A $T \bar{T}$ Deformation for Curved Spacetimes from 3d Gravity

TL;DR

This work proposes a curved-space generalization of the T Tbar deformation by defining a flow equation for the partition function and representing the deformed theory as a radial wavefunction of 3D gravity. The central construct, the Freidel kernel, maps a seed 2D QFT partition function to a deformed one while satisfying a local Wheeler–DeWitt equation, thereby linking TTbar to 3D gravity in a non-holographic (fake bulk) sense that recovers holographic behavior in the classical limit. The authors extend the framework to spatially varying deformations and interpret the kernel as a 3D annulus gravity path integral, providing exact results in special cases such as S^2 and exploring the large-c regime. They also address potential objections to curved-space deformations and outline future directions, including gauge-fixing analyses, loop corrections, and extensions to matter couplings.

Abstract

We propose a generalisation of the $T \bar{T}$ deformation to curved spaces by defining, and solving, a suitable flow equation for the partition function. We provide evidence it is well-defined at the quantum level. This proposal identifies, for any CFT, the $T \bar{T}$ deformed partition function and a certain wavefunction of 3d quantum gravity. This connection, true for any $c$, is not a holographic duality --- the 3d theory is a "fake bulk." We however emphasise that this reduces to the known holographic connection in the classical limit. Concretely, this means the deformed partition function solves exactly not just one global equation, defining the $T \bar{T}$ flow, but in fact a local Wheeler-de Witt equation, relating the $T \bar{T}$ operator to the trace of the stress tensor. This also immediately suggests a version of the $T \bar{T}$ deformation with locally varying deformation parameter. We flesh out the connection to 3d gravity, showing that the partition function of the deformed theory is precisely a 3d gravity path integral. In particular, in the classical limit, this path integral reproduces the holographic picture of Dirichlet boundary conditions at a finite radius and mixed boundary conditions at the asymptotic boundary. Further, we reproduce known results in the flat space limit, as well as the large $c$ $S^2$ partition function, and conjecture an answer for the finite $c$ $S^2$ partition function.

Figures (2)

• Figure 1: The main set-up of the deformation. The seed QFT lives on a dynamical base space, whose metric is coupled to the fixed target space. The target space is the space on which the deformed theory lives.
• Figure 2: $Z_{\lambda}[f]$ may be viewed as a transition amplitude between the states $\ket{\Psi_{\lambda}}=\int D\pi \left( \int De e^{\int_{\Sigma }\frac{\lambda}{2}\varepsilon^{ab} \pi_{a}\wedge \pi_{b}-\pi_a \wedge e^a} Z_{0}[e] \right) \ket{\pi}$ and $\ket{f}$. Geometrically, this corresponds to a 3d gravity path integral on an annulus, with particular choice of mixed boundary conditions on the outer edge and Dirichlet ones for the inner edge.