After mentioning in the last posting that Witten is giving talks in Berkeley and Cambridge this week, I found out about various recent developments in Geometric Langlands, some of which Witten presumably will be talking about.
Edward Frenkel has put a draft version of his new book Langlands Correspondence for Loop Groups on his web-site. In the introduction he describes the Langlands Program as “a kind of Grand Unified Theory of Mathematics”, initially linking number theory and representation theory, now expanding into relations with geometry and quantum field theory. The book is nearly 400 pages long, and to be published by Cambridge University Press. Frenkel also notes that recent developments in geometric Langlands have focused on extending the story from the case of flat connections on a Riemann surface to connections with ramification (i.e. certain point singularities are allowed). He has a new paper out on the arXiv about this, entitled Ramifications of the geometric Langlands program, and he writes that:
in a forthcoming paper [by Gukov and Witten] the geometric Langlands correspondence with tame ramification is studied from the point of view of dimensional reduction of four-dimensional supersymmetric Yang-Mills theory.
The title of the forthcoming Gukov-Witten paper is supposedly “Gauge theory, ramification, and the geometric Langlands program.”
Presumably people attending Witten’s talks in Berkeley and Cambridge will get to hear about this new story for the ramified case. For the rest of us, on his web-site David Ben-Zvi has notes from talks this summer by Witten at Luminy where he describes some of this. Ben-Zvi also has an announcement of a series of lectures on geometric Langlands that he’ll be giving at Oxford next April. The summary of the lectures says that he’ll “describe upcoming work of Gukov and Witten which brings together geometric Langlands and link homology theory.” Link homology theory is also known as Khovanov homology, and I wrote about this two years ago here, advertising Atiyah’s speculation that there may be a 4d TQFT story going on, something I always have found very intriguing. Ben-Zvi has recently lectured on Khovanov homology at Austin, and began his lecture by saying that this material relates “themes in 21st century representation theory” to 4d TQFT. He goes on to cover some of the ideas about 4d TQFT and “categorification” that I was very impressed by when I heard about them from a talk by Igor Frenkel a few months ago (described here).
At first I thought Ed Frenkel’s claim that geometric Langlands was going to give a Grand Unified Theory of mathematics was completely over the top, but seeing how some of these very different and fascinating relations between new kinds of mathematics and quantum field theory seem to be coming together, I’m more and more willing to believe that investigating them will come to dominate mathematical physics in the coming years.
Update: Slides from Witten’s Berkeley lectures are here. And many thanks to David Ben-Zvi for the informative comments!
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