Moving the CFT into the bulk with $T\bar T$

TL;DR

The paper investigates the solvable $T \bar{T}$ deformation of 2D CFTs and proposes a holographic dual where the CFT is encoded on a finite-radius AdS slice with a Dirichlet wall at $r_c$, related by $\mu = {24\pi\over c}\,{1\over r_c^2}$. By deriving an exact RG flow for the metric dependence of the deformed theory and matching it to the Hamilton-Jacobi equation of 3D gravity, the authors show precise correspondences in the light-cone propagation speeds, energy spectra, and thermodynamics between the deformed CFT and the bulk cutoff theory. They also connect the deformation to an exact Hubbard-Stratonovich formulation and to an equivalence with Nambu-Goto dynamics at $c=24$, illustrating a consistent bulk interpretation. This framework provides a controlled way to access bulk physics inside AdS and clarifies how holographic RG and radial cutoff emerge from a well-defined 2D QFT deformation.

Abstract

Recent work by Zamolodchikov and others has uncovered a solvable irrelevant deformation of general 2D CFTs, defined by turning on the dimension 4 operator $T \bar T$, the product of the left- and right-moving stress tensor. We propose that in the holographic dual, this deformation represents a geometric cutoff that removes the asymptotic region of AdS and places the QFT on a Dirichlet wall at finite radial distance $r = r_c$ in the bulk. As a quantitative check of the proposed duality, we compute the signal propagation speed, energy spectrum, and thermodynamic relations on both sides. In all cases, we obtain a precise match. We derive an exact RG flow equation for the metric dependence of the effective action of the $T \bar T$ deformed theory, and find that it coincides with the Hamilton-Jacobi equation that governs the radial evolution of the classical gravity action in AdS.

Figures (2)

• Figure 1: The energy levels $E_n$ at $L=2\pi$ and $J=0$ as a function of $\mu$ for different values of $E(0) = {{\Delta}}_n+\bar{{{\Delta}}}_n-{c\over 12}$. States with $E(0)>0$ that correspond to black holes in holographic CFTs are plotted in blue, while low-lying states are plotted in orange. For $\mu>0$ that is the relevant regime in our study we used solid lines, while for $\mu<0$ the spectrum is plotted with dotted lines. The levels exhibit a square root singularity at the critical value $\mu E(0) = 2\pi$. This indicates that, for given $\mu$, the energy spectrum of the deformed CFT is bounded by $E< {8 \over \mu}$, indicated on the plot by a dashed black line.
• Figure 2: As a localized wave approaches the horizon, the minimal RT surface that contains the excitation at time $t$ extends along the horizon over a distance $R(t)$ that grows linearly in time.