TT deformations in general dimensions

It has recently been proposed that Zamoldchikov's TT deformation of two-dimensional CFTs describes the holographic theory dual to AdS3 at finite radius. In this note we use the Gauss-Codazzi form of the Einstein equations to derive a relationship in general di-mensions between the trace of the quasi-local stress tensor and a specific quadratic combination of this stress tensor, on constant ra-dius slices of AdS. We use this relation to propose a generalization of Zamoldchikov's TT over bar 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 TT over bar . To describe gravity with gauge or scalar fields, the deforming operator needs to be modified to include appropriate terms involving the corre-sponding 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 en-ergy tensor trace constraint in holographic theories dual to vacuum Einstein gravity.

Automated summary

This study proposes a relationship between Zamoldchikov's TT deformation and the holographic theory dual to AdS3 at finite radius. The Gauss-Codazzi form of the Einstein equations is used to derive a relationship between the trace of the quasi-local stress tensor and a specific quadratic combination of this stress tensor on constant radius slices of AdS. The study also discusses the generalization of Zamoldchikov's TT over bar deformation to conformal field theories in general dimensions and the modification of the deforming operator to include appropriate terms for gravity with gauge or scalar fields.