Holographic Dictionary Overview

Updated 27 November 2025

Holographic Dictionary is a precise set of algorithmic correspondences that map bulk gravitational fields to boundary operators, sources, and symmetries.
It employs near-boundary expansions, counterterms, and renormalization techniques to extract source and expectation values across diverse geometries like AdS, Lifshitz, and flat spacetimes.
By incorporating elements of computational complexity and cryptographic analogies, it enables efficient operator reconstruction while concealing detailed bulk geometry information.

A holographic dictionary is a precise set of correspondences—often algorithmic or functional relations—mapping local or nonlocal data between a quantum gravitational bulk theory (such as gravity in asymptotically AdS, Lifshitz, flat, or Carrollian spacetimes) and a non-gravitational boundary field theory or related algebraic structure. These correspondences define the duality between bulk and boundary variables, operators, states, correlation functions, and symmetries. The construction and detailed content of the holographic dictionary depends on the geometry, matter content, and asymptotics of the bulk, as well as on the boundary conditions and variational principle, and is especially nontrivial for theories with non-AdS asymptotics, extended symmetry algebras, multi-trace deformations, and situations involving computational complexity.

1. Foundational Structure and General Features

The holographic dictionary specifies how to relate bulk fields (metric, matter, gauge fields, higher-spin fields, etc.) to boundary operators, sources, and expectation values. This process relies on a near-boundary expansion: for example, in standard AdS/CFT, a scalar field $\Phi$ in the bulk near the conformal boundary admits an expansion

$\Phi(z,x) = z^{d-\Delta} \phi_{(0)}(x) + z^\Delta \phi_{(\Delta)}(x) + \cdots$

where $z$ is the radial coordinate (with $z\to 0$ at the boundary), $d$ is the boundary dimension, and $\Delta$ is the scaling dimension. The non-normalizable mode $\phi_{(0)}(x)$ corresponds to the source for an operator $\mathcal{O}(x)$ in the dual field theory, while the normalizable mode $\phi_{(\Delta)}(x)$ corresponds to the vacuum expectation value (vev) $\langle\mathcal{O}(x)\rangle$ .

For each relevant holographic setting, the dictionary systematically assigns:

A mapping for metric or frame components to boundary stress tensors or conserved currents.
Identification of bulk gauge fields and their boundary behavior to conserved currents and chemical potentials.
Prescription for extracting boundary one-point functions as functional derivatives of the renormalized on-shell bulk action with respect to boundary sources.
A set of boundary conditions and counterterms (local in standard AdS, but with modifications for double holography, non-relativistic, or flat settings) that ensure a well-defined variational problem and finite observables.

This structure immediately extends to more complex cases, including multi-trace deformations (requiring mixed boundary conditions), theories with hyperscaling violation and/or non-relativistic scaling, and settings with additional symmetries, such as higher-spin or superconformal algebras.

2. Variants by Bulk Geometry and Asymptotics

The form of the holographic dictionary crucially depends on the asymptotic structure:

a. AdS/CFT and Higher-Curvature Extensions

In AdS, the canonical dictionary for fields and operators is well-established, with corrections for higher-curvature gravity (e.g., Gauss–Bonnet or Lovelock terms) entering in the mapping of bulk couplings to CFT correlators and the normalization of the stress tensor two- and three-point functions, as well as in the implementation of positivity, causality, and transport bounds (0911.4257, Sinamuli et al., 2017). For example, the two-point correlator normalization $T$ and energy flux coefficients $(t_2, t_4)$ are explicit functions of the Gauss-Bonnet coupling, with tight consistency constraints.

b. Double Holography

Double holographic models (e.g., Karch-Randall setups) require expanding bulk fields near a brane cutoff and reading off operator data from subleading coefficients, with the novel feature that the induced brane gravity is dynamical and metric fluctuations acquire a nonzero Pauli–Fierz mass via a mapping: $\langle\mathcal{O}(x)\rangle = (2\Delta_+ - d)\varepsilon_B^{\Delta_+} \Phi_{(\Delta)}(x) + \mathcal{O}(\varepsilon_B^{\Delta_+ + 1/2})$ establishing the direct relation between subleading bulk coefficients and brane/boundary operator expectation values (Neuenfeld, 2021).

