Dual Holography Framework

• Dual Holography Framework is a codimension-2 duality that maps wedge geometries in a (d+1)-dimensional bulk to a (d-1)-dimensional conformal field theory on the corner.
• It employs precise boundary conditions and metric uplifts to achieve exact equivalence of classical actions, anomalies, and entanglement prescriptions with traditional AdS/CFT.
• The framework generalizes to de Sitter, flat, and celestial holography, offering insights into operator spectra, effective matter, and unified gravitational dualities.

The Dual Holography Framework refers to a broad class of codimension-2 bulk/boundary dualities in gravitational theories, in which a $(d+1)$-dimensional bulk spacetime with appropriate boundary geometry (edges, corners, or branes) gives rise, in the classical or semiclassical limit, to a $(d-1)$-dimensional conformal field theory (CFT) or closely related structure on a codimension-2 locus—the “corner” of the wedge. This construction subsumes the standard AdS/CFT correspondence, wedge holography, dS/CFT, flat-space holography, and various higher-spin and celestial dualities, providing a powerful and unifying language for non-perturbative gravitational/CFT correspondences.

1. Codimension-Two Wedge Geometries and Classical Duality

The canonical dual holography setting is an asymptotically AdS$_{d+1}$ “wedge” spacetime $W_{d+1}$:

$$ds^2_{d+1} = dx^2 + \cosh^2\left(\frac{x}{L}\right)\bar h_{ij}(y)dy^i dy^j,\qquad x\in[-\rho,\rho].$$

Here, $\bar h_{ij}(y)$ solves the $d$-dimensional Einstein equations with (usually) negative cosmological constant, and $L$ is the AdS radius. Two end-of-the-world branes $Q_1, Q_2$ at $x = \pm \rho$ with tension $T = (d-1)\tanh(\rho/L)$ are introduced; Neumann boundary conditions (NBC) are imposed on these branes:

$K_{ij} - (K - T)h_{ij} = 0.$

The “corner” $\Sigma = Q_1 \cap Q_2$ is the codimension-2 locus supporting the dual CFT. The classical duality is formulated as

$$Z_{\mathrm{CFT}_{d-1}} = \exp(-I_{\mathrm{wedge}}),$$

where $I_{\mathrm{wedge}}$ is the (regulated) bulk gravitational action, including brane terms.

The bulk equations together with NBC force each $Q$ to have constant intrinsic Ricci scalar as a function of $T$. This geometric data, together with the wedge construction, forms the foundation for all higher analysis (Miao, 2020, Akal et al., 2020).

2. Exact Equivalence to AdS/CFT and Generalizations

A key result is that, for the “pure” wedge solutions of the type above, there is an exact map to a (dimensionally reduced) AdS$_d$/CFT$_{d-1}$ duality via the metric uplift

$$ds^2_{d+1} = dx^2 + \cosh^2(x/L)\bar h_{ij}(y)dy^i dy^j,$$

with any solution $\bar h_{ij}$ of the AdS$_d$ Einstein equations. Substitution into the bulk+brane action, followed by application of the equations of motion and NBC, yields

$I_{\rm wedge} = I_{\rm AdS_d}$

with an effective Newton constant

$$\frac{1}{G_N^{(d)}} = \frac{1}{G_N}\int_{-\rho}^\rho \cosh^{d-2}(x/L)dx.$$

This identification demonstrates that—at the level of classical action, partition function, conformal anomalies, entanglement entropies, and correlators—codimension-2 wedge holography is strictly equivalent to the traditional AdS/CFT duality for this subclass of solutions (Miao, 2020). More general wedge geometries (with $\bar h_{ij}$ solving more generic $d$-dimensional equations) correspond to CFTs coupled to effective matter, e.g., Kaluza-Klein towers.

3. Holographic Entanglement and Weyl Anomaly on the Corner

The dual CFT on the corner inherits all universal properties entailed by the standard holographic dictionary, with modifications dictated by the lower dimensionality and wedge geometry:

• Weyl Anomaly: The leading log-divergent piece of the wedge action (for even boundary dimension) yields the corner Weyl anomaly,

$$\mathcal{A} = \int_\Sigma \sqrt{\sigma^{(0)}} \left( \sum_n B_n I_n - 2(-1)^p A E_{2p} \right),\qquad 2p = d-1,$$

with explicit central charges determined by integrals over the $x$-direction.

