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The $T\bar{T}$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field theories (CFT) by an operator that is quadratic in the stress-energy tensor, called the $T\bar{T}$ operator.

Zamolodchikov [1] obtained analytic results for the expectation value of the operator, using the complex coordinates $z=x+it$ and $\bar z=x-it$ to express the 2d CFT coordinates:

$$ O\equiv T_{zz} T_{ \bar{z} \bar z} − T_{z\bar z}^2 $$

in a two-dimensional (2d) relativistic quantum field theory, where the $T_{µν}$ denote the components of the (euclidean) energy-momentum, or stress, tensor in complex coordinates. In particular, Zamolodchikov showed that it is well-defined by point-splitting, and that its vacuum expectation value $\langle O \rangle$ is proportional to $$ -\langle (T^{µ}_{µ})^2 \rangle $$

My question is that what are some possible generalizations of $T\bar{T}$ deformation in higher dimensions?

One possibility is consider this operator defined as a bilocal operator,

$$ TT¯(x, y) = T_{ij} (x)T_{ij} (y) − T_{ii}(x)T_{jj}(y) $$

where $T_{ij}$ is the stress-energy tensor. In a two-dimensional CFT this operator was shown by Zamoldchikov to have a remarkable OPE structure as $$ x \to y $$: $$ TT¯(x, y) = T (y) + \sum_α A_{α}(x − y) ∇y O_{α}(x). $$ where $O_α$ denote local operators and the function $A_α(x − y)$ can be divergent as $x \to y$; this relation implies that we can identify $TT¯$ as a local operator $T (y)$, modulo derivatives of other local operators. The $TT¯$ operator can be used to deform the CFT, generating a family of theories characterized by the coupling of this operator. While the deforming operator is irrelevant, its particular properties imply that the resulting theory is more predictive than a generic non-renormalizable quantum field theory (QFT).

My question again: are there other possibilities to motivate the (physical/geometrical) meanings of higher dimensional $d$ generalizations of $T\bar{T}$ deformation, say $d$ is larger than 2?

[1] A. B. Zamolodchikov, Expectation value of composite field T anti-T in two-dimensional quantum field theory, hep-th/0401146.

[2] M. Taylor, TT deformations in general dimensions, arXiv:1805.10287

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check https://arxiv.org/abs/1804.10967 (last section contains a discussion on this topic)

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