Holographic Consistency in AdS/CFT

• Holographic consistency condition is a set of strict algebraic, analytic, and geometric criteria that ensure the absence of divergences in correlators, operator algebras, and gravitational actions in AdS/CFT.
• It mandates constraints such as vanishing extremal couplings and isoperimetric inequalities for probe brane actions to prevent instabilities and ensure large-N factorization.
• Moreover, it governs aspects of dynamical consistency and quantum error correction, ensuring RG invariance and proper operator reconstruction, thereby demarcating viable bulk EFTs and dual CFTs.

The holographic consistency condition encompasses a class of stringent requirements that must be satisfied for a bulk gravitational theory—in particular, one in asymptotically AdS spacetime—to admit a consistent dual description as a large-$N$ conformal field theory (CFT). These conditions go well beyond the formal existence of a bulk-boundary map, instead imposing algebraic, analytic, and geometric constraints that guarantee the absence of pathologies in correlators, operator algebras, gravitational actions, or entropy measures. Their violation rules out the existence of a holographic dual or signals deep inconsistency in the proposed bulk or boundary theory.

1. Extremal Coupling Vanishing in AdS/CFT

A central example of a holographic consistency condition arises in the computation of three-point functions of scalar operators, where the conformal dimensions satisfy an "extremal" arrangement $\Delta_i = \Delta_j + \Delta_k$. In the standard AdS effective action, cubic scalar couplings are present of the form

$$S_3 \supset \int d^{d+1}x \sqrt{g} \left[ c_{ijk}\, \phi_i \phi_j \phi_k + d_{ijk}\, \phi_i \partial\phi_j \cdot \partial\phi_k \right]\,,$$

with canonically normalized bulk fields $\phi_i$. The relevant combination for correlators is

$$c'_{ijk} = c_{ijk} + \frac12 (m_i^2 - m_j^2 - m_k^2) d_{ijk}\,, \qquad m_i^2 = \Delta_i(\Delta_i-d)\,.$$

The crucial consistency condition is: If $\Delta_i = \Delta_j + \Delta_k$ (or any "super-extremal" generalization), then $c'_{ijk} = 0$.

This ensures that otherwise divergent Witten diagrams are absent, enforcing large-$N$ scaling and orthogonality between single- and multi-trace CFT operators. If this condition is violated, the scaling of three-point bulk diagrams is inconsistent with large-$N$ factorization in the dual CFT, making the bulk theory incompatible with a standard holographic dual (Bobev et al., 11 Dec 2025).

Verification in explicit string constructions (e.g., DGKT AdS$_4$ vacua) confirms nontrivial cancellations between potential and derivative couplings, highlighting that this constraint is not accidental but universal whenever the large-$N$ dictionary applies.

2. Geometric Consistency: Brane Actions, Isoperimetric Bounds, and Instabilities

Another core consistency criterion is geometric and relates the on-shell Euclidean gravitational bulk action $S_g^*$ to the extremal action $S_b^*$ of a probe brane (typically a $(d-1)$-brane) wrapping a bulk hypersurface homologous to the boundary:

$S_g^* = \frac{N}{\gamma} S_b^*$

or equivalently, for every hypersurface $\Sigma$,

$$\mathfrak{S}^E[\Sigma] = A(\Sigma) - \frac{d}{L} V(M_\Sigma) \geq 0\,,$$

where $A(\Sigma)$ is the area, $V(M_\Sigma)$ the enclosed volume, and $L$ the AdS curvature radius.

This isoperimetric inequality ensures that the action of any probe brane is always non-negative, preventing Seiberg–Witten-type instabilities (catastrophic brane nucleation) (Ong, 2022, McInnes et al., 2015). Violation of this inequality in Lorentzian signature leads to uncontrolled brane pair production (with rate $\Gamma \sim \exp[-\mathfrak{S}^{\rm L}(r)]$) and signals inconsistency of the bulk background, as static solutions become untenable.

Specializations of the inequality further constrain allowed parameters in modified-gravity bulk actions (e.g., requiring the Gauss–Bonnet parameter $\alpha > 0$ in Einstein–Gauss–Bonnet AdS gravity), in agreement with string-theoretic expectations (Ong, 2022).

