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 overline{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 overline{T} -deformed theories.
Highlights
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
Limit, the expectation [4, 5] is that the T Tflow equation maps to the bulk radial Wheelerde Witt equation (WdW), which is a quantization of the aforementioned constraint
We have derived a modification to the flow equation of the T Tdeformation which implements conformal boundary conditions in the bulk dual, rather than the Dirichlet boundary conditions of lore
Summary
We will parameterize the family of T T-deformed solutions by λ, the dimensionless T Tcoupling. The volume of the torus is given by: In this language, the T Tflow equation reads [9]:. Note that the domain of integration is the upper half plane H This expression appeared in [11,12,13] and [4]. Note that it can be derived from the prescription of [14], meaning that it can be seen as an expression for the path integral of 2d ghost-free massive gravity coupled to a conformal field theory. (2.4) leads to the following equation for the deformed energy levels En: 2λEn∂λEn + 4∂λEn + En2 = Jn2. In the analysis which follows, we derive expressions in terms of En(R) rather than En, to keep expressions simple
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