CFT-to-Bulk Reconstruction: Uniqueness and Ambiguities

Large-$N$ CFT correlators reconstruct bulk masses, spins, cubic couplings, and contact interactions only to the order and operator set that they probe. Local field redefinitions, crossing-symmetric contacts, states above the measured gap, finite-$N$ exponentials, global sectors, and boundary conditions leave distinct ambiguities. Finite four-point data therefore determine an EFT equivalence class, not a unique microscopic Lagrangian.

Required background. Mellin contact polynomials and exchange poles supplies the inverse map. Approximate locality criteria supplies the regime in which an EFT interpretation is licensed.

Helpful background. Finite-N and exponential sectors supplies invisible data. The local equivalence theorem supplies field-basis freedom.

What correlators determine

Single-trace dimensions and spins determine linearized bulk masses and representations. Three-point coefficients determine on-shell cubic couplings after field normalization. Four-point exchange poles and factorized residues check those couplings; polynomial Mellin pieces determine combinations of local quartic operators modulo equations of motion, integration by parts, and field redefinitions. Loop logarithms and cuts determine anomalous dimensions and nonlocal quantum data.

The reconstruction is triangular only with complete data. A finite derivative expansion leaves higher polynomials free; one external species probes only some couplings; a finite-$N$ asymptotic series misses $e^{-N}$ sectors. The large-gap construction proves perturbative locality order by order under its assumptions, not uniqueness of a completion Heemskerk et al. 2009, Fitzpatrick and Kaplan 2013.

First application: a scalar EFT through four derivatives

Suppose a normalized four-point Mellin amplitude is known through

$$M(s,t)=g_0+g_4(s^2+t^2+u_M^2) +g_\chi^2\sum_{m\ge0}\frac{R_m(t)}{s-(\Delta_\chi+2m)} +\text{crossed channels}.$$

The poles fix $m_\chi^2L^2=\Delta_\chi(\Delta_\chi-d)$ and $g_\chi^2$ after normalization. The constant and quadratic polynomial fix two on-shell quartic combinations. They do not choose among Lagrangians related by $\phi\to\phi+a\phi^2$, determine six-derivative terms, reveal a heavy state above the measured gap, or specify discrete gauge sectors.

Additional correlators with other external operators can reduce these ambiguities, while exact finite-$N$ global data are needed for a proposed nonperturbative bulk definition.

Adversarial control: two indistinguishable EFTs

Construct one action in which an EOM cubic operator is explicit and another in which a field redefinition replaces it by a quartic contact. Their separated-point correlators agree through the measured order. Next add a heavy field of mass $M\gg E_{\max}$ and tune its low-energy expansion to begin beyond four derivatives. The dataset cannot distinguish either pair. Declaring one action unique confuses a representative with the reconstructed equivalence class.

The evidence ceiling is an observable- and order-specific bulk EFT, including its field-basis, contact, gap, nonperturbative, and global ambiguities. Stronger uniqueness requires a complete operator algebra and global dictionary, not more confidence in the same finite data.

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.