Between conferences, (because two weeks worth of talks on the week days just wasn’t enough) Sergei Gukov is giving some talks on categorification and gauge theory. Today, we had about 3 hours of talks, which I mostly thought were good, though I’m blanking a little on their content.
Unfortunately, they were somewhat marred by slightly excessive (in my opinion) audience participation. It’s one thing as an audience to ask the speaker to stop an explain something for fully. It’s another to derail the talk for 10 minutes in an attempt to change the topic of the talk.
Anyways, the content was roughly as follows:
Inspired by mathematical physics, people (mostly Witten, but with plenty of others involved) have written down certain gauge theories for 4-manifold, by which I mean a Lagrangian on the space of connections of any principal bundle over your 4-manifold , possibly also using extra data, such as a section of the positive spinor bundle (as in the case of Seiberg-Witten theory, perhaps the most famous of these gauge theories) along with the action of gauge symmetries (this is the automorphism gorup of the principal bundle, that is, just smooth sections of .
You use this Lagrangian to construct a measure on the space of connections (in a very non-rigorous way, but that shouldn’t worry about this for now, since we won’t actually compute any integrals of these). As your parameter goes to 0, this measure becomes more and more concentrated on the locus where the Lagrangian has a critical point, which turns out to be the set of solutions to a particular differential equation. If you start with the Chern-Simons Lagrangian (this is in 3 dimensions), you end up with the space of flat connections on your 3-manifold. If you do Donaldson theory, (now we’re back in 4-d) you end up with the set of anti-self dual connections on your principal bundle. If you start with Seiberg-Witten theory, you get solutions to the “monopole equations.”
Now, if you quotient out by the action of your gauge symmetries, you end with a space that can be surprisingly nice, usually finite-dimensional, sometimes compact, sometimes smooth, sometimes with more exciting structures, such as symplectic, complex, or hyperkähler ones, which (if you like using confusing physics terminology) you might call “the moduli space of monopoles.”
If you have a 2-manifold or 3-manifold , you do dimensional reduction, by taking the product with or and look at the space of invariant solutions (you also want some kind of vanishing along fibers. I’ll admit, I didn’t quite understand this part), to get a couple of new moduli spaces .
So what does this have to do with categorification? Well, it’s not 100% clear yet. We basically got as far as the list:
closed 4-manifold goes to a -1-category
closed 3-manifold goes to a 0-category
closed 2-manifold goes to a 1-category
so associated to a TQFT, we have a hierarchy of categories. Hopefully, we’ll find out a little more about this tomorrow.
I think the most interesting thing I actually learned was the following somewhat remarkable theorem.
Theorem. Let be holomorphic symplectic, and let Lagrangian holomorphic submanifolds. Then is naturally isomorphic to the symplectic Floer homology .
Maybe the rest of you aren’t too impressed, but I think this is awesome. It makes me want to go compute intersections between core components of hypertoric varieties. But, instead I’m gonna go sleep.