3. Algebraic Constraints: Operator Algebras, Complementary Recovery, and Holographic Error Correction

The emergence of entanglement wedge reconstruction and bulk operator algebra reconstruction in AdS/CFT is governed by a set of algebraic consistency conditions. Formulated via quantum error-correcting codes and operator algebras, these require:

• The existence of a conditional expectation $E_A : M_A \to N_A$ mapping boundary von Neumann algebras $M_A$ (associated to boundary region $A$) to bulk algebras $N_A$ (entanglement wedge of $A$), with $E_A$ preserving modular data and satisfying normality, unitality, and complete positivity.
• Complementary recovery ("c-recoverability"): both $N_A$ and its commutant $N_A'$ must be reconstructable from $M_A$ and $M_A'$, respectively.
• The inclusion structure must match the Longo–Rehren net: $N_i \subset M_i \subset M_{\mathrm{can},i}$ for families of nested regions.
• For disjoint regions, a dual-additivity property determines entanglement wedge phase transitions and RT entropy "jumps."

Violation of these conditions leads to breakdowns in bulk operator reconstruction or unphysical additivity properties in overlapping regions, requiring resort to approximate (but not exact) conditional expectations (Faulkner, 2020).

4. Causal and Entropic Consistency: Entanglement Wedge, Mutual Independence, and Causality

A fundamental aspect of holographic consistency is the requirement that entanglement entropy, modular Hamiltonians, and wedge-reconstructable bulk regions respect causal and entropic constraints:

• Causal wedge inclusion: The causal wedge $\mathcal{W}_c(A)$ for any boundary region $A$ must obey $\mathcal{W}_c(A) \subset \mathcal{W}_e(A)$ (entanglement wedge). This prohibits trapped surfaces from being accessible to boundary causal evolution, guaranteeing that strong gravity regions are cloaked by horizons (Engelhardt et al., 2020).
• Mutual independence of wedges: In general spacetimes (beyond AdS), for families of bulk regions $a_i$, each must be outside the entanglement wedge of the union of all others:

$$a_i \subset \left[ E\left( \bigcup_{j \neq i} a_j \right) \right]' \,,$$

ensuring that entropy inequalities derived in the holographic entropy cone (HEC), such as strong subadditivity or MMI, hold. Without this, operator algebras for distinct bulk regions are not independent, and entropy inequalities may be violated (Bousso et al., 5 Feb 2025).

All these conditions enforce that boundary causality, unitarity, and entropy relations have well-defined duals in the bulk geometry, constrained by the null energy condition and the causal structure of extremal surfaces (Headrick et al., 2014).

5. Dynamical Consistency: RG Invariance and Inflationary Holography

In dynamical settings, such as holographic cosmology, the consistency of the bulk-boundary correspondence requires that time evolution in the bulk (e.g., of curvature perturbations $\zeta$ during inflation) matches the RG flow and Ward identities of the boundary QFT.

Two key consequences:

• RG-scale independence (cut-off independence) of stress-energy correlators in the dual field theory enforces the "freezing" of $\zeta_k$ modes on superhorizon scales,
• The so-called single-field inflationary consistency relation for the bispectrum emerges as a direct holographic counterpart of a Ward identity (dilatation/trace insertion) and the Callan–Symanzik (evolution) equation (Garriga et al., 2016, Schalm et al., 2012).

Failure of the RG or soft limits to match this requirement would indicate an inconsistency in the holographic realization of inflationary dynamics.

6. Physical Implications and Scope

The holographic consistency conditions tightly constrain both allowed bulk EFTs and candidate dual CFTs. Their nontrivial satisfaction in string theory constructions and their universal appearance in different guises—geometric, algebraic, causal, or entropic—underscore their role as necessary prerequisites for a well-posed AdS/CFT duality (Bobev et al., 11 Dec 2025, McInnes et al., 2015).

Conversely, their violation implies either the absence of a consistent dual description or the appearance of unresolvable physical pathologies (e.g., divergent correlators, instability, violation of entropy bounds, or breakdown of boundary unitarity).

Their continued refinement and generalization are central to delineating the swampland of inconsistent gravitational EFTs and sharpening the domain of validity for holographic duality.