TT deformations in general dimensions

TL;DR

This paper generalizes the $T ar{T}$ deformation from 2D CFTs to higher dimensions using holographic principles, deriving a quadratic stress tensor operator that captures finite-radius AdS holography.

Contribution

It introduces a new quadratic stress tensor deformation applicable in general dimensions, extending the $T ar{T}$ framework beyond two-dimensional conformal field theories.

Findings

Derived a relation between stress tensor trace and quadratic stress tensor in AdS slices.

Proposed a generalized $T ar{T}$-like operator for higher-dimensional CFTs.

Connected the deformation to holographic stress energy tensor constraints.

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.