Modified Holographic Dictionary

• Modified holographic dictionary is a set of controlled refinements to the standard AdS/CFT correspondence, adjusting asymptotic source/vev assignments for non-relativistic and finite-cutoff scenarios.
• It incorporates modifications like anisotropic scaling, hyperscaling violation, and running multi-trace couplings to address varying bulk asymptotics and boundary conditions.
• Operational reinterpretations in computational and double holography contexts bridge bulk geometry reconstruction with boundary complexity and encoding methods.

“Modified holographic dictionary” denotes a family of refinements of the bulk/boundary map used when the standard AdS/CFT identifications are no longer adequate. In the literature, the phrase covers several distinct but related programs: modifications forced by Lifshitz, hyperscaling-violating, Schrödinger, AdS$_2$, or braneworld asymptotics; refinements of source/vev assignments for multi-trace couplings and finite-cutoff theories; operational reinterpretations of what it means to “compute the dictionary”; and, in a different cosmological usage, phenomenological maps from infrared cutoff data to effective FRW dynamics rather than bulk/boundary duality in the AdS/CFT sense (Cong et al., 2024, Aharony et al., 2015, Akers et al., 2024).

1. Standard baseline and the logic of modification

In the standard AdS/CFT framework, the holographic dictionary is organized by asymptotic source/vev separation, symmetry matching, and, in thermodynamic settings, simple bulk-to-boundary rescalings. One explicit thermodynamic form quoted in the literature is

$$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$

In the quantum-error-correcting-code view of AdS/CFT, one similarly assumes a code-subspace embedding

$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$

so that bulk states and operators are mapped to boundary states and operators, at least approximately on the code subspace (Cong et al., 2024, Akers et al., 2024).

A modified dictionary is introduced when one or more of these baseline assumptions fail. The reasons vary. In non-relativistic holography, anisotropic scaling and hyperscaling violation alter the scaling dimensions of thermodynamic and operator data. In AdS$_2$, the dilaton and electric flux change the source/vev assignments. In multi-trace holography, bulk interactions induce cutoff-dependent boundary conditions rather than merely new source terms. In double holography, the leading near-brane mode is a dynamical brane field rather than a fixed source. In computational formulations, “the dictionary” ceases to be a single map and becomes a family of tasks with different input/output models and different complexity (Cvetič et al., 2016, Neuenfeld, 2021, Aharony et al., 2015, Akers et al., 2024).

This suggests that “modified holographic dictionary” is not a single formal object. It is a class of controlled departures from the relativistic, asymptotically AdS, fixed-source paradigm.

2. Anisotropic, hyperscaling-violating, and non-relativistic dictionaries

For asymptotically Lifshitz and hyperscaling-violating Lifshitz backgrounds, the standard relativistic dilatation expansion is insufficient. The general Hamilton–Jacobi construction therefore uses two commuting operators,

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$

and

$$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$

together with the asymptotic second-class constraint

$B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$

The resulting boundary operator content is not a relativistic stress tensor but a non-relativistic energy-momentum complex, and the source for the energy flux is set to zero by Lifshitz boundary conditions (Chemissany et al., 2014).

The same reorganization appears in Schrödinger holography. For $z<2$, the leading frame fields

$$e^+ = r^{-z} \hat e^+,\qquad e^- = r^{z-2} \hat e^-,\qquad e^I = r^{-1} \hat e^I$$

define the boundary geometry, and the dual operators form a Schrödinger stress-energy complex on $(t,\xi,\vec x)$. For $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$0, the scaling symmetry no longer acts on the extra null direction $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$1, so $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$2 is not treated as a field-theory direction. Instead one Fourier expands bulk fields in $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$3,

$$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$4

and interprets each Fourier mode as dual to an operator $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$5 in a theory living only on $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$6 (Andrade et al., 2014, Andrade et al., 2014).

A thermodynamic version of the same phenomenon appears for charged black holes with asymptotically Lifshitz and hyperscaling-violating behavior. There, energy and temperature no longer scale as $$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$7, but instead obey modified identifications fixed by anisotropic scaling and hyperscaling violation,

$$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$8

The same analysis gives a central charge

$$\tilde E=\frac{ML}{R},\qquad \tilde T=\frac{TL}{R},\qquad \tilde S=\frac{A}{4G},\qquad \tilde\Phi=\frac{\Phi}{R},\qquad \tilde Q=QL.$$9

which depends on the hyperscaling violating parameter but not on the Lifshitz dynamical exponent, and a generalized Smarr relation

$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$0

In this setting the bulk Smarr formula is dual to the boundary Euler relation

$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$1

provided one uses the modified dictionary and the Lifshitz/hyperscaling-violating equation of state (Cong et al., 2024).

