AdS/CFT Dictionary: Bridging Bulk & Boundary

• The AdS/CFT Dictionary is a formal set of correspondences linking bulk gravitational fields in anti-de Sitter space to operators in a boundary conformal field theory.
• It employs prescriptions like GKPW and BDHM, operator smearing, and RG flow mapping to translate bulk dynamics into well-defined boundary data.
• Recent advances integrate quantum information concepts, enabling robust error correction and complexity analysis to deepen our understanding of holographic duality.

The AdS/CFT dictionary constitutes the precise set of correspondences relating quantities in a gravitational or string theory on asymptotically anti-de Sitter (AdS) space (“the bulk”) to quantities in a conformal field theory (CFT) defined on its boundary. These rules translate bulk field asymptotics, partition functions, operator insertions, correlation functions, symmetry generators, and even dynamical or time-dependent behavior into well-defined field-theoretic data. The dictionary operates at multiple levels: as a linear map between Hilbert spaces, as an identification of operator algebras and correlators, as a mapping of renormalization-group flows, and as a vehicle for quantitative bounds linking bulk effective actions to CFT data.

1. Structure and Formulations of the AdS/CFT Dictionary

The foundational prescription for the AdS/CFT dictionary is twofold: (a) the GKPW prescription, where CFT n-point correlators are given by functional derivatives of the bulk partition function with respect to boundary conditions, and (b) the “extrapolate” (BDHM) prescription, where bulk correlators are rescaled and extrapolated to the AdS boundary to provide their CFT counterparts (Harlow et al., 2011). In Euclidean signature, for a bulk scalar $\phi(z,x)$ of mass $m$ in Poincaré coordinates $(dz^2 + dx^2)/z^2$, the expansion

$$\phi(z,x) \approx z^{d-\Delta} A(x) + z^\Delta B(x), \quad \Delta = \frac{d}{2} + \sqrt{\frac{d^2}{4} + m^2}$$

identifies $A(x)$ with the source coupling to the boundary operator $\mathcal{O}(x)$, and $B(x)$ with its expectation value.

The equivalence of the “differentiate” (GKPW) and “extrapolate” (BDHM) dictionaries persists in the presence of interactions provided composite operators are renormalized such that UV admixtures are subtracted, and local counterterms are incorporated at the boundary. This ensures the path-integral equivalence of both correlator definitions in the bulk (Harlow et al., 2011).

In Lorentzian signature, for non-equilibrium situations and real-time correlators, the “extrapolate” dictionary is shown to agree with the Schwinger–Keldysh (Skenderis–van Rees, SvR) prescription at the level of in-in (real-time) two-point functions (Keranen et al., 2014). For free fields, this agreement extends to time-dependent and thermalizing backgrounds, such as AdS–Vaidya geometries.

2. Bulk Operators, Smearing, and Bulk Reconstruction

Bulk operators in AdS can—at infinite $N$ and in horizon-free geometries—be exactly reconstructed on the CFT boundary as smeared, generally nonlocal, operator expressions. For a scalar field $\phi(r,t,\Omega)$ in global AdS, one can write

$$\phi(r,t,\Omega) = \int dt' d\Omega' K(r,t,\Omega|t',\Omega') \mathcal{O}(t',\Omega')$$

with a smearing function $K$ determined by the normal-mode decomposition. This explicit construction fails in geometries with horizons (such as AdS–Schwarzschild): the presence of trapped null geodesics or modes with large angular momentum and small boundary imprint preclude the existence of a state-independent smearing function everywhere, as the associated mode sum diverges (Leichenauer et al., 2013). In such cases, reconstructing bulk operators behind (or even outside) the horizon requires genuinely nonlocal CFT operators, such as Wilson loops or precursor constructions.

