TT deformations in general dimensions

TL;DR

The authors generalize the 2D ${\cal T}{\bar{\cal T}}$ deformation to higher dimensions by deriving a finite-radius trace identity for the holographic stress tensor from Gauss-Codazzi equations in AdS gravity. They propose a quadratic operator ${\cal T} = T_{ij} T^{ij} - \frac{1}{(d-1)} (T^{i}_{i})^{2}$ that deforms the boundary CFT, and show how to include R-current and scalar contributions for gauge and scalar fields. The resulting energy spectra from gravity and the deformed field theory are shown to agree for stationary, homogeneous states, with charged and scalar sectors providing consistent extensions. The work also links these deformations to the stress-energy constraint in holographic theories dual to vacuum Einstein gravity and discusses implications for fluids, correlation functions, and entanglement in deformed theories.

Abstract

It has recently been proposed that Zamoldchikov's $T \bar{T}$ deformation of two-dimensional CFTs describes the holographic theory dual to AdS$_3$ at finite radius. In this note we use the Gauss-Codazzi form of the Einstein equations to derive a relationship in general dimensions between the trace of the quasi-local stress tensor and a specific quadratic combination of this stress tensor, on constant radius slices of AdS. We use this relation to propose a generalization of Zamoldchikov's $T \bar{T}$ deformation to conformal field theories in general dimensions. This operator is quadratic in the stress tensor and retains many but not all of the features of $T \bar{T}$. To describe gravity with gauge or scalar fields, the deforming operator needs to be modified to include appropriate terms involving the corresponding R currents and scalar operators and we can again use the Gauss-Codazzi form of the Einstein equations to deduce the forms of the deforming operators. We conclude by discussing the relation of the quadratic stress tensor deformation to the stress energy tensor trace constraint in holographic theories dual to vacuum Einstein gravity.