Wedge CFT₂: Finite Bulk Holography & Dualities

Updated 20 October 2025

Wedge CFT₂ is a framework where a two-dimensional CFT resides on a finite bulk wedge using deformations like TT̄ and codimension-two constructions.
It establishes entanglement wedge subregion duality by linking minimal surface features to measures such as logarithmic negativity and reflected entropy.
The approach highlights how cutoff surfaces and RG flow govern holographic complexity and bulk-boundary operator reconstruction.

Wedge CFT $_2$ refers to a set of dualities and geometric correspondences in the context of two-dimensional conformal field theory (CFT $_2$ ) where the dual bulk region is a “wedge” or a finite bulk domain, typically bounded by a finite cutoff or by generalized end-of-the-world (EOW) branes, with the quantum field theory defined not on the asymptotic boundary but rather on a codimension-reduced locus (the “corner” or “wedge”). This paradigm captures a range of physical and holographic features, from deformations breaking conformal invariance to fine-grained subregion duality and entanglement geometry.

1. Geometric and Holographic Foundation

In the standard AdS $_3$ /CFT $_2$ correspondence, the CFT $_2$ is defined on the asymptotic boundary of AdS $_3$ . Wedge CFT $_2$ is distinguished by the CFT “living” at a finite bulk location, most canonically realized in several settings:

TT̄-Deformed CFT $_2$ and Finite Radial Cutoff: The TT̄ deformation adds an irrelevant operator built from the stress tensor to the action:

$S^{(\mu)} = S_{\text{CFT}} + \mu \int d^2x\, T\bar{T}$

with deformation parameter $\mu$ . Holographically, this shifts the CFT $_2$ from the asymptotic AdS boundary to a finite radial coordinate, effecting a UV cutoff. The position of the boundary is

$r_c^2 = \frac{6 R_{\text{AdS}}^4}{\pi c \mu}$

(Wang et al., 2018).

Generalized Wedge Holography (Codimension-Two): In “wedge holography,” gravity on a $(d+1)$ -dimensional AdS wedge, regulated by EOW branes (Neumann or mixed boundary conditions) placed at $\rho = \pm \rho_\ast$ , is dual to a CFT on the codimension-2 “corner” (intersection of the branes). For $d=3$ , this yields a wedge CFT $_2$ as the dual to AdS $_4$ gravity on a wedge (Akal et al., 2020, Miao, 2020).
Flat Space and Celestial Holography: Wedge-like constructions, especially in flat space limits, encode RG flows between different boundary CFTs triggered by double-trace or mixed boundary deformations (Fukada et al., 2023).

The key feature is the restriction of the dual gravitational geometry to a bulk wedge, either by physical cutoff or by geometric construction (EOW brane, regulated slice), resulting in a field theory that inherits the properties and the entanglement structure of the finite region.

2. Entanglement Wedge and Subregion Duality

The concept of the entanglement wedge is foundational for Wedge CFT $_2$ . Given a boundary subregion $A$ , the entanglement wedge $E(A)$ is the bulk region reconstructible from data on $A$ and bounded by the boundary and a minimal (or quantum extremal) surface (QES) homologous to $A$ (Kusuki et al., 2019, Bao et al., 7 Aug 2024). The wedge structure refines as follows:

Background Wedge: The maximal bulk region where the background geometry is extractable from nonlocal correlators in the reduced density matrix of $A$ (via, e.g., the WKB limit of boundary two-point functions) (Bao et al., 7 Aug 2024).
Operator Reconstruction Wedges: Depending on the information-theoretic protocol, e.g., causal wedge (reconstructibility via bulk causality), entanglement wedge (QES), or the minimal/maximal “one-shot” wedges (defined by smooth conditional min/max entropy), the region accessible to operator reconstruction admits a hierarchy (with $G(A)\subset EW(A)\subset R(A)$ ). A careful wedge classification is essential to resolving paradoxes regarding operator reconstructibility and the encoding of global symmetries (Linden, 2 Jan 2024, Bao et al., 7 Aug 2024).

This wedge structure underpins the ability of the boundary CFT $_2$ (either as a full theory or as a wedge CFT $_2$ living on a subregion) to encode local bulk physics, with implications for quantum error correction and the absence of bulk global symmetries.

