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Weakly Coupled Bulk Fields from Connected Correlators

Suppressed connected correlators support a perturbative bulk-field description when the relevant operators are normalized, spectrally isolated, and few enough that their cumulative effects remain small. A normalized three-point coefficient sets a cubic interaction at leading order; the resulting exchange contribution must appear at the corresponding order in the connected four-point function.

Required background. Large-N Factorization and Classical Bulk Scaling gives the connected hierarchy. Single-Trace, Multi-Trace, and Collective-Field Organization for Bulk Criteria identifies the candidate one- and multiparticle sectors.

Helpful background. Scalar Two- and Three-Point Functions fixes CFT normalization. Power Counting and Predictive Order explains how interaction order becomes a prediction.

From CFT coefficients to a cubic vertex

Section titled “From CFT coefficients to a cubic vertex”

Normalize scalar primaries by

⟨Oi(x)Oj(0)⟩=δijx2Δi.\langle {\cal O}_i(x){\cal O}_j(0)\rangle =\frac{\delta_{ij}}{x^{2\Delta_i}}.

Conformal symmetry fixes a three-point function up to CijkC_{ijk}. In a matrix large-NN CFT,

Cijk=cijkN+O(N−3).C_{ijk}=\frac{c_{ijk}}{N}+{\cal O}(N^{-3}).

For canonically normalized bulk fields φi\varphi_i, a cubic term

Sint=∫∣g∣ gijk3!φiφjφkS_{\mathrm{int}}=\int\sqrt{\lvert g\rvert}\, \frac{g_{ijk}}{3!}\varphi_i\varphi_j\varphi_k

produces the same boundary structure. After fixing bulk-to-boundary normalization,

gijkL(5−d)/2∝Cijk∼N−1,g_{ijk}L^{(5-d)/2}\propto C_{ijk}\sim N^{-1},

where the proportionality factor depends on dimensions and conventions. The reliable statement is the scaling and the normalized observable, not a convention-free numerical equality.

The perturbative reconstruction of AdS interactions from large-NN CFT correlators, including the role of contact ambiguities, is developed by Heemskerk et al. 2009 and Fitzpatrick and Kaplan 2013.

The four-point consistency check

Section titled “The four-point consistency check”

Two cubic vertices generate a tree exchange, so

G1234exchange∼g12ag34a∼N−2.{\cal G}^{\mathrm{exchange}}_{1234} \sim g_{12a}g_{34a} \sim N^{-2}.

This matches the connected four-point scaling. A local quartic vertex must enter at the same or a more suppressed order. Crossing symmetry then ties exchange and contact contributions across channels.

The split between cubic exchange and contact interaction is partly basis dependent: a bulk field redefinition can move terms proportional to equations of motion. Pole locations, conformal-block data, and the complete correlator are the invariant checks.

When small coefficients are not enough

Section titled “When small coefficients are not enough”

Suppose MM nearly degenerate operators Oa\mathcal O_a contribute to one channel. Even if every coefficient is small,

∑a=1MC12aC34a\sum_{a=1}^{M}C_{12a}C_{34a}

need not be suppressed when MM grows sufficiently rapidly with NN. Dense spectra can also make diagonalization ill-conditioned and invalidate a finite-field truncation. One must control both the size of each coupling and the spectral density.

An isolated set of light single traces with Cijk∼N−1C_{ijk}\sim N^{-1} supports weakly interacting bulk particles. A dense tower supports at most a more complicated collective or string-like description until a cutoff and resummation are established.

Orders of limits, evidence ceiling, and handoff

Section titled “Orders of limits, evidence ceiling, and handoff”

Infer g3∼N−1g_3\sim N^{-1} from a normalized three-point coefficient. The four-point exchange then scales as g32∼N−2g_3^2\sim N^{-2}, providing an independent order check. Introducing M∼N2M\sim N^2 comparable exchanges defeats the naive suppression and is the adversarial fixture.

The estimate takes N→∞N\to\infty at fixed operator set, dimensions, and cross-ratios. Letting the number of exchanged operators grow with NN, approaching a dense degeneracy, or entering a Regge limit can make the sum nonuniform. The evidence licenses weak interactions for the isolated tested sector; it does not prove a finite local field content or a nonperturbative bulk.

Volume IX owns the CFT coefficients, Chapter 3 the mass and spin dictionary, Chapter 7 their Witten-diagram extraction, and Chapter 12 reconstructed bulk operators.

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”