# Dispatches from the Conference on Gauge and Representation Theory

So, I got back from Scotland and my family Thanksgiving (mmm, turkey) just in time for an enormous conference here in Princeton on Gauge and Representation Theory. I don’t really have the energy at the moment for some full blown conference-blogging, but I thought I would mention what was going on for the benefit of those who couldn’t attend.

(EDIT: For those of you who find me too vague, David Ben-Zvi has posted some notes from the talks here.)

I’m afraid I can’t really comment on the physics talks. While my level of total lostness varied during them, the average was high enough that I have nothing intelligent to say. Peanut gallery?

On the subject of talks I did understand some of….Dennis Gaitsgory gave a lovely talk yesterday morning on the localization of $\hat{\mathfrak{g}}$-modules (which attracted attention from people who didn’t even see it. I don’t think Peter’s concerns about Dennis’s pedagogical style were really warranted. He shouldn’t believe everything he reads in the newspapers. The only complaint I had was the large quantity of distracting hand gestures). The starting point of the talk is every geometric representation theorist’s favorite theorem:

Theorem (Beilinson-Bernstein): The functor of global sections $\Gamma:\mathcal{D}(G/B)_\lambda \to\mathfrak{g}-\mathrm{mod}_\lambda$ from $\lambda$-twisted D-modules on the flag variety to the category of $\mathfrak{g}$-modules with the central character of the Verma module with highest weight $\lambda$ is an exact equivalence.

Since, lots of us have seen lots of nice consequences of this theorem, and would like to see a generalization of it to the affine level. In this case, there is a global sections functor, which goes between the appropriate categories, but there are a ton of problems with that. Unfortunately, this functor is not usually exact, and typically has no hope of being an equivalence, since the category on the right is just too big.

The exactness problem isn’t so bad, since you can always switch to the derived category (if that turns your crank). After doing this, at least at the critical level, we end up with a functor which is full and faithful, but still hits way too few objects. But, remarkably, one can enlarge the left hand side by “base change” to make the center big enough (as described by Frenkel and Gaitsgory here) to make a category that is (hopefully) big enough.

There are a bunch of different flavors of this correspondence, some of which work better than others, but all of which look pretty interesting.

Richard Thomas also gave a pretty interesting talk about giving a rigorous definition of BPS invariants of Calabi-Yau 3-folds, using some new and interesting moduli spaces sheaves on curves. Again, I’ll skip the details since I can’t seem to find my notes.

Victor Ginzburg talked about Calabi-Yau (non-commutative) deformations of $\mathbb{C}^3$. Basically, there are a bunch of interesting Poisson brackets on $\mathbb{C}^3$, defined by a function $\varphi$ (of the form $\{x,y\}=d\varphi/dz$, etc.), and these have natural deformations (which are Calabi-Yau in the non-commutative sense) where everything you could hope would survive does. Perhaps, the most interesting kink is that while there’s a non-constructive proof that the function $\varphi$ survives as a central element of the deformation (since it was in the Poisson center before), the actual formula is incredibly long (as in 2 pages long in one of the simpler cases). Anyways, non-commutative geometry marches on.

I’d love to take the time to write a good post about David Ben-Zvi’s talk, as for me, it was the most mathematical (and thus, for me, comprehensible) statement of the Langlands/field theory connection I’ve encountered. Sadly, that would be pretty time consuming, and at the moment, I have to be at Witten’s talk in about 9 hours, which means I should to bed now instead. Sorry!

## 4 thoughts on “Dispatches from the Conference on Gauge and Representation Theory”

1. There appears to be a verb missing in the Beilinson-Bernstein theorem.

2. Considerably more than a verb was missing…but not any more.

3. Scott Carnahan says:

Do D-modules on other varieties like G/P or G/N yield interesting categories?