These constructions all modify the dictionary for the same structural reason: anisotropic scaling and running scalars change which asymptotic data are independent, how derivatives are graded, and which observables remain relevant.

3. AdS$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$2, double holography, and finite-cutoff source/vev structure

In AdS$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$3 Einstein–Maxwell–Dilaton theory, the holographic dictionary splits into two distinct sectors: running dilaton solutions and constant dilaton solutions. For running dilaton asymptotics, the independent local sources are $$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$4, $$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$5, and $$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$6, and the one-point functions are

$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$7

$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$8

The trace identity becomes

$$V:\mathcal H_{\mathrm{code}}\to \mathcal H_{\mathrm{bdry}},$$9

The conformal anomaly is therefore induced by the dilaton source. For constant dilaton asymptotics, by contrast,

$_2$0

so the scalar operator dual to the dilaton carries the only nontrivial local observable, while the stress tensor vanishes identically (Cvetič et al., 2016).

AdS$_2$1 also modifies the treatment of the gauge field. When the conserved electric flux dominates asymptotically, the correct renormalization involves a Legendre transform and a counterterm written in terms of the canonical momentum $_2$2, rather than a naive local counterterm in $_2$3. The paper states that the correct boundary term is fixed by the requirement that it diagonalize the symplectic map between phase-space variables and asymptotic modes and preserve Ward identities already at finite radial cutoff (Cvetič et al., 2016).

In double holography, the modification takes a different form. For Karch–Randall braneworlds, there is no faithful bulk/brane dictionary analogous to ordinary AdS/CFT. The low-energy brane theory is instead described by a modified extrapolate-style dictionary: the induced brane field is the leading near-brane data, while the expectation value of the brane operator is encoded in the subleading Fefferman–Graham coefficient after subtracting source-like local terms. For the stress tensor,

$_2$4

to leading order in the critical limit (Neuenfeld, 2021).

A plausible implication is that finite-cutoff and lower-dimensional holography do not merely alter numerical coefficients. They can invert the usual meaning of leading and subleading data, or move the nontrivial physics from the stress tensor sector into scalar or flux sectors.

4. Multi-trace running and the dictionary for renormalization-group data

A distinct modification concerns the relation between bulk interactions and the beta functions of boundary multi-trace couplings. In the standard large-$_2$5 dictionary, multi-trace deformations are encoded by nonlinear mixed boundary conditions. For a scalar dual to $_2$6, with near-boundary coefficients $_2$7, a functional $_2$8 produces

$_2$9

That part of the dictionary was already established (Aharony et al., 2015).

The modification identified by Aharony, Gur-Ari, and Klinghoffer is that this does not yet capture the running of marginal multi-trace couplings generated by bulk interactions, even when the multi-trace coupling itself is set to zero. For a scalar interaction

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$0

with $$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$1, the near-boundary solution contains a logarithmic term,

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$2

The mixed boundary condition therefore becomes necessarily $$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$3-dependent,

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$4

and the corresponding field-theory coupling runs as

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$5

For marginal double-trace deformations, the modified dictionary similarly gives

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$6

in the presence of a bulk mass-shift interaction (Aharony et al., 2015).

The missing dictionary entry is therefore the map from specific classical bulk couplings to cutoff-dependent boundary conditions representing multi-trace beta functions. This is not a change in the bulk/boundary pairing of fields and operators; it is a change in how RG data are encoded holographically.

5. Computational and operational reinterpretations

A further use of “modified holographic dictionary” concerns computability rather than kinematics. In this view, the phrase does not mean a new bulk/boundary correspondence. It means that the practical content of the AdS/CFT dictionary depends on the task being attempted. The central distinction is between operator reconstruction and geometry reconstruction. Operator reconstruction asks for a boundary operator $$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$7 implementing a known bulk operator $$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$8 on a known code subspace,

$$\widehat\delta := \int d^{d+1}x\left( 2\gamma_{ij}\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right),$$9

whereas geometry reconstruction asks how hard it is to infer geometric properties of an unknown bulk dual from one or many copies of a boundary state (Akers et al., 2024).

