Conformal Boundary Conditions from Cutoff AdS$_3$

TL;DR

This paper introduces a new flow in 2D Euclidean QFTs on a torus, dual to certain 3D gravity solutions with conformal boundary conditions, and explores its properties and implications.

Contribution

It constructs a novel flow from a Legendre transform of the $T\bar{T}$ kernel, ensuring elliptic boundary conditions in Euclidean gravity and relating it to the Wheeler de-Witt equation.

Findings

The flow is equivalent to the Wheeler de-Witt equation in CMC slicing.

Derived a kernel for the flow and computed the ground state energy at low temperature.

High-temperature density of states exhibits Cardy-like behavior.

Abstract

We construct a particular flow in the space of 2D Euclidean QFTs on a torus, which we argue is dual to a class of solutions in 3D Euclidean gravity with conformal boundary conditions. This new flow comes from a Legendre transform of the kernel which implements the $T\bar{T}$ deformation, and is motivated by the need for boundary conditions in Euclidean gravity to be elliptic, i.e. that they have well-defined propagators for metric fluctuations. We demonstrate equivalence between our flow equation and variants of the Wheeler de-Witt equation for a torus universe in the so-called Constant Mean Curvature (CMC) slicing. We derive a kernel for the flow, and we compute the corresponding ground state energy in the low-temperature limit. Once deformation parameters are fixed, the existence of the ground state is independent of the initial data, provided the seed theory is a CFT. The high-temperature density of states has Cardy-like behavior, rather than the Hagedorn growth characteristic of $T\bar{T}$-deformed theories.