AdS/CFT as a Conditional Nonperturbative Definition

A completely defined finite-$N$ boundary CFT can serve as a nonperturbative definition of a bulk quantum-gravity sector with specified asymptotically AdS boundary conditions. The statement is conditional: the CFT, its global form and sectors, and a complete bulk dictionary must be known independently. A large-$N$ set of low-dimension correlators defines only an effective code subspace and need not select a unique ultraviolet bulk theory.

Required background. Dictionary completeness and global data supplies the information beyond local correlators. Nonperturbative definition criteria supplies the standard for completeness. The GKPW generating functional supplies the perturbative observable map.

Helpful background. Completeness and operator bases supplies the CFT Hilbert-space construction. Finite-N and exponentially small sectors supplies effects invisible to the genus expansion.

What the boundary definition must contain

Radial quantization supplies a Hilbert space on $S^{d-1}$, a Hamiltonian identified with dilatations, an operator algebra and OPE coefficients, and correlation functions satisfying unitarity and crossing. A complete definition also specifies the global symmetry group, line and surface operators, superselection sectors, anomalies, spin structure, and how the theory is placed on allowed manifolds. These data distinguish theories with identical local Lie algebras and perturbative correlators.

The familiar relation

$$Z_{\mathrm{bulk}}[\phi_{(0)}]=Z_{\mathrm{CFT}}[J=\phi_{(0)}]$$

is exact only after choosing which bulk boundary condition corresponds to each source, what is integrated over, and how both sides are regulated. In the classical limit it reduces to $Z_{\mathrm{bulk}}\simeq e^{iS_{\mathrm{ren}}}$, which is much weaker than the finite-$N$ equality Gubser, Klebanov, and Polyakov 1998, Witten 1998.

First application: from a finite-N CFT to bulk observables

Suppose the exact CFT partition function on $S^{d-1}\times S^1$ and its complete operator algebra are known. The spectrum fixes all energy levels in global AdS, including multi-particle and finite-$N$ states. Source derivatives produce boundary correlators. Thermal traces include competing saddles and exponentially small terms that a single classical geometry misses.

Additional steps remain. One must identify which operators create localized semiclassical bulk excitations, specify gauge-invariant relational observables, match extended objects and global charges, and determine whether apparently different bulk topologies are states or sectors of the same boundary theory. Bulk locality is approximate in a large-$N$, sparse-spectrum code subspace; it is not fundamental input to the exact boundary system Harlow 2018.

Adversarial control: identical low-energy dictionaries

Construct two bulk effective theories with the same light fields, masses, cubic couplings, and all correlators through some order in $1/N$ and $1/\lambda$, but different heavy branes, discrete gauge sectors, or corrections of order $e^{-N}$. Low-energy GKPW data cannot distinguish them. Calling that truncated data a complete definition is therefore unjustified. Exact finite-$N$ CFT global data could distinguish the completions if the dictionary reaches those sectors.

The evidence ceiling is conditional but strong: if a complete boundary theory and one-to-one global dictionary are established, the boundary supplies a nonperturbative definition for that AdS asymptotic sector. It does not automatically define flat-space cosmology, a sum over boundary conditions, or every duality frame. Background dependence and emergence makes those scope choices operational.

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

• Gubser, S. S., Klebanov, I. R., and Polyakov, A. M. (1998), “Gauge Theory Correlators from Non-Critical String Theory,” Physics Letters B 428, 105–114. arXiv:hep-th/9802109.
• Harlow, D. (2018), “TASI Lectures on the Emergence of Bulk Physics in AdS/CFT,” Proceedings of Science TASI2017, 002. arXiv:1802.01040.
• Witten, E. (1998), “Anti-de Sitter Space and Holography,” Advances in Theoretical and Mathematical Physics 2, 253–291. arXiv:hep-th/9802150.