3. Mixed-State Entanglement and Entanglement Wedge Cross Section

The entanglement wedge cross section (EWCS) is a central geometric object in Wedge CFT $_2$ , defined as the minimal area (or length) surface bisecting the entanglement wedge between two boundary subregions $A$ and $B$ : $E_W(A:B) = \frac{\text{Area}(\Sigma_{AB}^{\min})}{4G_N}$ This cross section plays a direct role in holographic computations of mixed-state entanglement quantities:

Logarithmic Negativity: For bipartite mixed states in CFT $_2$ , the logarithmic negativity is:

$\mathcal{E} = \frac{3}{2} E_W$

or, more generally, $\mathcal{E} = \mathcal{X}_d E_W$ , where $\mathcal{X}_d$ is dimension-dependent ($3/2$ in $d=2$ ) (Kusuki et al., 2019, Basak et al., 2021, Basak et al., 2020).

Reflected Entropy: The reflected entropy $S_R$ is related to EWCS as $S_R = 2 E_W$ . Negativity and reflected entropy are connected via twist operator replica techniques and the cosmic brane prescription (Kusuki et al., 2019, Basak et al., 2021).
Odd Entanglement Entropy: The odd entanglement entropy (OEE), designed for mixed-state configurations, admits a holographic dual in terms of HEE plus EWCS:

$S_o(A:B) = S(A \cup B) + E_W(A:B)$

This relation is preserved even under TT̄ deformations (with corrections computable in perturbation theory) (Basu et al., 2023, Biswas et al., 2023).

Mutual Information, Transitions, and Horizon-Crossing: In thermofield double states or thermal backgrounds, EWCS detects genuine geometric transitions, such as horizon crossing, which corresponds to EWCS connecting both boundaries of a Kruskal-extended BTZ geometry (Jiang et al., 20 Jan 2025).

The universality and robustness of this geometric correspondence make EWCS a key organizing principle for wedge entanglement dynamics, with equivalence shown among disparate holographic negativity prescriptions up to additive Markov gap constants (Basak et al., 2020).

4. Cutoff Surface, Complexity, and RG Flow

Wedge CFT $_2$ is inherently sensitive to the presence and geometry of bulk cutoffs:

Cutoff Dependence: The location and shape of the radial (or general bulk) cutoff determines the effective scale and the structure of the dual CFT state. This is captured by the embedding of networks of Wilson lines in SL(2, $\mathbb{R}$ ) $\times$ SL(2, $\mathbb{R}$ ) Chern–Simons theory; the density of complexity is set by the extrinsic curvature $K$ of the cutoff curve:

$d\mathcal{C} \propto K\, d\lambda$

Integration over the cutoff yields a complexity measure that matches the holographic complexity=volume proposal (Chen et al., 2020).

Deformations and RG Flow: Mixed Neumann/Dirichlet boundary conditions or double-trace deformations in Wedge CFT $_2$ drive renormalization group flows between fixed points (typically, “IR” Wedge CFT to “UV” celestial or boundary CFT) (Fukada et al., 2023). Cutoff and regularization parameters (e.g., $R$ and $\epsilon$ ) in bulk two-point functions encode the RG scale.

5. Casimir Effect, Stress Tensor, and Wedge Geometry

Wedge CFT $_2$ and its higher-dimensional analogues offer a controlled setting for studying boundary/corner effects:

Casimir Energy: In a wedge of opening angle $\Omega$ ,

$\langle T_{ij} \rangle_{\mathrm{wedge}} = \frac{f(\Omega)}{r^d} \operatorname{diag}(1, -(d-1), 1, ..., 1)$

with $f(\Omega) \sim \kappa_2 (\pi - \Omega)$ for smooth wedges and $f(\Omega) \sim \kappa_1/\Omega^d$ in the sharp wedge limit. The coefficient $\kappa_2$ is set by the displacement operator norm $C_D$ , establishing a universal connection between the Casimir effect and conformal defect data (Miao, 17 Apr 2024).

