$T\bar{T}$ Partition Function from Topological Gravity

TL;DR

This paper calculates the $T\bar{T}$ deformed finite volume spectrum using a first order JT gravity approach, providing a new exact method for understanding deformed 2D theories in finite volume.

Contribution

It introduces a first order formalism for JT gravity to compute the $T\bar{T}$ deformed partition function at finite volume, connecting gravitational and field theory perspectives.

Findings

Partition function is one-loop exact.

Reproduces the $T\bar{T}$ deformed spectrum.

Provides a new gravitational approach to finite volume deformations.

Abstract

The $T{\bar T}$ deformation of a relativistic two-dimensional theory results in a solvable gravitational theory. Deformed scattering amplitudes can be obtained from coupling the undeformed theory to the flat space Jackiw--Teitelboim (JT) gravity. We show that the JT description is applicable and useful also in finite volume. Namely, we calculate the torus partition function of an arbitrary matter theory coupled to the JT gravity, formulated in the first order (vierbein) formalism. The first order description provides a natural set of dynamical clocks and rods for this theory, analogous to the target space coordinates in string theory. These dynamical coordinates play the role of relational observables allowing to define a torus path integral for the JT gravity. The resulting partition function is one-loop exact and reproduces the $T\bar{T}$ deformed finite volume spectrum.