Holographic Consistency Conditions

Updated 22 November 2025

Holographic consistency conditions are a set of requirements ensuring that bulk spacetimes and boundary CFTs yield a unitary and physically meaningful quantum gravitational system.
They comprise geometric inequalities, causality constraints, and algebraic identities that limit bulk gravitational actions and boundary entanglement prescriptions.
These conditions also guide RG flows and moduli space potentials, enforcing information-theoretic and dynamical bounds while ruling out pathological bulk geometries.

A holographic consistency condition is a requirement that must be satisfied by the bulk theory, boundary theory, or their correspondence, in order for a proposed instance of holographic duality (typically AdS/CFT or its generalizations) to yield a well-defined, unitary, and physically meaningful quantum gravitational system. These conditions take varied forms—geometric inequalities, entropy/correlation constraints, algebraic identities, and causality criteria—and collectively constrain the allowable class of bulk spacetimes, boundary CFTs, and their couplings. They are often derived by demanding the mutual compatibility of boundary quantum field theory axioms, bulk gravitational dynamics, and the structural logic of the holographic dictionary.

1. Geometric and Isoperimetric Inequalities

A fundamental class of holographic consistency conditions is geometric in nature, most quantitatively crystallized in isoperimetric-type inequalities. For example, in Euclidean AdS gravity, the minimum value of the action of a probe brane homologous to the boundary, $S_{\rm probe}^E$ , is constrained to satisfy an inequality with respect to the (on-shell) bulk gravitational action $I_{\rm bulk}^E$ : $S_{\rm probe}^E \ge I_{\rm bulk}^E.$ This is equivalently stated via an auxiliary "isoperimetric functional" $\mathfrak{S}^E(\Sigma) = \mathrm{Area}(\Sigma) - \frac{d}{L} \mathrm{Vol}(M_\Sigma)$ , which must be non-negative for all hypersurfaces $\Sigma$ homologous to the boundary: $\mathfrak{S}^E(\Sigma) \ge 0.$ Violation of this bound implies an inconsistency either in the existence of the partition function of the dual CFT or in the stability of the bulk background (McInnes et al., 2015, Ong, 2022). In Lorentzian signature, analogous functionals regulate brane nucleation instabilities (“Seiberg–Witten” instabilities), with runaway nucleation destroying staticity if the bound is violated.

These geometric conditions are particularly nontrivial for higher-derivative (e.g., Gauss-Bonnet) gravity: the sign of the Gauss-Bonnet parameter $\alpha$ is fixed by the requirement that the isoperimetric functional remains non-negative throughout the bulk, enforcing $\alpha>0$ (and small enough for reality of the AdS vacuum), in agreement with string-theoretic $\alpha'\sim 1/\alpha$ expectations (Ong, 2022).

2. Causality, Entanglement, and Boundary QFT Requirements

Boundary causality and information-theoretic constraints rigidly restrict possible bulk spacetimes via wedge-inclusion relations. Any prescription for holographic entanglement entropy must satisfy “wedge-insensitivity” and “locality” compatibility:

The entanglement entropy $S[A]$ of a region $A$ can only depend on the causal domain of $A$ and $\partial A$ in the boundary (Headrick et al., 2014).
The HRT (Hubeny-Rangamani-Takayanagi) surface in the bulk must lie strictly within the causal shadow of the boundary region, enforced via the causal-wedge inclusion condition $C_W[R] \subseteq E_W[R]$ , where $C_W[R]$ is the causal wedge and $E_W[R]$ the entanglement wedge for region $R$ (Engelhardt et al., 2020).

If a bulk spacetime allows, for example, a trapped surface outside any event horizon, this directly violates the wedge-inclusion and is non-holographic: quantum gravity enforces a holographic cosmic censorship in classical AdS (Engelhardt et al., 2020). More generally, surfaces encoding von Neumann entropy in higher-curvature gravity must satisfy causality constraints, as well as Euler-Lagrange extremality, homology, and replica regularity (Erdmenger et al., 2014).

3. Topological and Transmission Constraints

The No Transmission Principle (NTP) sharply states that two boundary QFTs with disjoint Hilbert spaces cannot exchange signals; this is lifted to a strong constraint on the bulk: any traversable wormhole, cosmological bounce to a new AdS region, or a naked singularity allowing such nontrivial passage is forbidden. Consequently, any bulk geometry in which a singularity could be traversed by causal curves violating the independence of boundary QFTs must be ruled out, generalizing cosmic censorship and topological censorship to the holographic context (Engelhardt et al., 2015).

