TT deformations in general dimensions

Abstract

It has recently been proposed that Zamoldchikov's T 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 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 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.

Perturbative renormalization of the $T \bar{T}$-deformed free massive Dirac fermion

“…Hence, naively one can evaluate the u-channel loop-integrals L (u) , L (u) µ and L (u) µν directly from their s-channel counterparts (4.51) by replacing θ → iπ − θ. 9 However, notice that in the u-channel loop integrals (4.56) the functional dependence of the integrands on ζ is of the form ζ −q, while in the s-channel loop integrals (4.51) the corresponding functional 9 Replacing θ by iπ − θ is equivalent to replacing ξ 2 by ζ 2 .…”

Burgers Equation vs. Large $N$ Limit in $T\bar{T}$-deformed $O(N)$ Vector Model

“…One may also consider a generalization to higher-dimensional cases with a higherdimensional version of T T -operator [10,17,18]. It is curious to study whether the nontrivial cancellation would occur even in higher dimensions or not.…”

$T\bar{T}$ deformed partition functions

“…A holographic dual for T T deformed CFTs, for ℓ being purely imaginary, has been proposed in [4,11] in terms of AdS 3 with a finite radial cut-off [38]. This intriguing proposal reproduces several features of the deformation, including the spectrum, thermodynamics and signal propagation speeds.…”

“…Recently, a very interesting kind of irrelevant deformation called T T -deformation has been proposed [1,2] and it has attracted growing attention. The T T deformation can be defined for any quantum field theory in 1+1 dimensions (see also [3,4] for higher dimensional proposals). It defines a continuous family of theories along a trajectory in the field theory space whose tangent vector at each point is defined as the T T operator, which is nothing but − 1 2 det T µν where T µν is the stress-energy tensor of the theory.…”

“…We are grateful to Alex Maloney[24] and John Cardy for discussions which led to this subsection 4. It is important to compare coefficients of ℓ 2p and not λ 2p .…”