Geometrizing $T\bar{T}$

TL;DR

This paper provides a geometric interpretation of the ${T\bar{T}}$ deformation by formulating it as a dynamical coordinate (and Weyl) transformation induced by the stress tensor, and extends the construction to curved spaces via coupling to 2d topological gravity. It then establishes a precise holographic dictionary in AdS$_3$ with a finite radial cutoff, showing that the annular bulk action equals the ${T\bar{T}}$ operator integrated on either boundary and deriving the bulk-origin flow equations for the deformed stress tensor. The main technical advances include explicit classical deformed actions (via two complementary methods), the derivation of dynamical coordinates from gravity, and a concrete bulk derivation of the TTbar flow equation for the stress tensor. The results illuminate why the cutoff AdS$_3$ setup captures TTbar physics and suggest avenues to extend the framework to quantum corrections and correlation functions, potentially shedding light on nonlocal features and nonperturbative aspects of TTbar deformations.

Abstract

The $T\bar{T}$ deformation can be formulated as a dynamical change of coordinates. We establish and generalize this relation to curved spaces by coupling the undeformed theory to 2d gravity. For curved space the dynamical change of coordinates is supplemented by a dynamical Weyl transformation. We also sharpen the holographic correspondence to cutoff AdS$_3$ in multiple ways. First, we show that the action of the annular region between the cutoff surface and the boundary of AdS$_3$ is given precisely by the $T\bar{T}$ operator integrated over either the cutoff surface or the asymptotic boundary. Then we derive dynamical coordinate and Weyl transformations directly from the bulk. Finally, we reproduce the flow equation for the deformed stress tensor from the cutoff geometry.