## Symplectic duality slides November 24, 2008

Posted by Ben Webster in Algebraic Geometry, category O, crazy ideas, link homology, mathematical physics, QFT, talks.

I’ve been too lazy to write in detail about the progress in my research (well, I am writing six papers and applying to jobs, so it isn’t entirely due to laziness), but I did recently speak in the symplectic seminar at MIT, and have posted the slides on my webpage. Obviously, they’re less useful without someone to explain them, but given the current lack of an overarching paper on the subject (that’s no. 5 on the list, I promise), I thought it might be edifying. Executive summary below the cut.Tom Braden, Tony Licata, Nick Proudfoot and I have been working on understanding a certain kind of duality between holomorphic symplectic manifolds. At the moment, this lacks any really good overarching mathematical definition, but we have some really cool examples:

• the cotangent bundle to the flag variety $T^*G/B$ is dual to the cotangent bundle to the Langlands dual flag variety $T^*{^LG}/{^LB}$.
• the Hilbert scheme of points on $\mathbb{C}^2$ is self-dual.
• the quiver variety of two dominant weights $\lambda,\mu$ of a finite-dimensional, simply-laced Lie algebra is dual to the slice in $\overline{\mathrm{Gr}_\lambda}$ to the orbit $\mathrm{Gr}_\mu$ in the affine Grassmannian of said Lie algebra (it’s probably better to say Langlands dual, but that’s the same, since my algebra is simply-laced).

One point I got to at the end, but couldn’t cover in nearly the detail I would have liked (in part because I haven’t worked out the details) is the implications of this for knot theory. In particular, this seems to explain why Stroppel on one hand, and Seidel-Smith on the other, got completely different geometric definitions of the same knot homology theory; they were working in geometric situations (cotangent bundles of Grassmannians on one hand, and Slodowy slices in nilcones on the other) which are dual under this prescription.

All right, I’d better get back to writing the actual paper on this stuff, so that people have something more detailed to read than my slides.

1. Allen Knutson - November 25, 2008

…the implications of this for knot theory. In particular, this seems to explain why Stroppel on one hand, and Seidel-Smith…

And here I naively thought my students were working on unrelated subjects. You sneaks!

2. Ben Webster - November 25, 2008

Allen, you know me. I’m a uniter, not a divider.

3. Scott Carnahan - November 25, 2008

Hi Ben,
Could you say what a slice is? You said something about transversality during your talk, but I can’t tell if that defines a unique object.

4. Ben Webster - November 25, 2008

well, you have to be a little careful. The Slodowy slice is actually defined to be a particular choice of subspace which I didn’t want to take the time to write down.

In general, the slice is well defined analytically (it’s the unique variety such that an analytic neighborhood of your orbit is locally isomorphic to an affine space times the slice), which means that algebraically what’s well defined is the completion of the slice at the intersection point with the orbit. Thus, you can think of it as completing your space at point on the stratum, and decomposing that completion at the completion of the orbit times the completion of the slice.

There’s a detailed discussion of this point in my paper with Nick (read the proof of Lemma 2.4), since the fact that our slice wasn’t defined in the Zariski topology was important there.

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