A new perspective on dilaton gravity at finite cutoff

TL;DR

This paper explores finite-cutoff JT gravity using bulk and boundary perspectives, deriving exact equations and partition functions, and extends insights to general dilaton gravities, highlighting their UV properties and quantization.

Contribution

It introduces an exact Riccati equation for finite-cutoff boundaries and proposes a general finite cutoff partition function for dilaton gravities, advancing understanding of quantum gravity at finite boundaries.

Findings

Derived the trumpet wavefunction without asymptotic boundary conditions.

Obtained an exact Riccati equation for boundary extrinsic curvature.

Computed the finite cutoff partition function matching $T\bar{T}$ deformed theories.

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 Poincar\'e 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 $V(\phi)$ 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.