Large-N, Mellin, and Holographic CFT Interfaces
Large-N factorization makes conformal correlators resemble a weakly interacting system, but factorization and spectral sparsity are separate assumptions. This chapter develops their consequences in crossing and Mellin space, then examines the limits needed for a possible bulk interpretation. Its conclusions are statements about CFT data with explicit control parameters.
Enter this chapter
Section titled “Enter this chapter”Begin with Large-N and Sparse-Spectrum CFT Data to specify the operator normalization, factorization limit, and independent gap assumption.
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.
Topics
Section titled “Topics”- Large-N and Sparse-Spectrum CFT Data — Separates a hierarchy of connected correlators from sparsity of the light spectrum.
- Mellin-Space CFT Correlators — Relates OPE information to poles while retaining the transform’s measure and contour.
- Large-N Crossing, Double-Trace Data, and Contact Ambiguities — Solves crossing perturbatively without discarding homogeneous contact terms.
- Large-Gap Constraints and CFT-Side Locality Tests — Tests a hierarchy of low-energy data under gap and growth assumptions.
- Bulk-Point and Flat-Space Limits: CFT-Side Criteria — Controls Lorentzian continuations and scaling limits before interpreting them as scattering.
How the ideas fit together
Section titled “How the ideas fit together”Factorization fixes a leading correlator; crossing then relates exchange contributions to corrections in composite-operator dimensions and coefficients. Mellin space makes exchange poles visible, but a crossing-symmetric polynomial can leave those poles unchanged. Such contact terms encode information not determined by the exchanged spectrum alone. A large gap and a suitable boundedness condition are additional ingredients in an effective bulk description. Fitzpatrick and Kaplan 2013, §§ 2–3.
These assumptions must also hold in the limit used. A correction small at fixed Mellin variables can become important when those variables grow with the gap. A formal flat-space limit therefore needs a specified scaling sequence, analytic sheet, and external-state normalization.
Review the chapter
Section titled “Review the chapter”Can the poles of a Mellin amplitude uniquely determine the correlator?
Solution
No. Polynomial additions introduce no new poles and can still obey crossing. Their allowed degree is constrained by growth assumptions, but their coefficients require additional data. Reconstructing the exchange singularities therefore leaves a contact ambiguity unless another condition fixes it.
Where to go next
Section titled “Where to go next”Continue to Witten Diagrams and AdS Perturbation Theory to study the corresponding bulk calculations and their regime of validity.