The paper argues that these are different computational problems because they have different input/output models. Existing evidence, such as HKLL and the strong Python’s lunch conjecture, suggests that operator reconstruction is plausibly generically easy when the geometry is already known. By contrast, geometry reconstruction may be generically hard because geometry is encoded in entanglement, and entanglement structure can be computationally hidden (Akers et al., 2024).

The main theorem-level statement is that for any two bulk geometries $$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$0, there exist efficiently constructable ensembles approximately obeying Ryu–Takayanagi for $$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$1 that are computationally indistinguishable to any polynomial-time quantum algorithm given polynomially many copies. In the PRU-based construction,

$$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$2

while no polynomial-time quantum algorithm can distinguish the two ensembles with non-negligible bias. The paper calls such families pseudoentangled holographic state ensembles (Akers et al., 2024).

In this usage, the dictionary is modified operationally: accessibility, not formal existence, becomes the central issue. A common misconception is that a formally exact dictionary must be uniformly efficient to use. The cited work rejects that inference. It treats “compute the dictionary” as a complexity-sensitive statement rather than a yes/no property of AdS/CFT itself.

6. Alternative reconstructions and broader terminological scope

Several recent programs extend the term beyond standard asymptotically AdS holography. In flat/celestial holography, a single 4D bulk scalar on Minkowski space is mapped to two continuous families of operators on the celestial sphere, with dimensions on the principal series. The renormalized flat-space dictionary takes the schematic form

$$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$3

and the two operator families are interpreted as outgoing and ingoing sectors of the bulk field (Hao et al., 2023).

In asymptotically flat bulk reduction, the boundary theory at null infinity is specified by a Carrollian manifold, a fundamental field $$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$4, an infinite tower of descendant fields, constraints among the descendants, and a symplectic form

$$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$5

Bulk leaky fluxes become boundary Hamiltonians, and the quantum symmetry algebra acquires a divergent central charge interpreted as reflecting propagating degrees of freedom (Liu et al., 2024).

Cauchy-slice holography proposes another reorganization. There, a sufficiently large $$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$6-deformation of a Euclidean CFT produces a theory on a bulk Cauchy slice whose partition function is identified with a Wheeler–DeWitt wavefunctional,

$$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$7

The resulting kernel defines maps between bulk canonical quantum-gravity states and ordinary boundary CFT states, and the proposal argues for the equivalence

$$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$8

Here the modified dictionary is explicitly canonical: time rather than space is treated as the emergent bulk direction (Araujo-Regado et al., 2022).

An even more radical example is the $$\delta_B :=\int d^{d+1}x\left( 2Y^{-1}B_iB_j\frac{\delta}{\delta \gamma_{ij}} + B_i\frac{\delta}{\delta B_i} \right), \qquad Y:=B^2,$$9-dimensional Majorana generalized-free-field reconstruction of a $B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$0-dimensional bulk geometry with no assumed asymptotically AdS region and no symmetry matching. Starting from the boundary anti-commutator $B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$1, the bulk conformal factor is reconstructed as

$B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$2

with $B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$3 built from poles and zeros of $B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$4. The resulting near-horizon curvature can be negative, zero, or positive depending on the spectral data, and simple boundary models can produce de Sitter-like or anti-de Sitter-like near-horizon duals (Nebabu et al., 23 Feb 2026).

Finally, the term also appears in cosmological holographic dark-energy models, but there it has a different meaning. The “dictionary” is a phenomenological map from an infrared cutoff choice to an effective energy density and then to FRW evolution. Representative prescriptions are

$B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$5

and

$B_i\sim B_{oi}=\sqrt{-Y_o(\phi)}\,\mathfrak n_i.$6

The cited works explicitly state that this is not a formal AdS/CFT holographic dictionary, but a modified holographic energy-density prescription in cosmological phenomenology (Chakraborty et al., 2020, Granda et al., 2019).

Taken together, these usages show that the phrase “modified holographic dictionary” has become a marker for a specific kind of intervention: preserving the idea of bulk/boundary encoding, while altering the precise map because the asymptotics, operator content, quantization, computational task, or even the meaning of “holographic” itself has changed.