Question

I'm recently working on something called 3d mirror symmetry in QFT literature, which involves two hyperkähler manifolds. There seems to be a corresponding(?) mathematical theory called symplectic duality, pursued by Braden, Licata, Proudfoot and Webster.

Where can I read about it? The only thing I could find so far is the proposal by Proudfoot et al. I'm particularly intrigued by the fifth example in page. 7, which says

More generally, the moduli space of $G$-instantons on a crepant resolution of $\mathbb{C}^2/\Gamma$ is dual to the moduli space of $G'$-instantons on a crepant resolution of $\mathbb{C}^2/\Gamma'$, where $G$ is matched to $\Gamma'$ and $G'$ is matched to $\Gamma$ via the McKay correspondence.

Where can I read about this duality, in particular the case when neither $G$ nor $G'$ is of type $A$?

Answer 1

Nowhere. The paper is still in preparation, and looks to be for a few more months at least. Probably the best document at the moment is this (extremely long) set of talk slides of mine.

I should note: even when there is a paper, there won't be a definition that will tell you (by which I mean someone who studies mirror symmetry from the string theory side) anything new. Just an observation of a lot of very striking coincidences.

Also, it's not going to discuss the non-type A Mackay example in detail; it's not one of the ones we understand well, we just included it because the physicists told us to.