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

TL;DR

This paper extends the $T ar{T}$ deformation to curved spacetimes by formulating a flow equation for the partition function, linking it to 3d quantum gravity and the Wheeler-de Witt equation, with implications for holography and boundary conditions.

Contribution

It introduces a generalized $T ar{T}$ deformation applicable to curved spaces, connecting deformed CFT partition functions to 3d gravity path integrals and local Wheeler-de Witt equations.

Findings

Deformed partition function solves a local Wheeler-de Witt equation.

Partition function corresponds to a 3d gravity path integral.

Reproduces known flat space and large $c$ results.

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.