Holographic Dual: The wedge dual is not simply Poincaré-AdS but a cutout of an AdS soliton or similar geometry, with nontrivial EOW brane embedding satisfying specific Neumann conditions. The wedge contribution to holographic entanglement entropy and Casimir energy is monotonically increasing with the opening angle and brane tension.
Polygon Generalization: Schwarz–Christoffel maps provide a construction of holographic “polygons,” where near each vertex the bulk and boundary stress tensor captures the local wedge behavior.

6. Wedge Dynamics, Connected Wedge Theorems, and Information-Theoretic Implications

Recent developments have established a precise operational meaning for the connectedness of entanglement wedges:

Connected Wedge Theorem: In AdS $_3$ /CFT $_2$ , large mutual information between appropriately chosen boundary “decision regions” (relevant to scattering protocols or distributed quantum tasks) is equivalent to the connectedness of the bulk entanglement wedges, characterized geometrically as (Zhao, 27 Sep 2025, May et al., 2022):

$S_E = E(V_1 \cup V_2) \cap E(W_1 \cup W_2)$

with $S_E$ nonempty if and only if both entanglement wedges are connected. The existence of $O(1/G_N)$ mutual information is thus tightly linked to operational bulk regions supporting nonlocal tasks.

Task-Oriented Lower Bounds: Theorems bounding mutual information in terms of task performance probabilities (e.g., success in quantum information protocols) make explicit the deep link between entanglement wedge geometry and process capacity in the dual CFT $_2$ (May et al., 2022).

7. Wedge CFT $_2$ in Toy Models and Quantum Error Correction

Tensor network models (HaPPY codes, perfect tensors) provide explicit realizations of wedge-based subregion duality:

Causal, Greedy, and Minimum Entanglement Wedges: These models allow for direct visualization and computation of reconstructability thresholds, with the minimum entanglement wedge (computed as the minimal cut comparing both boundary region and its complement) best approximating the true geometric wedge (with a sharp $p_c=0.5$ threshold for bulk qubit reconstruction under erasure) (Linden, 2 Jan 2024).
Mutual Information Wedge Rejection: Numerical studies show that mutual information alone does not reliably indicate reconstructability, emphasizing the geometric (wedge) basis for robust error correction and bulk/boundary encoding.

8. Outlook and Unified Framework

Several recent works demonstrate that codimension-two wedge holography unifies AdS/CFT, dS/CFT, and flat space holography, distinguished by the tension and curvature of EOW branes and the corresponding effective theory on the brane/corner (Miao, 2020). The intrinsic Ricci scalar of the brane remains constant, encoding all data about the dynamic phase (AdS, dS, flat). For deformed theories, wedge CFT $_2$ provides a controlled setting to match geometric, thermodynamic, and quantum informational structures, and to paper dualities beyond the strict conformal regime.

Summary Table: Wedge CFT $_2$ Key Features

Feature	Geometric/Field Theory Manifestation	Reference(s)
Cutoff/Codim-2 "corner"	Finite-radius boundary or intersection of EOW branes	(Wang et al., 2018, Akal et al., 2020, Miao, 2020)
Entanglement wedge	Minimal surface-based reconstructable region	(Kusuki et al., 2019, Bao et al., 7 Aug 2024, Kusuki et al., 2019)
EWCS & mixed-state QIT	Geometric dual to negativity, OEE, reflected entropy, etc.	(Kusuki et al., 2019, Basak et al., 2021, Basu et al., 2023, Biswas et al., 2023)
Casimir effect	Boundary stress tensor controlled by displacement operator	(Miao, 17 Apr 2024)
RG flows/deformations	Double-trace, Neumann/Dirichlet mixing, cutoff dependence	(Fukada et al., 2023)
Operator recoverability	Minimum wedge ≠ guaranteed operator reconstruction	(Linden, 2 Jan 2024, Bao et al., 7 Aug 2024)
Mutual information/Tasks	Connectedness of wedge linked to process capability	(May et al., 2022, Zhao, 27 Sep 2025)

Wedge CFT $_2$ is thus a flexible and robust framework for analyzing the interplay of geometry, entanglement, and operator algebraic structure in deformed, cutoff, or codimension-reduced holographic dualities, providing a rigorous, geometric foundation for understanding how quantum field theory degrees of freedom encode finite or dynamical bulk domains.