On the symmetry of $T\bar T$ deformed CFT

TL;DR

This paper introduces a symmetry in $T\bar T$ deformed 2D CFTs that preserves the trace relation, derives the deformed conformal Killing equation, and connects it to boundary conditions in AdS$_3$ gravity, revealing insights into the deformation's geometric and physical effects.

Contribution

It proposes a new symmetry in $T\bar T$ deformed CFTs, derives the deformed conformal Killing equation, and links it to gravity boundary conditions and stress tensor properties.

Findings

Deformed conformal Killing equation accounts for background metric running.

Deformation acts as a renormalization group flow of the metric.

Stress tensor matches Brown-York tensor with counterterm.

Abstract

We propose a symmetry of $T\bar T$ deformed 2D CFT, which preserves the trace relation. The deformed conformal killing equation is obtained. Once we consider the background metric runs with the deformation parameter $\mu$, the deformation contributes an additional term in conformal killing equation, which plays the role of renormalization group flow of metric. The conformal symmetry coincides with the fixed point. On the gravity side, this deformed conformal killing equation can be described by a new boundary condition of AdS$_3$. In addition, based on the deformed conformal killing equation, we derive that the stress tensor of the deformed CFT equals to Brown-York's quasilocal stress tensor on a finite boundary with a counterterm. For a specific example, BTZ black hole, we get $T\bar T$ deformed conformal killing vectors and the associated conserved charges are also studied.