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

TL;DR

The paper establishes a concrete bridge between $T\bar{T}$-deformed 2D QFTs and flat-space JT gravity by formulating the JT sector in a first-order (vielbein) language. It computes the torus partition function of matter coupled to JT gravity, showing the result localizes to constant backgrounds and reproduces the deformed finite-volume spectrum through a one-loop exact analysis, supported by diffusion and localization arguments. The work solidifies the equivalence between the $T\bar{T}$ deformation and a topological gravitational coupling, and it points to extensions to other geometries and related deformations. Overall, it provides a transparent path-integral derivation of the finite-volume data associated with $T\bar{T}$, highlighting the role of dynamical coordinates and localization in gravitationally dressed theories.

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.

Figures (1)

• Figure 1: A torus can be represented as a parallelogram with identified opposite sides. Its angle is determined by modular parameters, $\measuredangle BAC=\arctan{\tau_2\over \tau_1}$. The overall orientation of the parallelogram is determined by the angle $\phi$, which does not affect the internal geometry of the torus.