CFT Criteria for Approximate Bulk Locality

Approximate bulk locality is supported by a conjunction of CFT properties: large-$N$ factorization, a sparse light single-trace spectrum, a gap to higher-spin states, controlled Mellin growth, causal Regge behavior, and correlators consistent with local propagation. The conclusion must state the code subspace, energy and spin range, bulk region, norm, and error; locality is not a yes-or-no property of finite-$N$ data.

Required background. Sparse spectra and large-gap criteria supplies the spectrum. Bulk-point diagnostics and finite-gap errors supply kinematics and accuracy. Large-gap locality tests supplies the boundary analysis.

Helpful background. Necessary, sufficient, and heuristic criteria supplies logical classification.

A quantitative locality scorecard

Large-$N$ factorization makes connected $k$-point functions perturbative and supplies a bulk coupling. Sparsity limits the number of light fields. A higher-spin gap $\Delta_{\mathrm{gap}}$ supplies a derivative cutoff. Mellin polynomial boundedness and factorized poles resemble local vertices and exchanges. Regge causality constrains high-spin completion. Bulk-point and wavepacket tests check approximate local propagation in selected states. The perturbative reconstruction makes these assumptions explicit Fitzpatrick and Kaplan 2013.

No item is redundant. Generalized free fields factorize but need not contain an interacting stress-tensor bulk. Vector models have large $N$ and local higher-spin duals but no Einstein higher-spin gap. A sparse low spectrum without Regge control can still have unacceptable high-energy growth Heemskerk et al. 2009.

First application: assess a candidate large-N CFT

Suppose a unitary CFT has connected correlators $O(N^{-2})$, one stress tensor, finitely many light single traces, and $\Delta_{\mathrm{gap}}=20$. Its Mellin amplitudes are polynomially bounded through the measured order and its Regge phase has the causal sign. For scalar packets with $EL=3$ when the first omitted contact has four derivatives, a natural gap estimate is

$$\epsilon_{\mathrm{gap}}\sim(3/20)^4\simeq5\times10^{-4},$$

to be combined separately with $N^{-2}$, OPE uncertainty, and wavepacket-curvature errors. The licensed claim is a local bulk EFT for those packets and operators in the central region—not exact locality at $EL\sim20$ or a unique UV theory.

Adversarial control: omit the higher-spin gap

Keep factorization, a sparse set of low scalars, attractive large-spin data, and a visible bulk-point feature, but allow conserved currents of arbitrarily high spin at low dimension. The appropriate bulk can be higher-spin and non-Einstein; a finite low-derivative graviton EFT is not justified. Likewise, omitting a Regge bound leaves high-energy causality inconclusive.

The evidence ceiling is an approximate EFT statement with an explicit error budget and code subspace. None of these finite-order criteria is a theorem of microscopic uniqueness or nonperturbative locality. CFT-to-bulk reconstruction lists the surviving inverse-problem ambiguities.

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. L., and Kaplan, J. (2013), “AdS Field Theory from Conformal Field Theory,” Journal of High Energy Physics 2013(02), 054. arXiv:1208.0337.
• Heemskerk, I., Penedones, J., Polchinski, J., and Sully, J. (2009), “Holography from Conformal Field Theory,” Journal of High Energy Physics 2009(10), 079. arXiv:0907.0151.