Cauchy Slice Holography

• Cauchy Slice Holography is a formulation where the bulk state on a Cauchy slice, interpreted as a Wheeler–DeWitt wavefunctional, serves as the primary object for encoding gravitational data.
• It employs a T²-type deformation—transforming the local Weyl generator into a Hamiltonian constraint—to establish a concrete bulk-to-boundary map across both AdS and de Sitter settings.
• The framework offers practical insights into emergent semiclassical spacetimes, entropy bounds, and the information-theoretic structure of quantum gravity by integrating canonical gravity methods with deformed field theories.

Cauchy Slice Holography (CSH) is a formulation of holography in which the dual theory is associated with a bulk Cauchy slice $\Sigma$ rather than only with an asymptotic boundary, and its partition function is interpreted as a Wheeler–DeWitt (WDW) wavefunctional $\Psi[g,\Phi]$. In the foundational AdS formulation, the construction starts from a Euclidean CFT deformed by a sufficiently large $T^2$ irrelevant deformation so that time rather than space is the emergent dimension, while in de Sitter an analogous $T^2$-type deformation produces a holographic theory living on finite-time Cauchy slices, with conformal time identified with an RG scale (Araujo-Regado et al., 2022, Araujo-Regado et al., 6 Nov 2025). Across these realizations, CSH reformulates holography in canonical quantum-gravity language, making the bulk state on a slice primary and the emergent spacetime secondary.

1. Foundational definition and canonical dictionary

The basic object in CSH is a WDW state on a bulk Cauchy slice, satisfying the Hamiltonian and momentum constraints

$$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$

The central move is to construct a field theory whose partition function is itself such a state,

$Z[g]=\Psi[g],$

with the spatial metric $g_{ab}$ playing the role of a source and the stress tensor $T^{ab}$ identified with the conjugate momentum $\Pi^{ab}$ (Araujo-Regado et al., 2022).

In this formulation, the usual asymptotic-boundary emphasis of AdS/CFT is inverted. The deformed theory is interpreted as living directly on a Lorentzian bulk Cauchy surface $\Sigma$, or more generally on arbitrary bulk Cauchy slices anchored on the boundary Cauchy slice $\Psi[g,\Phi]$0. The proposal is intended to encode bulk data deep inside spacetime, including regions behind horizons, in a way naturally adapted to canonical gravity rather than to radial reconstruction (Araujo-Regado et al., 2022).

A concise expression of the boundary-to-bulk map appears in the later background-independent formulation,

$\Psi[g,\Phi]$1

This dictionary treats the boundary state as the input and the WDW state on $\Psi[g,\Phi]$2 as the primary bulk object. A classical $\Psi[g,\Phi]$3-dimensional spacetime is not assumed at the start; it emerges only when a semiclassical approximation is valid (Khan, 2023).

2. $\Psi[g,\Phi]$4 deformation, constraint algebra, and Hilbert-space interpretation

The deformation mechanism is a path-ordered irrelevant flow,

$\Psi[g,\Phi]$5

with a local operator $\Psi[g,\Phi]$6 built from the stress tensor and matter operators, and with counterterms chosen so that the final constraint takes the standard ADM form (Araujo-Regado et al., 2022).

Its structural role is to turn the local Weyl generator of the seed CFT into the Hamiltonian constraint of gravity. The undeformed theory satisfies

$\Psi[g,\Phi]$7

while the deformed theory is engineered to satisfy

$\Psi[g,\Phi]$8

The construction generalizes the two-dimensional $\Psi[g,\Phi]$9 case and, according to the foundational proposal, extends to arbitrary local Hamiltonian constraints that close with the momentum constraint in the ADM algebra, provided the CFT has suitable anomalies (Araujo-Regado et al., 2022).

CSH is also a Hilbert-space statement. The deformed partition function defines a map between the boundary CFT Hilbert space and the bulk canonical quantum-gravity Hilbert space, with a concrete bulk-to-boundary map $T^2$0 for a chosen slice metric and a converse boundary-to-bulk map obtained by integrating the deformed partition function against a boundary state. Within this framework, the identity

$T^2$1

is presented as a consistency condition equating boundary time evolution with the ADM generator of bulk time evolution (Araujo-Regado et al., 2022).

