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.](http://pages.uoregon.edu/njp/sd.pdf) 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 $A$?



What is the "symplectic duality" between holomorphic symplectic manifolds? Where can I read more about it?
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](http://pages.uoregon.edu/bwebster/austin-dual.pdf) 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](http://arxiv.org/abs/hep-th/9611063v2).


