# What is the “symplectic duality” between holomorphic symplectic manifolds? Where can I read more about it?

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 $A$?

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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.

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Thank you very much, Ben! I'm looking forward to the full paper ... it's also good to know which string theory paper you had in mind. –  Yuji Tachikawa Jun 5 '11 at 13:53
You didn't follow the link? –  Ben Webster Jun 5 '11 at 18:03
Hi, Ben. Do you have more precise statements of, for example, duality between quiver varieties and affine Grassmann ? I am not fully satisfied with your evidences in p.38. –  Hiraku Nakajima Jun 6 '11 at 1:25
@Ben I did, thank you. (I meant to write "it was good to be informed which string paper...") –  Yuji Tachikawa Jun 6 '11 at 1:57
I think there is a bit more evidence now (those slides are about a year and a half old now, but it's still a little pie in the sky outside the type A case (where things are pretty well established). What sort of evidence would be more satisfying for you? –  Ben Webster Jun 6 '11 at 4:58