Gauge Theory and Representation Theory, Day II

Today was the second day of Gauge Theory and Representation Theory. I understood a bit more today, and so this post will be longer and with more math.

Today we started out with the Institute’s own Jaun Maldacena. Now, first off a bit about what he’s famous for: the AdS/CFT Conjecture. This is the claim that there is a nice correspondence between $SU(4)$ Super Yang Mills Theory and String Theory on $AdS^5\times S^5$, that is, the five dimensional anti-deSitter Space cross the five-sphere. And honestly, beyond getting that much, I didn’t follow a lot of his talk, so I’ll move on to the first one that I got something out of:

Victor Ginzburg from the Univesity of Chicago spoke about Calab-Yau Algebras and noncommutative geometry, though the noncommutative geometry was somewhat hard to see, though the algebra was quite nice. This stuff was joint work with Pavel Etingof, another big name in math these days. So first he gave the definition of an anti-dualizing complex for an associative algebra $A$ with a finite projective resolution as an $A$ bimodule. This is just $A^!=RHom_{A-bimod}(A,A\otimes A)$. So I hid a lot of complexity in the word “just”…really this is something I’m just learning about now, which is that derived functors should be thought about as functors between derived categories, and so that’s what this is. We can just think of it as the $Ext$ complex, though.

So next he said that $A$ is Calabi-Yau of dimension $d$ if $A^!\cong A[-d]$, that is, has only one nonzero term and that is shifted by $d$ from 0. So then the deformation theory started. The short of it is that if $A$ is a deformation of $\mathbb{C}[X]$ for some variety $X$, then $A$ is CYd (Calabi-Yau of dimension $d$) requires that $X$ is also, in the geometric sense. The first order stuff gets a Poisson Structure on top of things. He followed these ideas and eventually constructed a full family of deformations for the coordinate ring of a Del Pezzo Surface, almost all of which are noncommutative. It was great, and he was able to make sense (mostly) of the derived category of coherent sheaves over the spectrum of a noncommutative ring (this last bit is the tricky part).

Moving on, I got to have lunch with several fellow bloggers who are attending the conference, notably Peter Woit of Not Even Wrong and AJ Tolland and Ben Webster from the Secret Blogging Seminar. Fun people. But now back to the talks.

Phillip Boalch of many places substituted today with “Some Geometry of Irregular Connections on Curves.” This talk I didn’t follow very much of, but it was talking about Hyperkahler manifolds, surface groups, and quaternionic curves, and I wish I’d been able to keep up and explain some of these things (or even just have a vague understanding myself).

In the penultimate talk of the day, Andre Losev of ITEP gave a talk that didn’t resemble in the slightest his title. He did some stuff involving Morse Theory, Mirror Symmetry and (this I didn’t expect to see) some Tropical geometry. He then generalized to higher dimension, and spoke about higher dimensional holomorphic maps between complex manifolds.

Finally, David Ben-Zvi from the University of Texas spoke about “Langlands Duality for Character Sheaves.” This was rather nice for me, because I wasn’t feeling well at the beginning of the semester when he gave the colloquium here, and so I missed a chance to catch him speaking, but got a second chance at this conference. His stuff I understood a chunk of too, and so I’ll explain a bit of it.

First we’re going to look in two dimensions. Let $G$ be a finite group, and then $Z_G$, a two dimensional topological field theory can be associated to it. Now, to study the representations of $G$, we look at $\mathbb{C}G$, the group algebra, which is generally noncommutative, and is a Frobenius algebra with the trace map given by evaluation at the identity (just take your element and take the coefficient of the identity element of the group).
So representations of the group are the same as modules over the algebra. So now out TFT takes a circle and spits out the center of the group algebra, which is just the set of class functions. So we take a surface with boundary (really a cobordism between sets of circles) and it gives a map from the tensor product of a bunch of centers to the tensor product of a bunch of centers, with the first number being the starting circles and the second being the finishing circles. This much was actually known by Frobenius and Schur.

If we look at the open string version, however, then we also have to associate something to points. We take points to the category of $\mathbb{C}G$-modules, which is a Frobenius category, that is, there are trance maps on the endomorphisms of each object. Rather than typing up the whole talk, I’ll just mention some highlights, like the fact that going up to three dimensions doesn’t require three dimensional manifolds, but rather, we increase the category level. We switch from polynomials to $\mathscr{D}$-modules, and then also tack on the derive category, and things still work out in mostly the same ways (with appropriate natural changes). This led to some connections with braided monoidal categories that I’m not really qualified to talk about (I’d recommend John Baez’s “This Week’s Finds” or the n-Category Cafe for that).

And before I go, I should mention that I won’t be blogging tomorrow, because I have to leave the conference early, but I’ll be back on the job Thursday.