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\(J\bar T\) deformations: how a non-local theory stays UV-complete and conformally symmetric

Primordial gravitational waves: A combo, when BICEP3 is added, claims that there aren't any so far and \(r_{0.05}\lt 0.036\). Google News.
Two or three decades ago, theoretical high energy physics was much more lively. An aspect of this liveliness was the constant stream of uprisings and minirevolutions. Sourballs and crackpots may have called them fads and they may have criticized the fact that many researchers were joining them because of FOMO. But it was an overwhelmingly good phenomenon. People were sufficiently excited and the system motivated them to work. So when someone found something new – about twistors, BMN string bits, and dozens of other topics (later – many people (including young ambitious people) worked on them. Researchers could have been sure that someone cared about their paper on the fashionable topic and it further stimulated their research.

It's not necessarily Monica's cup of tea but Ozone's Dragostea Din Tei is the greatest product of Romanian culture since Count Dracula. ;-) Leony was recently inspired (what is the polite word for plagiarism?) by that song, the new one is in English.

I am confident that the number of such signs of collective excitement and widespread activity has significantly dropped in a recent decade or so, and especially in the recent five years (some sourballs and crackpots probably celebrate). But there are still examples of mini-industries that look as healthy as we used to know them, their number is just smaller. On Saturday, it will have been four years since the launch of one such mini-industry.
An integrable Lorentz-breaking deformation of two-dimensional CFTs
Monica Guica – whom I have known as a brilliant Harvard grad student (and who has been to Pilsen once) – has found a new class of UV-complete quantum field theories that are actually... not local. Well, this broader industry was founded by Smirnov and Zamolodčikov (Sorbonne+Rutgers; the latter approved my spelling of his name when he was my instructor LOL) in August 2016, i.e. a year earlier. They added some composite operators \(T\bar T\) to the two-dimensional CFT (conformal field theory) action. The action has some extra term that is bilinear in the stress energy tensor.

You might intuitively protest that it is a random contrived construction that messes the good properties of the CFT. But most of the magical traits of these deformations are all about the fact that the virtues are not really messed up at all. Instead, the \(T\bar T\) deformation of a CFT was found to be solvable. The spectrum is obtained from the original one by a universal formula, too.

OK, in October 2017, Monica Guica (Saclay-Paris; sometimes add Nordita plus either Stockholm or Uppsalla University) presented a new deformation, the \(J\bar T\) deformation (don't get confused by the letters JT, the JT gravity means Jackiw-Teitelboim gravity, a different thing, which Wikipedia also tells you not to confuse with Liouville/CGHS gravity, not sure why you should). It is similar to \(T\bar T\) except that the first stress energy tensor is replaced with a \(U(1)\) current \(J\). The other factor \(\bar T\) is still the generator of translations in a null direction (or the antiholomorphic one, if you have a Euclidean world sheet). Back in 2017, she noticed that her deformation is somewhat less disruptive and preserves a finite subalgebra of the Virasoro-style algebra.

These "dipole" deformations may look ad hoc but what is nicely surprising is that despite the non-locality, the theories seem to be UV-complete QFTs. There are no inconsistency at any energy scale (energy in the 2D world sheet). This UV-completeness is surprising (I would say) because one wants to think that if the theory makes sense at arbitrarily short distances, it should admit some local lattice-like construction, and it's therefore local. But this reasoning is too sloppy and indeed, these deformations are examples of nonlocal but UV-complete quantum field theories.

Some 250+ followups of Smirnov-Zamolodčikov were written; and over 100+ of Monica's 2017 \(J\bar T\) paper. One of the important papers was the 2019 article by Adam Bzowski and Monica which asked what happens to the holographic bulk dual of the CFT (the AdS theory) if the CFT is deformed. The answer is that it is still gravity in an AdS space but it has some modified boundary conditions that mix the metric with the Chern-Simons gauge field.