These principles yield additional theorems: the absence of singularity–free bounces, the nonexistence of traversable wormholes, and the necessity that any bulk singularity either be hidden behind an event horizon or completely cut off causal evolution to avoid cross-boundary signals, covering all values of ’t Hooft coupling and $N$ (Engelhardt et al., 2015).

4. Holographic Consistency in RG Flows and Quantum Anomalies

The structure of the boundary renormalization group induces stringent algebraic consistency requirements known as Holographic Wess-Zumino (HWZ) conditions. These guarantee that the algebra of local Weyl variations closes onto diffeomorphisms: $[\delta_{\sigma_1}, \delta_{\sigma_2}] W = 0,$ which, in the bulk, is realized as the Dirac algebra of Hamiltonian and momentum constraints. Enforcing HWZ conditions uniquely fixes the bulk theory to be Einstein-Hilbert gravity (plus matter, for $D\leq4$ ), and ensures that holographic Weyl anomalies are compatible with the dual RG structure (e.g., $a=c$ in $D=4$ CFTs with gravity duals) (Shyam, 2017).

In lower dimensions, e.g., $d=3$ , consistency conditions derived from the local RG and Wess-Zumino integrability further constrain possible anomalies and their coefficients, factorizations, and scheme ambiguities as verified in explicit bulk models (Nakayama, 2013).

5. Dynamical and S-matrix Constraints

Holography also imposes nonlocal bounds on bulk degrees of freedom accessible via the boundary. The number of localized bulk wavepackets that can be encoded or “created” by the boundary CFT within an AdS volume is rigorously bounded, scaling with $(\omega R)^{d-1}$ for wavepackets of frequency $\omega$ and AdS radius $R$ (Gary, 2012). This ensures that the dual CFT is just sufficient to encode the full Hawking evaporation S-matrix for black holes whose decay times are less than $R$ , with the bound softening below the Planck scale and saturating the Bekenstein-Hawking entropy at Planckian energies.

Violation, for example by trying to encode more than $N_{\max}(\omega)$ independent bulk modes, signals an incompatibility of the holographic map with unitarity and bulk locality (Gary, 2012).

6. Landscape and Moduli Space Constraints

Consistency conditions derived from holography severely restrict possible scalar potentials in the string landscape. The requirement of a well-defined dual CFT, interpretable as a decoupled brane theory, enforces:

The Trans-Planckian Censorship Conjecture (TCC) for positive potentials (C1), forbidding slow-roll plateaus and insufficiently steep tails.
An “Asymptotic No-Scale Separation” (ANSS) condition (C2), demanding that negative potentials admit an infinite-distance direction in moduli space along which gravity decouples, with explicit algebraic bounds, e.g., $|\nabla V/V|\le2\sqrt{\frac{d-1}{d-2}}$ (C4) (Bedroya et al., 19 Nov 2025).
The necessity of an explicit brane-worldvolume realization, precluding very shallow, very steep, or certain “crossing” potentials; these criteria unambiguously exclude various flux vacuum scenarios such as DGKT and KKLT as admitting consistent holographic duals (Bedroya et al., 19 Nov 2025).

7. Additional Classes and Applications

Other notable classes include:

Gravitational “corner” conditions, implementing the infinite sequence of relations constraining the analytic expansion of the boundary metric at the initial-value surface. These guarantee that initial spacelike (bulk) data and boundary Dirichlet data are not independent, but must be matched at the AdS boundary “corner” for precise bulk-to-boundary reconstructibility (Horowitz et al., 2019).
Consistency conditions for non-AdS/non-CFT “holographic” models, including null-energy constraints for generic hyperscaling-violating metrics, the scaling-dimension structure for observables, universal ratios (e.g., Wilson and Wiedemann-Franz), and membrane-paradigm–boundary checks. The simultaneous satisfaction of such relations is a minimal requirement for phenomenologically credible holographic models in condensed matter (Khveshchenko, 2014).

Collectively, holographic consistency conditions form a tightly interlinked web of geometric, analytic, algebraic, dynamical, and information-theoretic constraints. They ensure that not only does the dictionary between bulk/boundary exist, but that it respects field-theoretic causality, quantum complexity, dynamical stability, and topological obstruction. These conditions are now widely understood as foundational: violation in any regime signals either a breakdown of the proposed duality or necessitates new physical mechanisms—e.g., higher-spin corrections, quantum gravity nonlocality, or novel boundary sectors—to restore consistency.