Large-N, Mellin, and Holographic CFT Interfaces

Large-N conformal data are unusually structured: connected correlators are small, products of low-dimension operators generate predictable towers, and a sparse set of additional primaries can control the first corrections. The crossing consequences of that hierarchy were developed systematically by Heemskerk et al. 2009 and sharpened into a conditional effective-description criterion by Fitzpatrick and Kaplan 2013. Mellin space makes some of that structure visible as poles, polynomial residues, and contact ambiguities Mack 2009; Penedones 2011. These are intrinsic statements about CFT data. Interpreting them as particles, interactions, locality, or an S-matrix requires additional input and belongs to the holography volume.

Helpful background. Large-N CFT Data and Vector Models develops controlled examples. Large-N Factorization and Master-Field Claims fixes the dynamical meaning of factorization. Dispersion Relations for CFT Correlators explains bounded reconstruction and subtraction ambiguities.

Enter the CFT-side interface

The chapter follows five questions in order:

1. Large-N and Sparse-Spectrum CFT Data asks what the expansion parameter, operator normalization, factorization law, and spectral gap actually say.
2. Mellin-Space CFT Correlators defines the transform, Gamma-function measure, contours, poles, residues, and polynomial terms.
3. Large-N Crossing, Double-Trace Data, and Contact Ambiguities solves crossing order by order while retaining homogeneous solutions and low-spin ambiguity.
4. Large-Gap Constraints and CFT-Side Locality Tests compares Regge growth, contact hierarchies, anomalous dimensions, and finite-gap errors as conditional diagnostics.
5. Bulk-Point and Flat-Space Limits: CFT-Side Criteria specifies Lorentzian sheets, smearing, normalization, scaling sequences, and order of limits.

The natural stopping point is a statement such as: “this family of CFT correlators has a polynomially bounded Mellin expansion with corrections suppressed by the declared higher-spin gap.” It is not: “therefore a particular local bulk Lagrangian exists.”

One expansion, several independent assumptions

Write a normalized four-point function schematically as

$$\mathcal G(U,V;g,\Delta_{\rm gap}) =\mathcal G^{(0)}(U,V) +g^2\mathcal G^{(1)}(U,V) +O(g^4),$$

where $g\to0$ is the factorization limit. The notation deliberately avoids identifying $g$ with a particular power of $N$: vector, matrix, and tensor large-N limits use different counting. A complete claim separately fixes:

• the normalization that defines $g$;
• which operators remain light as $g\to0$;
• whether a higher-spin single-trace gap $\Delta_{\rm gap}$ is assumed and how it scales;
• the spin range and Lorentzian sheet on which a Regge bound holds;
• the Mellin contour and Gamma-function convention;
• the order of the $g\to0$, $\Delta_{\rm gap}\to\infty$, large-Mellin-variable, and any flat-radius limits.

Changing the order can change the answer. A term that is suppressed at every fixed Mellin variable may dominate when that variable grows with the gap. A perturbative bulk-point singularity may be smoothed at any fixed nonzero $g$. Nonuniformity is a physical qualification, not a technical footnote.

Strength of the conclusion

Established CFT statement	Additional condition	Strongest conclusion here	Further interpretation deferred
connected correlators scale with $g$	a stable low-operator basis	factorized CFT expansion	approximate multiparticle language
Mellin amplitude has declared poles	contour and Gamma measure are fixed	OPE twists and residue polynomials are recovered	exchanged bulk fields
crossing leaves a polynomial addition	degree and Regge growth are bounded	contact ambiguity is parameterized	local interaction vertex
higher-spin gap is large	$g$ and gap limits are uniform	gap-suppressed CFT hierarchy	low-energy bulk locality
a Lorentzian singular scaling appears	sheet, smearing, and limit order are controlled	bulk-point-type CFT diagnostic	a bulk scattering event
a Mellin scaling distribution converges	normalization and wavepackets are fixed	flat-space-type CFT limit	a physical S-matrix

Necessary and sufficient conditions must never trade places. A pole is not by itself a particle, polynomial boundedness is not by itself locality, and a formal scaling formula is not by itself an observable scattering amplitude.

Chapter review

Before carrying a proposed holographic inference forward, you should be able to answer all of the following:

• What is the small parameter, and at what order is the correlator known?
• Which primaries are called single-trace, and is that definition intrinsic or basis-dependent at finite $g$?
• What precisely is gapped: all additional primaries, only spin $J>2$, or a specified sector?
• Which Mellin poles come from the amplitude and which double-trace poles come from the Gamma measure?
• What contact polynomial can be added without changing the declared exchange poles?
• Which Regge bound limits its degree?
• Are the gap, large-variable, Lorentzian, and flat-radius limits uniform and taken in a stated order?
• Does the conclusion remain a CFT statement, with the optional bulk interpretation explicitly deferred?

References

• Fitzpatrick, A. L., and Kaplan, J. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 2013, 054 (2013). arXiv. DOI.
• Heemskerk, I., Penedones, J., Polchinski, J., and Sully, J. “Holography from Conformal Field Theory.” Journal of High Energy Physics 2009, 079 (2009). arXiv. DOI.
• Mack, G. “D-Dimensional Conformal Field Theories with Anomalous Dimensions as Dual Resonance Models.” Bulgarian Journal of Physics 36 (2009): 214–226. arXiv.
• Penedones, J. “Writing CFT Correlation Functions as AdS Scattering Amplitudes.” Journal of High Energy Physics 2011, 025 (2011). arXiv. DOI.