Witten gave a lecture on the beach at Stony Brook on the topic of gauge theory and the Langlands program two months ago, and lecture notes are now available. Lubos Motl has a posting about this, where he promotes the idea that people should stop referring to the “Langlands Program” and just refer to “Langlands duality”. Somehow I suspect that mathematicians will keep doing what they have always done, using “program” to refer to the general, well, program, and “duality” to refer to the more specific, well, duality, that one would like to prove as part of the program.

An earlier posting of mine contains a lot of relevant links, to which should be added the notes from David Ben-Zvi’s talk in Seattle this summer.

It should probably be pointed out that what seems to show up in the physics is Geoemetric Langlands. The full Langlands program is a lot more than just that.

I am trying to synopsize this conversation so I understand this issue.

I was actually looking for “culminating mathematical visualizations” that could take us directly to the geometric design, and work backwards.

Any idea here?

DID YOU NOT LISTEN TO ME? WHO GIVES A CRAP ABOUT GEOMETRIC LANGLANDS WHEN WE COULD BE TALKING ABOUT THE “IMAGING OF THE BRAIN” ARTICLE IN THE NOVEMBER NOTICES OF THE AMS. WITTEN HAS BETRAYED PHYSICS WITH HIS SUPPORT OF STRING THEORY THEREBY DIVERTING GLOBAL ATTENTION FROM BETTER THEORIES LIKE LQG. YOU SHOULD STOP GIVING UNDUE PUBLICITY TO WITTEN HOWEVER INTELLIGENT AND PRAISEWORTHY HIS MATHEMATICAL WORK HAS BEEN, FOR TRULY HE IS METHUSELAH INCARNATE. HE IS THE FIGUREHEAD OF THE STRING THEORY CULT AND MUST NOT BE GIVEN UNDUE PUBLICITY BY RIGHT-MINDED FOLKS.

NOW DO SOMETHING USEFUL AND BLOG ABOUT THE AMS MATHEMATICAL SOCIETY NOVEMBER NOTICES PLEASE. I NEED VALIDATION FOR MY INSECURITY BY HAVING MY NEEDS ATTENDED BY PROMINENT PUBLIC MATHEMATICIANS. THANX.

Sorry qwerty, but if you’re looking for someone who knows something about imaging the brain (or prominent public mathematicians for that matter..), you’re in the wrong place. And I’d much prefer that commenters attend to my needs and insecurities than expect me to do that for them.

As for Witten, I don’t think he’s old enough to be Methuselah, and doesn’t really deserve to be compared to Mephistopheles, which I think is what you had in mind. His recent work on geometric Langlands mercifully has nothing to do with string theory unification or the Landscape, and a lot to do with the relation of quantum field theory and mathematics, which is a hopeful sign.

Dear Peter,

one of the reasons why I don’t see why it’s “program” is that it is not clear what kind open questions and eventual “big goals” i.e. future in general does it offer. Can you write something about it?

Let me remind you that N=4 super Yang Mills, for example, is exactly equivalent to type IIB string theory on AdS5 times a five-manifold, a completely standard string theory with 10 dimensions, strings, D-branes, and all other objects that you so fervently dislike. So maybe your idea that something goes away from strings is confused after all. 😉

All the best

Luboš

Hi Lubos,

The Langlands Program is (amongst other things) an attempt to understand the Galois groups of number fields. The Galois group of a field extension, you’ll recall, is the group of automorphisms of the extension field which fixes the original field. They’re the central objects of number theory; understand all the Galois groups and you can answer almost any question in number theory. Understanding the Galois groups would be _huge_; it would completely change number theory.

The nice thing about Galois groups is that they fit together into hierarchies, which mirror the hierarchies of field extensions — and even better, there is, for every number field, an uber Galois group which governs this entire hierarchy. This is the absolute Galois group, the Galois group of a number field’s maximal field extension.

Unfortunately, the absolute Galois group G is a pretty hard object to get your hands on. So mathematicians usually approach it by trying to understand its category of finite dimensional representations. This is an easier problem: representations are nice linear objects, and we can sort them by their rank. So, we can look at a simpler problem: trying to understand all representations G -> GL_n

Langlands insight was one can understand the n-dimensional representations of the Galois group by relating them to certain representations of the group of adeles of the original number field. By “certain” above, what I really mean is automorphic; we study the representations which exhibit a kind of modular behavior. A huge amount of work has gone into understanding these conjectures; the case of two-dimensional representations was basically the Taniyama-Shimura conjecture which implied Fermat’s Last Theorem. We don’t have anything resembling a complete solution for number fields yet though.

So we try to learn more about this Langlands correspondence by translating it into other fields. We can realize a number field as the field of rational functions on an arithmetic curve. When we do this, the Galois group gets reinterpreted as the fundamental group of our curve! So maybe we can learn something by studying the n-dimensional representations of the fundamental groups of algebraic curves? But these are the flat connections on principal GL_n-bundles on our algebraic curve! They’re nice geometric objects. In particular, they make sense even when our curve is not an arithmetic curve. So we can study the Langlands phenomena over the complex numbers for instance. We don’t even need to restrict ourselves to maps from pi_1(curve) to GL_n. We can replace GL_n by G. At this point, things start to look physics-y. Flat connections on a Riemann surface are instantons of Yang-Mills theory, the automorphic representations can be thought of as sections of certain line bundles living on the space of G-bundles, and the Langlands map starts to look a lot like mirror symmetry.

All of which is a long winded way of saying, we’re borrowing ideas from physics to understand questions in algebraic geometry that originally arose in number theory. For myself, the main fun is that I get to think about relations between quantum field theory and the theory of moduli spaces.

Hope that was a satisfactory answer.

–A.J.

Thanks A.J. for the explanation of the Langlands program.

Some other comments: while the Langlands program for number fields is a central idea in modern number theory and has proved its importance in many ways (e.g. the proof of Fermat), what geometric Langlands is good for is less clear. I know of number theorists who are dubious that it has anything useful to tell them, and other kinds of mathematicians who are also skeptical about it. Personally I’ve always found it fascinating because of its relationship to QFT. It involves many of the same mathematical structures that come up in 1+1 d QFT, including affine Kac-Moody representations, the moduli space of flat connections on a complex curve, etc. The kind of relationship to 4d QFT that Witten is talking about is yet a very different kind of relationship to QFT.

What has always fascinated me about the Langlands program in general is that one of the basic ideas is to look at the cohomology of a moduli space (actually a Shimura variety in the number field case), which gives you both an automorphic representation and a representation of the Galois group, establishing a relation between these. The TQFTs that come out of 2 and 4d gauge theories also are basically all about the cohomology of a related moduli space. And these TQFTs are just twisted versions of supersymmetric gauge theories not that different than the standard model QFT. There’s some still not understood relationship here between perhaps the deepest ideas in physics and one of the deepest ideas in mathematics, and neither side of this story is well understood. If we understand this relationship better, we’re likely to learn something new about QFT, number theory, or both.

Hi Peter,

I have to admit: I don’t care too much about the number theory side of Langlands. It’s interesting in so far as its geometry, but the relations to physics are a lot more interesting. It’s quite tempting to think that quantum field theories are in fact the cohomology theories of derived stacks.