At the KITP in Santa Barbara there’s a wonderful program on Gauge Theory and Langlands Duality starting up this week, with some of the talks beginning to become available. The main topic will be the relations between S-duality in quantum field theory and geometric Langlands duality that Witten and collaborators have been working on the past few years.
I started trying to watch the talks, but the fact that the video quality is such that one can almost but not quite tell what is being written on the blackboard makes this a bit of a trial. I’m hoping that David Ben-Zvi or someone else will make available notes, which would help a lot. I did very much like Edward Frenkel’s description of the Langlands story as a “Grand Unified Theory of Mathematics”, and was interested to hear that he still feels that there are two different stories about the relation to QFT here, whose relationship is not at all understood (S-duality in 4d QFT is one, 2d CFT and vertex algebras is the other). It seems that A.J. Tolland is there, maybe he or someone else will do some blogging. As I get time to take in the lectures, I hope to write some more about them here.
Update: Notes for the talks are now also being posted, making following them on-line much more feasible. The quality of the talks is excellent, with Ed Frenkel so far giving a beautiful introduction to the roots of the Langlands program in number theory, David Ben-Zvi explaining the structures in topological quantum field theory that mathematicians are trying to exploit, David Morrison and Paul Aspinwall explaining mirror symmetry, D-branes, and the relation to N=(2,2) superconformal field theory, with examples, and Anton Kapustin starting on the 4d N=4 TQFT used to turn S-duality into a mirror symmetry.
there is a funny buzzword sometimes when I read something of witten’s most recent words…
What is wild ramification?
In the context of geometric Langlands, `wild ramification’ refers to the case when the connections or Higgs fields are allowed to have `irregular singularities’. Basically this means they can have poles of order two or more. Heuristically having a meromorphic connection with simple poles on a curve is akin to having punctured the curve at the poles, whereas having higher order poles is more complicated. The word “wild” arises from an analogy with wild ramification (of maps between curves) in characteristic p. One can also look at the growth of horizontal sections approaching such a pole: with simple poles such sections have polynomial growth (“tame”) whereas with higher order poles one can get exponential growth (or decay) as the pole is approached in different directions.
Thank you very much.
Do you know any source that introduces “wild ramification”?
I am also curious about “Langlands conjectures for GLn of a function field”. What is that? Why is that so important? I found this, but I would want something more complete and with applications
For more about “wild ramification” in the geometric context, see references in Witten’s paper with that in the title (what is at issue are “irregular connections”).
To answer your question about Langlands conjecture would be a major effort. Best source I can recommend is expository articles by Edward Frenkel, as well as the talks he is giving now at the KITP, where he is addressing exactly the topic you are asking about.
Opening the pdf notes by Frenkel and then listening to the audio works well.
Grand unified theory of nothing.
If you want to see what mathematicians think is important in mathematics, you should look at the Millenium Problems. Except possibly for the one on Yang-Mills theory, the geometric analogue of Langlands theory has nothing to say about any of them. The Langlands program is very profound with deep arithmetic content. In comparison, its geometric analogue appears to be more superficial and without arithmetic interest.
If you want to see what mathematicians think is important in mathematics, you should look at the
Millenium ProblemsFields Medals.
It’s true that as far as I know geometric Langlands has not led to any new insight into the arithmetic case, and because of this some mathematicians are skeptics. But there are quite a few extremely good mathematicians working on it, and the field does have all sorts of connections to deep questions in mathematics, as well as the connections to quantum field theory and physics. Personally I think it’s definitely fascinating and still mostly unexplored territory, and may turn out to be one of the great themes of 21st century mathematics. Time will tell….
Of course Frenkel’s comment on the “grand unified theory”
is tongue in cheek, but in any case he was referring to the
entire Langlands program, not just to its geometric
Also the claims about the disconnect with arithmetic are
not very accurate. First the Langlands program for function
fields has profound connections both with the arithmetic
and the geometric Langlands programs.
And most spectacularly, Ngo Bao-Chau has
used ideas originating from the geometric setting
(in fact from the study of Yang-Mills!) – namely
the Hitchin system – to resolve one of the major outstanding
conjectures in the arithmetic setting, the so-called
“Fundamental Lemma”, which has numerous immediate
number-theoretic consequences (work which to many
suggests he should be honored with a Fields medal).
The geometric Langlands program itself doesn’t claim
to the profundity and impact of the arithmetic conjectures
(few areas in math do), but it has had spectacular successes,
for example the work of Bezrukavnikov and collaborators
resolving various conjectures of Lusztig on the deep relations
between quantum groups, loop groups and
algebraic groups over finite fields, one of the central
problems in representation theory.
As the term “geometric Langlands” should suggest, a cardinal motivation for the development of the theory by Drinfeld and Beilinson was to apply and refine the analogies between number fields and function fields. A big part of the issue is learning how to properly formulate and generalize the picture in the arithmetic setting and the geometric picture is very valuable for that. Hitchin fibrations are presumed to be the first of many examples of this.
Putting the mathematical worth of geometric Langlands aside, it seems to me that the Physics is actually closer to the arithmetic and finite field channels anyway. By Physics I mean the science that actually has some bearing on testable predictions in the real world.
And Kea2? Cute.