c. Lifshitz, Hyperscaling-Violating, and Schrödinger Geometries

For models with nontrivial dynamical exponent $z$ and/or hyperscaling violation $\theta$ , the dictionary involves a refined scaling analysis in which all thermodynamic, operator, and geometric observables scale with dimension dictated by $(z, \theta)$ . For example, the central charge $C$ in the boundary thermodynamics scales as $C\propto L^{d-1-\theta} r_F^\theta/G$ , where $L$ is the relevant length scale and $r_F$ is associated with the hyperscaling violation (Cong et al., 21 Oct 2024, Papadimitriou, 2014). Both Fefferman–Graham expansion and the identification of source and vev data are constructed via recursive solutions of the radial Hamilton–Jacobi equation, with covariance under anisotropic or non-relativistic scaling (Andrade et al., 2014, Andrade et al., 2014).

d. Flat Holography, Carrollian Limits, and Celestial CFT

In asymptotically flat spacetimes, dictionary entries translate bulk radiative modes (Bondi mass, angular momentum, news, and shear) at null infinity to conserved fluxes and moments of a coupled Carrollian field theory, with energy and momentum densities on the boundary identified as the leading coefficients in the large- $r$ expansion of physical fields (Fiorucci et al., 30 Apr 2025, Liu et al., 20 Jan 2024). On the celestial sphere, each bulk field decomposes into an infinite family of boundary operators with principal-series scaling dimensions, and correlation functions are constructed via harmonic (spherical conical) mode expansions (Hao et al., 2023).

3. Operator Map, Sources, and VEVs: Algorithmic Extraction

The dictionary assigns the precise identification of bulk and boundary data for all operator types. Concrete prescriptions include:

Non-normalizable (leading) mode $\leftrightarrow$ source for dual operator; normalizable (subleading) mode $\leftrightarrow$ vev $\langle\mathcal{O}\rangle$ .
For each field sector (scalar, vector, tensor), tables specify boundary operator correspondence, their scaling dimensions, and explicit functional dependence (see, e.g., the summary “dictionary table” in (Hao et al., 2023), the frame-based map in (Andrade et al., 2014), and detailed Hamilton–Jacobi expansions in (Papadimitriou, 2014)).
For non-relativistic geometries (Schrödinger, Lifshitz), leading frame fields correspond to energy, momentum, particle currents, and extra scalar operators, with subleading data giving vevs and capturing Ward identities. At $z=2$ , the boundary stress tensor complex descends directly from the $k_\xi=0$ sector (Andrade et al., 2014, Andrade et al., 2014).
In the presence of multi-trace deformations, mixed boundary conditions are imposed, with the associated RG running directly computable holographically:

$\beta_f = -\lambda_n c_n + O(\lambda_n^2)$

where $\lambda_n$ is the $n$ -trace coupling in the bulk (Aharony et al., 2015).

For complex boundary algebraic structures (superconformal, $\mathcal{W}$ -algebras), the dictionary matches bulk Chern–Simons connection components to boundary current multiplets, including nontrivial OPEs, spectral flows, and hermiticity constraints (Castro et al., 2020, Özer et al., 24 Jul 2024).

4. Ward Identities, Symmetries, and Anomalies

The dictionary is constructed to enforce all physical Ward identities:

Boundary conservation and trace (Weyl or scaling) Ward identities are derivable from the bulk equations and the variational structure of the renormalized action. For running dilaton AdS $_2$ solutions, for example, the trace Ward identity is anomalous:

$\langle T^t{}_t\rangle + \langle O_\phi\rangle = \mathcal{A}(t), \quad \mathcal{A} = \frac{\ell}{8\pi G_2} \partial_t(\alpha'/\alpha)$

signifying a nontrivial conformal anomaly in the dual quantum mechanics (Cvetič et al., 2016).

In non-relativistic and flat settings, anisotropic/diffeomorphism/CFT algebraic structure is mirrored by the corresponding bulk gauge or frame invariances, producing generalized conservation laws, scaling identities, and anomalies (or their absence where absorbed by source redefinitions) (Andrade et al., 2014, Liu et al., 20 Jan 2024).
In extended asymptotic symmetry contexts (e.g., BMS, celestial or Carrollian field theory), the energy and angular-momentum flux-balance laws emerge as local consequences of Carroll, Weyl, and diffeomorphism invariance at null infinity, with leading and subleading expansion coefficients realizing the boundary "momenta" (Fiorucci et al., 30 Apr 2025).