For nonlocal bulk theories, it is possible to define an AdS/CFT dictionary by relating the nonlocality scale to shifts in the conformal dimension of the dual CFT operator. For example, for a bulk action $$S_{\rm bulk}[\phi] = \frac{1}{2}\int d^{d+1}x \sqrt{-g}\left[(\nabla\phi)^2 - \lambda^2 \phi\, \Box^{-1}\phi\right]$$, the relevant operator dimension is

$$\Delta_{\rm NL} = \frac{d}{2} + \frac{1}{2}\sqrt{d^2 - 4\lambda}$$

and the standard extrapolate-dictionary limit of the bulk two-point function recovers the modified CFT correlator, reverting to the local case as $\lambda \to 0$ (Kajuri et al., 2018).

3. Wilsonian Renormalization, Radial Flow, and Operator Mapping

A central insight of the AdS/CFT dictionary is its connection to renormalization-group (RG) flow. The holographic direction $z$ is mapped to the inverse Wilsonian cutoff $\varepsilon=1/\Lambda$ of the boundary theory. The RG evolution of single-trace couplings in the boundary QFT can be encoded via a generalized Hubbard–Stratonovich transformation, resulting in a flow equation for a functional $I[\varphi;\Lambda]$ that takes the Hamilton–Jacobi form:

$$\varepsilon \, \partial_\varepsilon I[\varphi; \varepsilon] = H_{\rm bulk}(\phi, \pi; \varepsilon)$$

where $H_{\rm bulk}$ coincides, up to identifications, with the Hamiltonian generated by radial Hamilton–Jacobi flow in the bulk (Radicevic, 2011). This formalism allows direct translation between bulk boundary conditions, normalizable/non-normalizable modes, and QFT sources and expectation values:

• $\Phi^a(z\to0,x) \leftrightarrow$ source $g^a(x)$ for operator $\mathcal{O}_a(x)$
• Non-normalizable mode $\sim z^{d-\Delta_a}$ identifies the coupling, normalizable mode $\sim z^{\Delta_a}$ the vacuum expectation value.

In this context, $\beta$-functions for QFT couplings directly encode the canonical momenta for bulk fields.

4. Quantum Information, Pseudoentanglement, and the Complexity of the Dictionary

Viewing the AdS/CFT dictionary as a linear embedding $$V: \mathcal{H}_{\rm code} \to \mathcal{H}_{\rm bdy}$$, where $\mathcal{H}_{\rm code}$ is a “code subspace” of low-energy bulk states, brings quantum error-correction and computational complexity into focus. The quantum information-theoretic paradigm distinguishes two key tasks:

• Operator reconstruction: Find, for a given bulk unitary $U$, a boundary unitary $U_{\rm bdy}$ such that $U_{\rm bdy} V \approx VU$—shown to be efficiently computable (in $\mathrm{BQP}$) when the bulk geometry is given.
• Geometry reconstruction: Determine geometric data (areas, curvatures) from an unknown boundary state. This is provably hard: one can construct ensembles of boundary states, individually efficiently preparable, that are cryptographically indistinguishable (up to BQP algorithms) yet correspond to macroscopically distinct geometries as judged by the Ryu–Takayanagi entropy formula (Akers et al., 7 Nov 2024). This is a manifestation of pseudoentanglement.

The analogy with quantum fully homomorphic encryption (QFHE) is exact: the dictionary enables efficient homomorphic evaluation of bulk operators on encoded states, but generic state properties (i.e., geometry) are computationally hidden. This complexity-theoretic separation underpins recent “quantum gravity in the lab” thought experiments and imposes intrinsic cryptographic obstacles to geometry learning from boundary data.