A recurring point of interpretation is that the $T^2$2-deformed theory is not reflection positive. The foundational AdS work and the closed-slice de Sitter work both argue that this does not preclude bulk unitarity. In the AdS formulation, bulk unitarity is tied to the Hermiticity of the bulk Hamiltonian with respect to the appropriate inner product; in the closed-slice de Sitter formulation, the dynamical inner product is positive-semidefinite even though the Euclidean field theory on the slice can fail reflection positivity (Araujo-Regado et al., 2022, Araujo-Regado, 2022).

3. Closed slices and de Sitter realizations

For closed spatial slices with $T^2$3, CSH is formulated directly in Wheeler–DeWitt language on a closed manifold $T^2$4 with $T^2$5. In the large-volume regime, the state takes the asymptotic two-branch form

$T^2$6

and away from that regime each branch is obtained by a specific $T^2$7-type deformation flow. In $T^2$8 dimensions with $T^2$9, the associated central charge becomes imaginary, the two branches are CPT-dual complex conjugates, and explicit minisuperspace solutions reproduce Hartle–Hawking-like and Vilenkin-like wavefunctions as particular superpositions (Araujo-Regado, 2022).

Because the slice is closed, there is no place to specify extra boundary conditions for the QFT. The state is therefore encoded entirely by the choice of branch superposition,

$T^2$0

This brings the formalism close to the wavefunction-of-the-universe problem. The same work further conjectures a relation between the choice of branch superposition and the class of geometries summed over in the gravitational path integral, expressed in minisuperspace as a choice of contour in the complex lapse plane (Araujo-Regado, 2022).

A distinct de Sitter realization places the deformed theory on flat finite-time Cauchy slices $T^2$1. Starting from a bottom-up dS/CFT model and deforming it by the quadratic stress-tensor combination

$T^2$2

with additional terms depending on dimension and matter content, one obtains a holographic theory in which conformal time $T^2$3 is identified with an RG scale. In this picture, the dS/CFT point is an IR fixed point rather than a UV fixed point: early bulk times correspond to the UV regime of the deformed boundary theory, while late times correspond to the IR fixed point (Araujo-Regado et al., 6 Nov 2025).

The de Sitter case also introduces specifically Lorentzian features. The relevant holographic counterterms are purely imaginary when expressed in terms of wavefunction coefficients, because de Sitter bulk evolution is governed by a Lorentzian path integral $T^2$4. The deformed theory reproduces the finite-time bulk wavefunction coefficients for both scalar and graviton fluctuations, with formulas analytic in the scalar operator dimension $T^2$5, allowing continuation to generic complex $T^2$6, including the principal series. Cosmological observables are then obtained by the Born rule, and the shift induced by counterterms is nonlinear because the wavefunction coefficients are complex (Araujo-Regado et al., 6 Nov 2025).

4. Emergence of semiclassical spacetime and subregion classicalization

Within CSH, a classical background is an emergent approximation to a WDW state rather than the starting point. The semiclassical analysis expands the state as

$T^2$7

with $T^2$8 obeying the Einstein–Hamilton–Jacobi equation. A classical geometry emerges only on the semiclassical superspace

$T^2$9

where the WKB phase is appropriately minimized and the classical momentum is read off from $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$0 (Khan, 2023).

A central diagnostic is the gravitational Wigner functional. Under the strong factorization condition stated in the paper, the reduced Wigner functional becomes sharply peaked in momentum around the classical trajectory, which is presented as the gravitational analogue of the Halliwell criterion. This yields a backreacted Einstein–Hamilton–Jacobi equation containing $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$1, and thereby a semiclassical Einstein equation in Hamilton–Jacobi form (Khan, 2023).

Once the classical background exists, the next WKB order defines a bulk QFT state

$$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$2

and the next-order WDW equation gives the Tomonaga–Schwinger equation for matter on the emergent curved spacetime. The resulting pipeline is explicit: boundary CFT state, WDW state on $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$3, emergent classical geometry from $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$4, and emergent bulk QFT from $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$5 (Khan, 2023).

The same framework extends to situations in which gravity is semiclassical only outside a finite region. The interface is an emergent timelike boundary called a WDW screen, carrying an induced WDW state that encodes the quantum-gravitational data inside the region. This subregion classicalization is applied to evaporating black holes, where the paper argues that the near-horizon semiclassical approximation may fail around the Page time, under its stated assumptions, because the required classicality conditions are incompatible with black-hole–radiation entanglement of order $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$6 (Khan, 2023).