5. Holographic Renormalization, Counterterms, and Regularization Schemes

To extract finite, renormalized correlation functions and well-defined operator map, the dictionary incorporates a systematic procedure:

Addition of boundary counterterms (local in standard settings, nonlocal or mixed in multi-trace or double holographic cases), ensuring finite on-shell action and well-posed variational principle. For example, in AdS $_2$ , the running dilaton class requires (Cvetič et al., 2016):

$S_{\mathrm{ct}} = -\frac{1}{8\pi G_2} \int dt\sqrt{-\gamma} \left(\ell^{-1} e^{-\phi} - u_0 \ell\, \Box_t e^{-\phi}\right)$

In certain settings, particularly in dimensional regularization frameworks, the source field is redefined to absorb all secular terms, and the renormalized coupling is equated to the bulk field at holographic RG scale $z=L$ :

$\kappa \psi_L (z=L, x) = g_L (x) \equiv L^\epsilon \varphi_L(x)$

with the corresponding $\beta$ -function extractable from the bulk flow generator (Bzowski et al., 2019, Aharony et al., 2015).

For flat holography, a fundamental bulk-to-boundary reduction leads to a boundary theory with a symplectic form and constraints implementing the infinite tower of descendant fields, flux Hamiltonians, and (upon regularization) central extensions proportional to the number of radiative DOFs (Liu et al., 20 Jan 2024).

6. Advanced Entries: Complexity, Computational Tasks, and Cryptography

The content of the holographic dictionary can be viewed through computational complexity. Two crucial tasks are distinguished and mapped into computational classes (Akers et al., 7 Nov 2024):

Operator Reconstruction: Given explicit access to the encoding map $V$ , reconstruct boundary operators corresponding to bulk unitaries. Unless "Python’s lunch" emerges in the geometry, this task lies in BQP (poly-size quantum circuit) given bulk and code data.
Geometry Reconstruction: Given only black-box access to $k$ copies of the boundary state $V \rho_{\text{bulk}} V^\dagger$ , extract a geometric property (e.g., minimal Ryu–Takayanagi area). Using cryptographic pseudoentanglement and pseudorandom unitaries, it is shown that geometry reconstruction is generically hard—computationally indistinguishable ensembles of boundary states can have distinct underlying bulk geometries but no efficient quantum algorithm can distinguish them.

This creates a pronounced separation, suggesting that even quantum laboratory simulation of the boundary CFT does not generically allow efficient extraction of bulk geometry, which can be cryptographically hidden (Akers et al., 7 Nov 2024).

A deep analogy to quantum fully homomorphic encryption (QFHE) is drawn: the holographic code implements a map that allows efficient homomorphic operator application (operator reconstruction), but not decryption (geometry reconstruction), unless a secret key is provided.

7. Extension to Specialized Physical Systems and Generalized Settings

The dictionary framework applies equally to systems with nontrivial condensed-matter analogs, higher-dimensional boundaries, non-essentially AdS backgrounds, and new algebraic structures:

In holographic metallic ferromagnets, magnetization and magnetic susceptibility are encoded in near-boundary expansion coefficients, reproducing mean-field exponents near $T_c$ and both magnon and conduction-electron contributions at low temperature (Yokoi et al., 2015).
For conductors, beyond the standard current/conserved charge, novel dictionary entries correspond to the carrier density $n$ and mean carrier velocity $v$ , computable from worldvolume horizon data, thus extending the standard AdS/CFT map to dynamic transport observables (Hoshino et al., 2017).
In topological quantum information, a pictorial dictionary relates planar para algebra diagrams to $n$ -qudit linear maps, string Fourier transforms, and multipartite entangled resource states, creating a direct computationally actionable duality between algebraic operations and topological manipulations (Jaffe et al., 2016).

The holographic dictionary thus not only establishes an explicit operator/state correspondence but also encodes the interplay between geometric, algebraic, computational, and symmetry data in a logic that generalizes the original AdS/CFT paradigm to a wide array of physically and mathematically rich settings. Across these, the dictionary serves as both a computational tool and a guiding principle for extracting, protecting, or hiding geometric and dynamical information between bulk and boundary theories, reinforcing the nontriviality and subtlety of holographic duality in modern high-energy and mathematical physics.