One of my minor hobbies over the years has been trying to understand something about the Langlands conjectures in number theory, partly because some of the mathematics that shows up there looks like it might be somehow related to quantum field theory. A few days ago I was excited to run across a web-page for a workshop held in Princeton earlier this year on the topic of the Langlands Program and Physics. Notes from some of the lectures there are on-line.
Unfortunately, after reading through the notes, I’m afraid there’s relatively little there about the potential intersection of the ideas of the Langlands Program with Physics. From the physics end of things there are some pretty illegible notes of a lecture by Witten about the Langlands Dual Group in Physics. Part of this story involves the Montonen-Olive duality of N=4 supersymmetric Yang-Mills. This duality interchanges the coupling constant with its inverse, whiile taking the gauge group G to the Langlands dual group (group with dual weight lattice). The symmetry that inverts the coupling constant is actually part of a larger SL(2, Z) symmetry.
One possible explanation for this SL(2,Z) symmetry is the conjectured existence of a six-dimensional superconformal QFT with certain properties. Witten explains more about this in his lectures at Graeme Segal’s 60th birthday conference in 2002. His article from the proceedings volume, entitled “Conformal Field Theory in Four and Six Dimensions” doesn’t seem to be available online, but his slides are, and they cover much the same material. There has been a seminar going on at Berkeley this past semester in which Ori Ganor has been giving talks on this topic.
While the occurence of the Langlands dual group and SL(2,Z) symmetry are suggestive, the relation of this to the full Langlands program seems to be a bit tenuous. There is however a much closer relation between 2d conformal field theory and the Langlands program, a relation which is part of the story of what is now known as “Geometric Langlands”.
Some of the other lectures at the Princeton workshop give a good explanation of the standard Langlands duality conjectures, although I’m not convinced that many physicists will find them easy going. These conjectures posit a duality between two very different kinds of group representations associated to a one-dimensional field (a number field or function field of a curve over a finite field). On the one side one has an analytic object, an “automorphic representation” on a space of functions on a group G(A), where G is a group over A, the adeles of the field. On the other side one has an arithmetic object, representations of the absolute Galois group of the field in the Langlands dual group to G. Typically this duality is used to get information about arithmetic objects using the more tractable analytic objects. The most famous example of this is the Taniyama-Shimura-Weil conjecture relating the arithmetic of elliptic curves to modular forms, which Wiles (with Taylor) was able to prove enough of to use it to prove Fermat’s last theorem.
In general the Langland conjectures for the case of number fields remain an open problem, but for the case of function fields of a curve, they have been proven for G=GL(n) by Drinfeld for n=2 and Lafforgue for general n (which got both of them Fields medals). The geometric Langlands program involves reformulating the function field case in such a way that it still makes sense when you replace the curve over a finite field by a curve over the field of complex numbers. This idea goes back to Drinfeld and Laumon in the 1980s, and has evolved into a specific conjecture which was recently proved by Frenkel, Gaitsgory and Vilonen.
I confess to still being pretty mystified by this subject. The analog of the arithmetic side is clear enough, it’s a homomorphism of the fundamental group of the curve into the Langlands dual group, or equivalently a vector bundle with holomorphic flat connection. But I still don’t understand the analog of the analytic side, which is some sort of D-module over the moduli space of bundles over the curve, broken up into “Hecke eigensheaves”. My colleague Michael Thaddeus explained to me today over lunch what a “Hecke eigensheaf” is supposed to be, but there’s a whole web of relations of this to representations of affine Lie algebras, CFT and vertex operator algebras that neither of us understands very well.
While I don’t understand this material, I do hope to find time in the future to try and figure some of it out. Various sources that seem to explain this are the following:
Edward Frenkel’s web-site at Berkeley contains a lot of interesting material. Many of his papers are on this topic, especially relevant is his Bourbaki seminar report on Vertex Algebras and Algebraic Curves.
Another relevant web-site is that of David Ben-Zvi at Texas. Look at his very informal surveys of Langlands theory written in 1995 before he gets too embarassed by the mistakes in them and takes them down. He is joint author with Frenkel of a book Vertex Algebras and Algebraic Curves.
There’s an on-going seminar on geometric Langlands at the University of Chicago which has a web-page.
Kari Vilonen has a web-site devoted to geometric Langlands and its relation to physics.
As usual, Witten has a hand in all of this, see his remarkable paper “Quantum field theory, Grassmanians and algebraic curves”, Communications in Mathematical Physics, 113 (1988) 529-600, and his contribution to the 1987 conference “The Mathematical Heritage of Hermann Weyl” entitled “Free fermions on an algebraic curve”.
For a different conjectural relation between Langlands and QFT, see:
Mikhail Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, in Functional Analysis on the Eve of the 21st Century, Vol. 1, Birkhauser, Boston, pp. 119-151.