Cauchy Slice Holography: A New AdS/CFT Dictionary

TL;DR

The paper introduces Cauchy Slice Holography, a framework in which time emerges as the bulk dimension through a T^2 deformation of a boundary CFT. It constructs a generalized holographic principle (GHP) linking a T^2-deformed boundary theory to a bulk gravitational path integral with Dirichlet data, and shows how bulk Wheeler–DeWitt states can be mapped to boundary CFT states, with ADM Hamiltonian matching the CFT Hamiltonian at large N. By building a bulk Hilbert space from gravitational path integrals and defining explicit bulk-boundary maps, the authors provide a coherent AdS/CFT dictionary that remains valid on arbitrary Cauchy slices and across Lorentzian signatures, while addressing contour choices and unitarity issues. The work suggests that a UV-complete T^2 theory could serve as a nonperturbative definition of quantum gravity, and discusses extensions to holographic cosmology and potential UV completions.

Abstract

We investigate a new approach to holography in asymptotically AdS spacetimes, in which time rather than space is the emergent dimension. By making a sufficiently large T^2-deformation of a Euclidean CFT, we define a holographic theory that lives on Cauchy slices of the Lorentzian bulk. (More generally, for an arbitrary Hamiltonian constraint equation that closes, we show how to obtain it by an irrelevant deformation from a CFT with suitable anomalies.) The partition function of this theory defines a natural map between the bulk canonical quantum gravity theory Hilbert space, and the Hilbert space of the usual (undeformed) boundary CFT. We argue for the equivalence of the ADM and CFT Hamiltonians. We also explain how bulk unitarity emerges naturally, even though the boundary theory is not reflection-positive. This allows us to reformulate the holographic principle in the language of Wheeler-DeWitt canonical quantum gravity. Along the way, we outline a procedure for obtaining a bulk Hilbert space from the gravitational path integral with Dirichlet boundary conditions. Following previous conjectures, we postulate that this finite-cutoff gravitational path integral agrees with the T^2-deformed theory living on an arbitrary boundary manifold -- at least near the semiclassical regime. However, the T^2-deformed theory may be easier to UV complete, in which case it would be natural to take it as the definition of nonperturbative quantum gravity.

Figures (13)

• Figure 1: Illustration of the usual finite-cutoff AdS/CFT duality. The $T^2$ deformed theory lives on a finite radius brick wall, labelled by $\partial\mathcal{B}^{(\mu)}$.
• Figure 2: The evolution of the embedding of the slice $\Sigma$, with a given fixed metric, under the deformation parametrised by $\mu$, in a saddle-point approximation. The illustration shows a (complexified) $\text{AdS}_2$ cross-section of the $\text{AdS}_{d+1}$ vacuum geometry. The Euclidean CFT boundary is labelled by 0, in a conformal frame where its geometry is hyperbolic. Three different stages (0, 1 and 2/$\textbf{2}^\textbf{*}$) of the deformation are shown. Stage $\textbf{0}$ corresponds to no deformation of the CFT. As we perform the $T^2$ deformation the slice moves inwards along the imaginary time direction, while still being embedded in a Euclidean bulk space (stage 1). For a sufficiently large deformation there is a phase transition to a Lorentzian saddle-point bulk geometry. By time-reversal symmetry, the slice $\Sigma$ can be embedded in two different ways in the same bulk geometry (with the same intrinsic metric). This corresponds to stages 2 and $\textbf{2}^\textbf{*}$. From the perspective of the field theory, this phase spontaneously breaks time-reversal (and therefore $CPT$).
• Figure 3: The figure on the left represents the partition function of the $T^2$-deformed field theory living on $\partial \mathcal{M}$ (the closed curve) with a background metric $g$. The shaded figure on the right represents the gravitational path integral over the space filling manifold $\mathcal{M}$. The bulk metric $\mathbf{g}$ satisfies the Dirichlet boundary conditions (i.e $\mathbf{g}|_{\partial\mathcal{M}} = g)$.
• Figure 4: The third case, in which $\Sigma_2$ straddles $\Sigma_1$, is shown. We are using the convention that we fix the time orientation of $\mathcal{M}$ to always be upward so that the time-ordering of the two slices is opposite in the two regions.
• Figure 5: For the case of the slices consistent with Lorentzian embedding, we get 4 saddles. The top two "gibbous" saddles are time reversals of each other and so they together contribute as a complex conjugate pair to the transition amplitude. Similarly for the bottom two "crescent" saddles. In this figure the time orientation is directed upwards and the $K=0$ dashed line is the time symmetric slice. It can be seen that when $g_1 = g_2$, the crescent saddles go to zero lapse, leading to divergences of the 1-loop quantum determinant $\Delta_\text{crescent}$, which will be discussed in section \ref{['prop']}.