Why is $T\overline{T}$ deformation so exciting? I keep an ear to high energy physics discussions, and one of the things I've heard a lot about recently in these channels is the TTbar deformation (stylized $T\overline{T}$)$^1$.  Wikipedia is lacking an explanation of this idea, so I thought it would be useful to get an expert to chime in here.  

What is the $T\overline{T}$ deformation, and why do people care about it?

I think the target audience for the present question should be someone with the usual graduate level knowledge of QFT (including basic knowledge of CFT and RG flows).  For those looking for a nice technical introduction to $T\overline{T}$, this paper suffices. 



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*Useful tip: the latex for $T\overline{T}$ is "T\overline{T}".

 A: After watching a colloquium by Alexander Zamolodchikov on this topic, I think I can answer this question in broad strokes now.
At a basic level, $T\overline{T}$ deformation refers to a flow on the space of 2D QFTs in a direction given by the determinant of the stress-energy tensor $T_{\mu \nu}$.  Explicitly, the infinitesimal change in the Lagrangian density $\mathcal{L}$ along the flow is given by
$$\mathcal{L}\rightarrow \mathcal{L}+\delta t \det T_{\mu \nu} = \mathcal{L}+\frac{\delta t}{\pi^2} T\overline{T}$$ hence the name $T\overline{T}$ deformation.  More generally, there are similar deformations generated by any pair conserved currents associated with $\mathcal{L}$.
The interesting thing about these deformations is that, when you do the math, the effect of the deformation on the observables of the theory is totally solvable. Thus one of the main things these deformations are useful for is understanding what the space of all possible QFTs "looks like".  For instance, the traditional point of view in QFT focuses on field theories described by well-behaved, quasi-local actions.  However, we have reason to believe that not all QFTs should be described in this way (because of something known as the "UV completeness problem", which is where RG flows take you out of this space of "nice" QFTs).  It seems that $T\overline{T}$ deformations might be a way to generate such exotic QFTs, which would nevertheless have understandable properties.
Alexander Zamolodchikov's main interest in $T\overline{T}$ deformations was for this reason, to understand better the space of all possible QFTs.  However, he mentioned that there are also connections to quantum gravity, holography, world-sheet dynamics of strings, and non-relativistic deformations.
A: A $T\overline T$-deformation of a quantum field theory defines a one-parameter family of quantum field theories as
$$
\frac{\partial S_{(\mu)}}{\partial\mu}=-\int\mathrm{d}^2z\ \mathcal O_{T\overline T}\big|_\mu
$$
where $|_\mu$ denotes that we employ the stress-energy tensor of the QFT defined by action $S_{(\mu)}$, and $\mathcal O_{T\overline T}$ is formally defined in a 2D theory as
$$
\mathcal O_{T\overline T}=\lim_{\varepsilon\to0} T_{zz}(z+\varepsilon)T_{\bar z\bar z}(z)-T_{z\bar z}(z+\epsilon)T_{z\bar z}(z)\sim\det T
\\\Rightarrow\langle\mathcal O_{T\overline T}\rangle=\langle T_{zz}\rangle\langle T_{\bar z\bar z}\rangle - \langle T_{z\bar z}\rangle^2
$$

Broadly, the resulting QFT is seen to be solvable since the deformation is topological, and calculating its observables is in fact tractable. So why do we find $T\overline T$-deformation exciting?

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*$T\overline T$-deforming a QFT deforms its S-matrix, introducing a phase that manifests as a universal scattering time delay even in the presence of a classical background. In particular, $\Delta t\propto E$, which is characteristic of a gravitational theory. So $T\overline T$-deformations, in a sense, can turn ordinary quantum field theories into a quantum gravitational theory - this has not yet been fully understood and so is ripe for further development. (It is also rendered plausible by the fact that no conclusive off-shell observables have been discovered in such theories).


*$T\overline T$-deformed QFTs have a very peculiar interplay between locality and UV-completeness: the coupling $\mu$ introduces a length scale $\sqrt\mu$ where the deformed QFT is seen to become non-local and, despite the fact that the $T\overline T$ operator is irrelevant, the S-matrix is ostensibly UV-finite. The most interesting thing here is that there is no UV fixed point that saves the RG flow, as per the usual paradigm of QFT! A different mechanism, dubbed asymptotic fragility, kicks in to generate a non-local UV completion of our base EFT. It is worth investigating these esoteric UV properties, not least because they may manifest in physical theories (and also that the UV behaviour is similar to that in string theory!). Dubovsky has a nice paper on this in Jackiw-Teitelboim gravity: arXiv:1706.06604.


*Studying the $T\overline T$ deformation (on a wide class of initial EFTs, owing to the universality of the deformation) enables us to develop the requisite tools to study other members of the Zamolodchikov-Smirnov class of integrable local deformations. This includes, for instance, $J\overline T$ deformations where $J$ is a chiral $U(1)$ current. Such a deformation is also solvable and generates a UV-complete theory. These are exciting in their own right since, for instance, $J\overline T$-deforming a 2D CFT can shed light on the microscopic description of extremal Kerr black holes and warped BTZ black holes. See e.g. arXiv:1902.01434.


*Speaking of 3D quantum gravity, a double trace $T\overline T$-deformation of certain 2D CFTs be regarded as holographically dual to an $\mathrm{AdS}_3$ theory where (roughly) the boundary no longer lies at asymptotic infinity, but is instead brought inwards to finite radial distance. This is known as "cutoff holography". Equivalently, the deformation modifies the boundary conditions obeyed by the bulk fields. Among other things, this could provide valuable insights on the resolution and emergence of bulk locality. See e.g. arXiv:1807.11401.


*A single trace $T\overline T$-deformation acting on a symmetric product orbifold CFT can "interpolate" (in an RG sense) between the holographic bulk $\mathrm{AdS}_3$ and an asymptotically linear dilaton spacetime. So it provides a concrete, tractable example of holography where the bulk theory is not AdS, and hence may be a step towards formulating holography in flat spacetime. The most well-known example of this construction is the holographic description of a 2-dimensional compactification of little string theory (whose geometry involves a system of NS5 branes). See e.g. arXiv:1911.12359 and references therein.
