Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $A$?

share|improve this question

1 Answer 1

up vote 6 down vote accepted

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.