JT gravity at finite cutoff

TL;DR

This work shows that Jackiw-Teitelboim gravity in AdS$_2$ at finite boundary cutoff yields a partition function that exactly matches the Schwarzian theory deformed by a TTbar-like operator, supporting the holographic picture of TTbar as moving the AdS boundary inward. The authors achieve this via two complementary bulk approaches: (i) a canonical Wheeler-DeWitt analysis in radial quantization and (ii) an exact Euclidean path-integral computation on disk topologies, including Hartle-Hawking boundary conditions. The results establish a precise, nonperturbative link between 1D TTbar-type deformations and 2D gravity with finite cutoff, and they clarify issues of unitarity and nonperturbative corrections, with extensions to de Sitter space and to more general topologies. Overall, the paper provides a concrete bridge between TTbar deformations in quantum mechanics and holographic finite-cutoff gravity, offering new tools to study UV completions of holography.

Abstract

We compute the partition function of $2D$ Jackiw-Teitelboim (JT) gravity at finite cutoff in two ways: (i) via an exact evaluation of the Wheeler-DeWitt wave-functional in radial quantization and (ii) through a direct computation of the Euclidean path integral. Both methods deal with Dirichlet boundary conditions for the metric and the dilaton. In the first approach, the radial wavefunctionals are found by reducing the constraint equations to two first order functional derivative equations that can be solved exactly, including factor ordering. In the second approach we perform the path integral exactly when summing over surfaces with disk topology, to all orders in perturbation theory in the cutoff. Both results precisely match the recently derived partition function in the Schwarzian theory deformed by an operator analogous to the $T\bar{T}$ deformation in $2D$ CFTs. This equality can be seen as concrete evidence for the proposed holographic interpretation of the $T\bar{T}$ deformation as the movement of the AdS boundary to a finite radial distance in the bulk.

Figures (4)

• Figure 1: (a) We show the slicing we use for Euclidean JT gravity in asymptotically $AdS_2$, which has disk topology (but not necessarily rigid hyperbolic metric). (b) Frame where the geometry is rigid $EAdS_2$ with $r$ increasing upwards and a wiggly boundary denoted by the blue curve.
• Figure 2: In orange, we show the undeformed density of states $\sinh(2\pi \sqrt{E})$ of JT gravity at infinite cutoff. In dashed black, we show the density of states of the theory with just the branch of the root, $\mathcal{E}_-$, that connects to the undeformed energies, until the energy complexifies. In blue, we show the density of states $\rho_{\lambda}(E)$ of the deformed partition function \ref{['Znp']} which includes non-perturbative corrections in $\lambda$. Above we have set $\widehat{\rho}(E)=0$ and $\lambda = 1/4$ and $C = 1/2$, the black line therefore ends at $E = \frac{1}{4\lambda} = 1$. The vertical dashed line indicates the energy beyond which $\widehat{\rho}$ has support.
• Figure 3: Frame in which geometry is rigid $dS_2$. Time runs upwards. We show the wiggly curve where we compute the wavefunction in blue (defined by its length and dilaton profile).
• Figure 4: Plot of the wavefuntion $\Psi_{\rm{HH, real}}$ for $\phi_b = 1/4$. The vertical dashed line indicates the location, $L = 2\pi$, where the expanding branch $\Psi_+$ of the wavefunction \ref{['K2wavefunction']} diverges, but $\Psi_{\rm{HH, real}}$ remains finite.