5. Dispersive CFT Sum Rules and Quantitative AdS/CFT Dictionaries

Sharp quantitative AdS/CFT dictionaries leverage dispersive sum rules in the CFT, specifically functionals $C_{k,\nu}[G]$ built from the double discontinuity of four-point correlators. These sum rules can be directly related, in the bulk-point limit, to twice-subtracted S-matrix dispersion relations:

$$\mathcal{C}_{k,u}[M] = -\oint \frac{ds}{2\pi i} \frac{M(s,u)}{s[s(s+u)]^{k/2}} = 0$$

matching CFT and bulk quantities as follows (Caron-Huot et al., 2021):

Flat-space (S-matrix)	AdS/CFT Correspondence
Amplitude $A(s)$	Correlator $G(z,\bar{z})$ via $d\mathrm{Disc}[G(u',v')]$
Energy $s=m^2$	Dimension $m^2=(\Delta-J-d+1)(\Delta+J-1)$
Impact parameter $b$	AdS invariant $\beta$
Transverse momentum $u$	$u=-\nu^2$

For holographic CFTs with large twist gap $\Delta_{\text{gap}}$ and a conserved stress tensor, the bootstrap-derived bounds on CFT data yield explicit large-$\Delta_{\text{gap}}$ bounds on local bulk Wilson coefficients $g_n$ via

$g_n \sim O(1/\Delta_{\text{gap}}^{2n-4})$

with controlled corrections. Gravitational IR divergences are automatically regulated in AdS$_4$ by replacing $8\pi G/u$ with $8\pi G/(\nu^2 + 1)$.

6. Time-Dependent, Lower-Dimensional, and Special Cases

The dictionary has specific realizations for time evolution, lower-dimensional models, and special geometries. In 2D Einstein–Maxwell–Dilaton theory, explicit expansion in radial (Fefferman–Graham) gauge gives:

• For running-dilaton solutions: boundary data $(\alpha(t), \nu(t), \mu(t))$ serve as sources for $(T_{tt}, J^t, \mathcal{O}_\phi)$, with corresponding Ward identities and anomalous trace (the Schwarzian).
• For constant-dilaton solutions: only the electric charge remains dynamically nontrivial, with the dilaton’s expectation value serving as the sole local observable (Cvetič et al., 2016).

In the context of AdS$_3$/CFT$_2$ at $k=1$, vertex operator correlators localize via Ward identities to covering maps of the worldsheet onto the boundary sphere. The structure constants, genus expansion, and correlator normalizations precisely match those computed in the symmetric orbifold CFT via the Lunin–Mathur covering-surface technique. This alignment manifests even in topological truncation and large-$N$ scaling, validating the dictionary in detail for string backgrounds with $k=1$ NS–NS flux (Eberhardt et al., 2019).

For emergent temporal AdS dimensions, T$^2$-deformations relate boundary partition functions to bulk canonical wavefunctionals via a map that realizes the Wheeler–DeWitt equation and canonical Hamiltonian algebras. The T$^2$ parameter is dual to placing a Dirichlet wall at a finite radial position, and the partition function is mapped to the gravitational path integral with Dirichlet cutoff (Araujo-Regado et al., 2022).

7. Summary Table of Principal Dictionary Entries

Bulk Quantity	Boundary Quantity	Remarks
$\Phi(z\to0, x)$	Source $g^a(x)$ for $\mathcal{O}_a$	Non-normalizable mode
$z^{\Delta_a}$ coefficient	v.e.v. $\langle \mathcal{O}_a\rangle$	Normalizable mode
On-shell bulk action	QFT generating functional	Classical limit
Bulk S-matrix	CFT four-point correlator functionals	Via dispersive sum rules
Radial RG flow	Boundary Wilsonian RG flow	Hamilton–Jacobi eqns.
Smearing function ($K$)	Boundary operator nonlocality	Fails at horizons
Bulk operator algebra	CFT nonlocal operator algebra	Code subspace
Bulk entanglement surfaces	Boundary RT entropy structure	Entanglement–geometry

The AdS/CFT dictionary thus encompasses linear mappings, operator correspondences, dynamical and algebraic structures, renormalization flow, and computational complexity. Advanced quantum information paradigms, such as QFHE and pseudoentanglement, are now recognized as essential in understanding the subtlety and limitations of bulk reconstruction and geometry learning (Akers et al., 7 Nov 2024). Quantitative refinements continue to be developed through conformal bootstrap, Mellin-space techniques, and holographic RG methods, providing a versatile toolkit for translating gravitational phenomena into rigorous quantum field theoretic data.