Conformal Boundary Conditions from Cutoff AdS$_3$

TL;DR

This work proposes a modified TTbar flow that implements conformal (elliptic) boundary conditions in AdS3 holography by a Legendre transform of the TTbar kernel. The flow is shown to be equivalent to a Wheeler–deWitt constraint in Constant Mean Curvature slicing, and it maps to a 3D gravity problem on a torus with a well-defined ground state and a Cardy-like high-energy density of states, rather than Hagedorn behavior. The authors derive the kernel governing the transformed ensemble, express the transformed partition function through Laplace transforms, and analyze the spectrum and thermodynamics in low- and high-temperature limits, including a connection to Jackiw–Teitelboim gravity in a degenerate torus limit. The results illuminate how conformal boundary conditions alter the holographic dictionary for finite-cutoff gravity and open avenues for exact spectral data and generalizations to broader holographic contexts, including potential dS analogues and worldsheet constructions.

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.