5. Entropy bounds, simple wedges, and information on a slice

One information-theoretic development of CSH is a new holographic covariant entropy bound (HCEB), defined for a codimension-2 cut $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$7 as the logarithm of the maximum number of states of the Cauchy-slice theory that can pass through $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$8. In pure three-dimensional GR with negative cosmological constant, where the slice theory is the $$\mathcal{H}(x)\Psi[g,\Phi]=0,\qquad \mathcal{D}^a(x)\Psi[g,\Phi]=0.$$9-deformation of the dual CFT, the bound is computed explicitly and shown to depend only on codimension-2 data on $Z[g]=\Psi[g],$0, including the area or length, the null expansions, and the twist. It agrees with area at certain marginal and extremal surfaces, but it can exceed the area by an arbitrarily large amount for trapped and anti-trapped surfaces, such as surfaces inside a black hole (Soni et al., 2024).

That same work emphasizes that imaginary energy eigenstates in the spectrum of the deformed theory are essential in Lorentzian signature. They are needed for a valid bound on slices entering black-hole interiors, and they underwrite a tensor-network interpretation in which the bond dimension is governed by the HCEB rather than by area alone (Soni et al., 2024).

A complementary geometric development defines the simple or outermost wedge in arbitrary globally hyperbolic spacetimes by a zigzag of antinormal lightsheets. Starting from an input wedge $Z[g]=\Psi[g],$1, one alternates maximal future and past lightsheet wedges to construct an increasing sequence $Z[g]=\Psi[g],$2 and a corresponding preferred piecewise-null Cauchy slice $Z[g]=\Psi[g],$3, with limits $Z[g]=\Psi[g],$4 and $Z[g]=\Psi[g],$5. The resulting simple wedge is unique, is accessible from $Z[g]=\Psi[g],$6, is contained in every other throat accessible from $Z[g]=\Psi[g],$7, and lies inside the generalized entanglement wedge or max-hologram $Z[g]=\Psi[g],$8. The construction is explicitly not purely spacelike; the authors state that no spacelike construction is known to reproduce these results, even in time-symmetric settings (Bousso et al., 1 May 2025).

A related AdS/CFT result, formulated in bilocal holography, states the holography of information principle: a copy of all the information available on a Cauchy slice is also available near the boundary of the Cauchy slice, and this redundancy is already visible in the low-energy theory. In the free $Z[g]=\Psi[g],$9 vector model dual to higher-spin gravity in AdS$g_{ab}$0, the mechanism is the OPE expansion of bilocal operators into local single-trace primaries and descendants, showing that bulk information deep in the interior can be represented in a near-boundary operator algebra (Koch et al., 2022).

6. Conformal reformulations, candidate states, and unresolved issues

A recent reformulation proposes an alternative phase space for asymptotically AdS gravity in which the Hamiltonian constraint is replaced by a real Weyl-anomaly constraint,

$g_{ab}$1

while the momentum and matter-gauge constraints remain unchanged. Under the assumptions that every classical solution in the relevant domain admits a unique maximal volume slice $g_{ab}$2 and that the matter Hamiltonian satisfies the stated structural conditions, the reduced phase space of this formulation is symplectomorphic to the reduced phase space of standard ADM gravity (Khan, 19 Jul 2025).

In this conformal version of CSH, partition functions of certain CFTs with imaginary central charge, defined on bulk Cauchy slices, satisfy the operator gauge constraints and therefore provide candidate quantum-gravity states in the alternative phase space. The key analytic step is continuation in the central charge $g_{ab}$3, which turns the standard conformal Ward identity into the quantum constraint $g_{ab}$4 while preserving the momentum and matter-gauge constraints (Khan, 19 Jul 2025).

This reformulation sharpens an existing theme in CSH: the appearance of CFT partition functions with imaginary central charge and the replacement of the standard Hamiltonian constraint by a structure tied to the Weyl anomaly. It also leaves several issues open. Among those stated explicitly are the construction of a proper inner product and the normalizability of the candidate states, the definition of the dynamical Hamiltonian in the alternative phase space, the extension beyond the low-energy Einstein sector, and the establishment of the exact AdS/CFT dictionary in this formulation (Khan, 19 Jul 2025).

More broadly, the overall CSH program remains partly conjectural. The generalized holographic principle equating the $g_{ab}$5-deformed partition function on arbitrary boundary manifolds with the bulk gravitational path integral is proposed most strongly near semiclassical saddles and at large $g_{ab}$6, and the gravitational contour problem, factorization subtleties, and nonperturbative UV completion remain open (Araujo-Regado et al., 2022). Taken together, these works define CSH less as a single fixed model than as a canonical reorganization of holography around bulk Cauchy data, WDW states, and deformed field theories that live on the slices themselves.