TITLE: What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?
QUESTION [18 upvotes]: 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?

REPLY [11 votes]: 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.