A new perspective on dilaton gravity at finite cutoff

TL;DR

This work advances the understanding of dilaton gravity at a finite bulk cutoff by developing two self-consistent routes in JT gravity: a bulk trumpet path integral and a boundary curve path integral governed by an exact Riccati equation for the extrinsic curvature. The trumpet calculation yields a closed-form finite-cutoff trumpet wavefunction that, when glued to a cap, reproduces the finite-cutoff disk partition function, while the boundary analysis derives a one-loop boundary action and shows precise agreement with the bulk results and with TT-deformed Schwarzian theory. The authors then extend the insights to general dilaton gravities with arbitrary potentials V(φ), proposing a nonperturbative, contour-based partition function and highlighting TT-like structures in the finite-cutoff regime. These findings illuminate signatures of UV completeness, discuss higher-topology generalizations, and propose a canonical open-channel framework that may hint at spacetime discretization. Overall, the paper provides a robust gravitational foundation for finite-cutoff holography in two dimensions and a pathway toward a unified treatment of TT-like deformations across general dilaton gravities.

Abstract

The formulation of two-dimensional quantum gravity at finite cutoff remains an open problem. We revisit this question in JT gravity from two perspectives: the closed-channel bulk path integral and the path integral over boundary curves. First, we study the radial evolution of a closed universe and derive the trumpet wavefunction as a transition amplitude between a geodesic boundary and a finite Dirichlet boundary. Our analysis recovers the Hartle--Hawking wavefunction without imposing asymptotic boundary conditions, allowing the trumpet to be glued to a cap wavefunction to reconstruct the smooth disk. Second, we derive an exact Riccati equation for the extrinsic curvature of a finite-cutoff boundary curve in the Euclidean Poincaré disk. A WKB expansion of this equation yields all perturbative corrections in the cutoff parameter and captures nonperturbative effects. From this, we compute the quadratic boundary action and the one-loop partition function at finite cutoff, finding agreement with both the bulk approach and the expected one-loop effective action for the $T\bar{T}$ deformation of the Schwarzian theory. Extracting lessons from JT gravity, we then argue that similar relationships hold for general dilaton gravities with arbitrary potentials $V(φ)$ and propose an exact expression for their finite cutoff partition functions. We finally investigate several signatures of UV completeness in these settings, introducing a canonical quantization approach within the finite cutoff framework.

Figures (3)

• Figure 1: In this figure we represent the geometry of the spacetime \ref{['geom']} that is a solution of the equation of motion and interpolates between the initial and final time circles, at $r=0$ and $r=1$, which are subject to the boundary conditions \ref{['bcc']}. The blue circle of length $b$ is a geodesic, the neck of the trumpet, while the other circle is the boundary of length $L$.
• Figure 2: In this figure we show the process of constructing the disk partition function by gluing together the trumpet wavefunction $\Psi_b(L,\phi)$ and the cap amplitude $\rho_{\text{cap}}(b)$.
• Figure 3: Numerical evaluation of the $T\bar{T}$-deformed two point function \ref{['improvement']} for $\Delta=1$, $\phi_r=20$, $\beta=2\pi$. Orange dots correspond to $\varepsilon=0.5$, blue dots to $\varepsilon=0.8$, red dots to $\varepsilon=0.9$ and green dots to $\varepsilon=1$. The line represents the semiclassical conformal answer $\langle \mathcal{O}(\tau)\mathcal{O}(\beta-\tau)\rangle=\frac{1}{\sin (\pi \tau/\beta)^{2\Delta}}$ obtained by evaluting \ref{['undeformed']} on the Schwarzian thermal saddle. All data sets have been normalized such that the amplitude is equal to $1$ at $\tau=\beta/2$.