• Entanglement/Rényi Entropy: For a region $A$ on $\Sigma$, the Ryu-Takayanagi formula generalizes via insertion of cosmic branes in the wedge:

$$S_{\mathrm{EE}} = \frac{\mathrm{Area}(\mathrm{minimal\ surface})}{4G_N} = \frac{V_{H_{d-2}}}{4G_N} \int_{-\rho}^{\rho} \cosh^{d-2}(x/L) dx,$$

demonstrating that universal (e.g., logarithmic) terms and scaling laws match corresponding CFT$_{d-1}$ predictions (Miao, 2020, Akal et al., 2020).

4. Two-Point Functions, Operator Spectrum, and Effective Field Content

Two-point functions of CFT primaries and conserved operators on the corner are produced by solving for normalizable modes of bulk fields (with Dirichlet/Neumann boundary data on the branes) and projecting via the holographic bulk-to-boundary prescription:

$$\langle T_{ab}(y)T_{cd}(y')\rangle = C_T \frac{\mathcal I_{ab,cd}(y-y')}{|y-y'|^{2(d-1)}}$$

with $C_T$ determined by wedge integrals and

$$\Delta_n = \frac{d-1}{2} + \sqrt{\frac{(d-1)^2}{4} + M_n^2}$$

for towers of operators from bulk fields (Miao, 2020, Akal et al., 2020). More general wedge solutions with non-trivial $h_{ij}(y)$ correspond to CFTs coupled to effective $d$-dimensional matter, dictated by the brane curvature and asymptotic conditions.

5. Extensions to dS/CFT, Flat Holography, and Unified Frameworks

The “Gauss-normal” construction with

$ds^2 = dx^2 + f(x)\bar h_{ij}(y)dy^i dy^j$

where $f(x)$ satisfies

$$\frac{f''}{2f} - \frac{1}{4}\left(\frac{f'}{f}\right)^2 = 1$$

allows three real solutions:

• $f(x) = \cosh^2(x/L)$: ($|T| < d-1$) AdS wedge and AdS$_d$/CFT$_{d-1}$
• $f(x) = \sinh^2(x/L)$: ($|T| > d-1$) dS wedge and dS$_d$/CFT$_{d-1}$
• $f(x) = e^{\pm 2x/L}$: ($|T| = d-1$) flat wedge and flat/CFT correspondence

All three holographic correspondences—AdS, dS, and flat—arise as limits distinguished by the induced brane geometry and tension (Miao, 2020). This unification enables new perspectives on the structure of holographic dualities in diverse backgrounds.

6. Flat Space and Celestial Holography: Wedge Realization

In the flat-spacetime limit, wedge holography adopts a geometry in which a region of Minkowski is bounded by two $d$-dimensional hyperbolic or de Sitter branes, and the $(d-1)$-sphere at their intersection supports a CFT with non-unitary properties (e.g., imaginary central charge, complex operator dimensions) (Ogawa et al., 2022). The on-shell bulk action produces the (typically non-unitary) CFT partition function on $S^{d-1}$, the entanglement entropy follows a generalized RT prescription, and scalar two-point functions display principal series spectra. Gluing hyperbolic and dS wedges along null surfaces reconstructs the full Minkowski spacetime and suggests a natural connection to the celestial holography program, in which conformal primary wavefunctions live on the celestial sphere.

7. Effective Matter, Nontrivial Brane Geometries, and Broader Implications

Beyond pure “uplifted” solutions, general wedge holography accommodates effective matter content on the branes arising from bulk Kaluza-Klein towers or other curvature-induced sources. The Hamiltonian constraint enforces that each brane has constant scalar curvature, and the Einstein equation on the brane takes a modified form:

$$R_{h\,ij} - \frac{1}{2}R_h h_{ij} - \frac{(d-1)(d-2)}{2\cosh^2\rho} h_{ij} = 8\pi G_N^{(d)} T^{(\mathrm{eff})}_{ij},$$

with $T^{i}{}_i = 0$. This provides a systematic approach to model CFTs with matter (e.g., Kaluza-Klein, higher-spin, generalized boundary conditions), broadening the landscape of dualities accessible within the wedge construction.

---

In summary, the dual holography framework systematizes a class of codimension-2 bulk/boundary correspondences encompassing AdS/CFT, dS/CFT, flat holography, and celestial constructions. Its essential features are the reduction to $(d-1)$-dimensional dual field theories on codimension-2 loci, exact matching of classical partition functions and anomalies, and a geometry-driven mechanism for unifying diverse holographic dualities. This approach both consolidates and extends the conventional understanding of holography, serving as a launch point for further investigations into gravitational duals of generalized boundary field theories, non-unitary holographic dualities, and celestial amplitudes (Miao, 2020, Ogawa et al., 2022, Akal et al., 2020).