traceless conserved current of spin s≥1s\ge1 has

Δs=s+d−2.\Delta_s=s+d-2.

Under the AdS dictionary it corresponds to a massless spin-ss bulk gauge field. An infinite tower of such currents therefore requires an infinite tower of light bulk fields. The rigidity of exactly conserved higher-spin symmetry is analyzed by Maldacena and Zhiboedov 2013. If the currents are weakly broken,

Δs=s+d−2+γs,γs≪1,\Delta_s=s+d-2+\gamma_s, \qquad \gamma_s\ll1,

the bulk fields are light rather than absent.

An Einstein regime instead requires the first new single-trace operator with J>2J>2 to satisfy

ΔHS≫1,\Delta_{\mathrm{HS}}\gg1,

so that higher-spin and string-scale effects enter above an AdS EFT window.

Matrix and vector large N

Section titled “Matrix and vector large N”

Matrix large-NN CFTs with strong coupling can have CT∼N2C_T\sim N^2 and a large higher-spin gap. Their candidate low-energy bulk contains a metric and finitely many light fields.

Vector models have CT∼NC_T\sim N and factorization, but also approximately conserved currents of arbitrarily high spin. Their proposed Vasiliev-type duals are legitimate holographic theories with very different locality properties. This comparison proves that factorization is not a sufficient Einstein criterion.

Causality and finite-gap corrections

Section titled “Causality and finite-gap corrections”

Higher-derivative graviton couplings can produce causality problems in high-energy scattering unless new higher-spin states enter near the correction scale Camanho et al. 2016. In AdS/CFT this yields a schematic relation

∣a−cc∣≲1ΔHS2\left|\frac{a-c}{c}\right| \lesssim \frac{1}{\Delta_{\mathrm{HS}}^2}

in the class of theories and conventions where the bound applies. A large higher-spin gap suppresses non-Einstein graviton structures; a low gap announces the new tower needed to restore consistency.

Hold the gap fixed while N grows

Section titled “Hold the gap fixed while N grows”

Take N→∞N\to\infty with ΔHS\Delta_{\mathrm{HS}} fixed. Connected correlators and quantum loops vanish in their usual powers of 1/N1/N, but

EΛHS∼ELΔHS\frac{E}{\Lambda_{\mathrm{HS}}} \sim\frac{EL}{\Delta_{\mathrm{HS}}}

does not improve. At fixed ELEL comparable with the gap, higher-derivative or higher-spin effects remain order one. The strongest conclusion is a classical bulk with its full higher-spin content, not Einstein gravity.

This scaled limit shows the evidence ceiling: a large higher-spin gap is a necessary condition for the specified Einstein window, not a sufficient condition for a bulk dual and not a requirement for higher-spin holography. Volume IX owns the weakly broken higher-spin data, Chapter 22 the corresponding dualities, and Chapter 8 the detailed Regge and causality analysis.

Evidence cutoff. General status statements are 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

Section titled “References”