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Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$ superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $\mathcal X$. I don't really know what that would entail; apparently we have partial information about this theory, suggesting that it's difficult to describe even in a physics sense (e.g. non-Lagrangian), let alone to mathematically formalize.

But I have also heard from mathematicians who are interested in Theory $\mathcal X$ for what I assume are entirely mathematical reasons. I came away with the impression that even though we can't construct it, there are ways to study it to yield interesting results in pure mathematics; but I don't know any examples of such results.

So my question is: what are some purely mathematical takeaways from the story of Theory $\mathcal X$? And, if it's known, what would be some expected mathematical consequences of a construction of Theory $\mathcal X$ in a physics sense?

I get the impression that various dimensional reductions of Theory $\mathcal X$ should include several commonly-studied TQFTs and QFTs, so in a sense studying Theory $\mathcal X$ generalizes the study of those TQFTs. So one possible answer is that there could be theorems about those TQFTs whose proofs were inspired by some conjectured aspect of Theory $\mathcal X$ — but is this accurate?

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    $\begingroup$ There is a convenient chart at ihes.fr/~celliott/workshop to use in addition to the links you provided. $\endgroup$ – AHusain Sep 6 '18 at 8:04
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    $\begingroup$ Those slides by David Ben-Zvi are pretty enlightening, especially the picture at the end summarizing the various dimensional reductions you mention and to which cool mathematical topics they are related: web.ma.utexas.edu/users/benzvi/clay092216.pdf $\endgroup$ – Adrien Sep 6 '18 at 10:17
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    $\begingroup$ There were two workshops recently precisely on the topic of your question, funded by an FRG grant precisely on this topic (from whence the charts/pictures cited originate): A BIRS workshop birs.ca/events/2015/5-day-workshops/15w5154 (the lectures are all available as video) and the workshop aimed at grad students that @AHusain mentioned, ihes.fr/~celliott/workshop -- which has the advantage of typed lecture notes by Qiaochu Yuan. $\endgroup$ – David Ben-Zvi Sep 6 '18 at 16:24
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If you take the (2,0) theory and put it on a manifold which is $T^2 \times M_4$, it is known to reduce to $\mathcal{N} = 4$ super-Yang Mills theory on $M_4$. That theory exhibits S-duality, which has been shown to be related to geometric Langlands. In particular, S-duality is part of an $SL(2,\mathbb{Z})$ symmetry, and the $G \leftrightarrow \widehat{G}$ duality in both geometric Langlands and $\mathcal{N}=4$ SYM is related to the $\left({0\ -1 \atop 1\ \ 0}\right)$ element in the $SL(2,\mathbb{Z})$ symmetry of the torus above. The name S-duality arises because this operates as the $\tau \leftrightarrow -\frac{1}{\tau}$ Mobius transformation on the coupling and exchanges strong and weak coupling. The somewhat mysterious S-duality in four dimensions then becomes something geometric in six dimensions. Presumably, then, a mathematical understanding of the six dimensional theory would give new insight into the mathematics of geometric Langlands (and maybe, if one is lucky, the Langlands program more broadly).

The rub, however, is that the interesting case is when the gauge group in four dimensions is non-Abelian, and even on the physics side, there is no good understanding of the six dimensional theory that gives rise to this upon compactification. String theoretically, it should arise as the theory on a stack of M5-branes. It must be something novel, because it has to reduce to a theory with two different (Langlands dual) gauge groups simply by changing one's perspective on the torus.

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  • $\begingroup$ This is really interesting; thank you! When you say "the interesting case is when the gauge group in four dimensions is non-Abelian, and even on the physics side, there is no good understanding of the six dimensional theory that gives rise to this upon compactification," does that mean the abelian case is understood? if so, do you know where I could read about it? $\endgroup$ – Arun Debray Sep 11 '18 at 17:01
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    $\begingroup$ I’d say it’s better understood, at least (just as S-duality is better understood in the Abelian case). I’m not sure of a good recent review, though. Maybe someone more up to speed can chime in. $\endgroup$ – Aaron Bergman Sep 12 '18 at 1:14
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In general, mathematical outputs of SUSY field theories often become more accessible after performing some twist, and the same is true of the 6d (2,0) SCFT. Considering the theory on $\Sigma\times M_4$, it admits a twist (first studied by Beem-Rastelli I believe) that's holomorphic along $\Sigma$ and topological along $M_4$. Some discussion of the difficulties in mathematically describing this twist in perturbation theory can be found in these notes from a talk of Kevin Costello: https://math.berkeley.edu/~qchu/Notes/6d/Day%205,%20Talk%202,%20Costello.pdf

One interesting application of this twist to representation theory is the AGT correspondence. Specializing $M_4=\mathbb{R}^4$, the rough idea is that after putting suitable $\Omega$-backgrounds in the $\mathbb{R}^4$ direction, the degrees of freedom of the 6d theory localize to $\Sigma$ where we find a Toda field theory. We then get several relationships between this Toda field theory living on $\Sigma$ and the 4d $\mathcal{N}=2$ gauge theory gotten by compactifying on $\Sigma$. In particular, the conformal blocks of the Toda field theory should match the partition function of the 4d $\mathcal{N}=2$ theory, and vertex operator insertions on $\Sigma$ should correspond to including certain defects in the 4d $\mathcal{N}=2$ theory. In the math literature, this connection (at least for pure gauge theory) was established through work of Maulik-Okounkov and Schiffman-Vasserot who showed that the affine W-algebra (i.e. the local observables of the above Toda field theory) acts on the equivariant cohomology of the moduli of instantons on $\mathbb{A}^2$.

Let me also very briefly comment that there's a 6d string theory (dubbed the little string theory) that becomes the (2,0) SCFT in the limit of infinite string mass. This string theory appears to be just as mathematically rich as its SCFT limit. These slides: https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2016/main/Aganagic%20v2.pdf from Mina Aganagic's 2016 String-Math talk contain some statements in this direction.

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    $\begingroup$ Welcome to MO!I believe the holomorphic-topological twist of the (2,0) theory was first studied by Gaiotto-Moore-Neitzke in 2009 (Beem-Rastelli-van Rees is 2014) - or at least they certainly studied the reduction of the (2,0) theory on a complex surface, which then has the standard Donaldson-Witten twist as an N=2 theory. The twist is explicitly studied in 6d in Witten's Fivebranes and Knots from 2011. $\endgroup$ – David Ben-Zvi Sep 7 '18 at 19:43
  • $\begingroup$ This is also a great answer, and I wish I could accept both. $\endgroup$ – Arun Debray Sep 11 '18 at 17:02

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