$T\bar T$ and the mirage of a bulk cutoff

TL;DR

This work derives a first-principles holographic dictionary for the universal $T\bar{T}$ deformation at large $N$ by applying a variational-principle approach to the bulk gravity side. It shows that the deformation induces nonlinear mixed boundary conditions for the boundary graviton while leaving matter-boundary conditions undeformed, and that the deformed stress tensor matches the Brown–York tensor with a fixed counterterm, yielding the correct deformed energy spectrum. The analysis reveals that the asymptotic symmetry group remains two commuting Virasoro algebras with the same central extension as the original CFT, albeit generated by state-dependent diffeomorphisms. A concrete matter example demonstrates that the TTbar deformation encodes bulk information entirely via asymptotic data rather than enforcing a hard bulk cutoff, arguing for a broader, spacetime-wide holographic dictionary beyond Verlinde’s finite-radius picture.

Abstract

We use the variational principle approach to derive the large $N$ holographic dictionary for two-dimensional $T\bar T$-deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non-dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial "bulk cutoff" is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT.

Figures (2)

• Figure 1: The values of $\mathcal{L}_0, \bar{\mathcal{L}}_0$ that lead to real $\mathcal{L}_\mu, {\bar{\mathcal{L}}}_\mu$. The domain can be divided into two regions: the solution in region I is given by the upper sign in \ref{['rellmul0']}, while in region II it is given by the lower sign.
• Figure 2: The $\bar{\mathcal{L}}_0 \rightarrow 0$ limit of $E_\mu$ for the two solutions in \ref{['rellmul0']}. The green dotted line is the CFT answer, $E=\mathcal{L}_0$. To converge to it, one must choose the upper sign (continuous blue line) for $\rho_c \mathcal{L}_0 < 1$ and the lower sign (dashed orange line) for $\rho_c \mathcal{L}_0 > 1$.