I have always been uncertain about the basic status of these deformations. Are they a new, "more generic" class of QFTs that people should actually study? Or just classes of "fudged up" CFTs that actually lose some properties? Some trivial variations of the CFTs? Or even simple transformations of the CFTs to new variables? Even after the latest paper, I am uncertain but the latest paper surely changes a lot. Today, one second after the previous deadline (18:00:01 UTC), Monica released a new paper
\(J\bar T\)-deformed CFTs as non-local CFTs.
She worked in the Hamiltonian formalism and the paper is full of Poisson brackets (and then commutators). But she adds some \(J\bar T\) deformations to the Hamiltonian and analyzes possible symmetries. Note that \(J\bar T\) consists of two factors, a holomorphic one and an antiholomorphic one. In the Minkowski signature, it means that the left-moving and right-moving sectors of the CFT are treated differently. One of them preserves a Kač-Moody algebra (she adds Witt- at the beginning). But the shocking thingg is that she can find the whole infinite-dimensional algebra on the other side as well. There are Kač-Moody algebras on both sides.

Despite the seemingly nonlocal deformation, the theory seems to have the same symmetry algebra as a rather normal CFT. But there is a catch. The Virasoro algebra has a zero mode and we automatically think that it is the same thing as the Hamiltonian, the generator of translations. However, in her deformation, the Kač-Moody zero mode generator is actually a quadratic function of the Hamiltonian!

Well, I think that this could have been found earlier and maybe by someone else than the mother of the \(J\bar T\) deformations. But she is just very bright and many people no longer feel the FOMO – or the incentives to work hard – so she had to find it herself. It's all very interesting because some of the old-fashioned assumptions we had about "every good UV-complete QFT" (and we surely tended to think that "most" good UV-complete QFTs are CFTs in the UV) are strictly speaking invalidated, but relatively mildly so.

A natural guess would be that all these theories are just some weird change of variables and they are otherwise totally equivalent to the undeformed CFT. This could lead to a rather trivial explanation of all these virtues. Well, it seems to me that it cannot be the case. The AdS dual of these deformed theories are just "slightly modified" but exactly because the modification looks so innocent (a linear mixing of bulk fields), we may see that the modification is non-empty. After all, it is just some deformation of the boundary conditions. When you change the allowed periodicity of world sheet scalars on a closed string, you are changing the radius of a circle in the target space which is a nonzero change. Well, you "unfreeze" some spacetime scalar field that you could have frozen or overlooked but you are surely doing something "new". In the same sense, the \(J\bar T\) deformation has to be "somewhat new". The deformation parameter \(\lambda\) is on par with the vev of something analogous to a field in the target spacetime or the dual gravitational bulk spacetime. Well, the parameter modifying the mixed boundary conditions has an easy interpretation but it isn't necessarily the only one.

I think that many HEP theorists have the same (almost?) black-and-white perspective on the field theories that are worth studying. There should be a unique list of "good properties" and only the theories that obey all of them are "truly interesting". Tens of thousands have been written that contributed something to the clarification of these precise rules but we're still uncertain about the "right virtues" and the "size of the space of the good theories", not only in string theory but even in field theory. If the theories with all the virtues don't have to be conformal, how many extra deformations and modifications preserve the kosher character of the QFTs? How big is the space that we actually remain ignorant of?

String theory really has an advantage because the string vacua are physically connected with each other. In the case of field theory, we might say that the questions in the previous paragraph are ill-defined because there may be many layers of "field theory" and the list of virtues doesn't have to be unique. In string theory, it is arguably unique and the list of vacua is well-defined. However, there may still be superselection sectors that are "just a step away from each other" from the perspective of one physical interpretation; but infinitely far from another vantage point.

These confusions will probably continue for some time but people should still try to settle similar more localized questions. Either the \(J\bar T\) deformations are a truly new original class of theories that open totally new possibilities, and in that case, the research of QFT should really be upgraded to that larger class; or all the new theories are really just "derivative objects" and cherries on a pie that only "quantatively and modestly change" something about the properties of the parent theories. People should better know which viewpoint is the more correct one.

Also, people should master the methods to rigorously prove whether such theories are unequivalent to the original ones by a redefinition of variables; and whether they are really nonlocal in all variables. We know other examples where the nonlocality is just an artifact of a choice of variables and using other variables, the locality is completely restored.

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