$T \bar{T}$ deformations in general dimensions

It has recently been proposed that Zamoldchikov’s $T \bar{T}$ deformation of two-dimensional CFTs describes the holographic theory dual to $\mathrm{AdS}_3$ 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 $\mathrm{AdS}$. 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 $\mathrm{R}$ 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.

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This work is funded by the STFC grant ST/P000711/1. This project has received funding and support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 690575.

MMT would like to thank the Kavli Institute for the Physics and Mathematics of the Universe and the Banff International Research Station for hospitality during the completion of this work.

Published 13 July 2023