Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$
superconformal field theories ([Wikipedia](https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory), [nLab](https://ncatlab.org/nlab/show/6d+%282%2C0%29-superconformal+QFT), [PhysicsOverflow](https://www.physicsoverflow.org/25127/2-0-superconformal-field-theory-in-6-dimensions)) 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?



What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?
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.


