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.

中文翻译：

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一般尺寸中的 $T \bar{T}$ 变形

最近有人提出，二维 CFT 的 Zamoldchikov 的 $T \bar{T}$ 变形描述了有限半径处 $\mathrm{AdS}_3$ 对偶的全息理论。在本文中，我们使用爱因斯坦方程的高斯-科达齐形式来推导在 $\mathrm{ 的恒定半径切片上准局部应力张量的迹与该应力张量的特定二次组合之间的一般维度关系。广告}$。我们利用这种关系提出将 Zamoldchikov 的 $T \bar{T}$ 变形推广到一般维度的共形场论。该算子在应力张量中是二次的，并且保留了 $T \bar{T}$ 的许多但不是全部特征。为了用规范场或标量场描述重力，需要修改变形算子以包含涉及相应 $\mathrm{R}$ 电流和标量算子的适当项，我们可以再次使用爱因斯坦方程的高斯-科达齐形式来推导变形算子的形式。我们通过讨论与真空爱因斯坦引力对偶的全息理论中二次应力张量变形与应力能量张量迹约束的关系来得出结论。