Geometric Langlands and QFT

Wednesday’s session at the IAS Conference on Gauge Theory and Representation Theory was mostly devoted to talks by Witten and his collaborators about their latest work on the approach to relating geometric Langlands and QFT that he has pioneered over the last couple years. The talks were quite understandable, giving a general overview rather than details of what are some very technical topics, about which the speakers have produced recently some very long papers. Before discussing the talks, I’ll try and explain the background of this line of inquiry into the borderlands between mathematics and physics.

The history of this subject goes back thirty years, to a 1977 paper of Goddard, Nuyts and Olive entitled Gauge Theories and Magnetic Charge. In the GNO paper the authors noted that in a gauge theory with group G, while the electric charges take values in the weight lattice of G, the magnetic charges take values in the weight lattice of a “dual” group, which is now generally called the Langlands dual group ^LG. This group was used by Langlands in a crucial way in conjectures about number theory that go back to a letter of his to André Weil in 1967. Also in 1977, Montonen and Olive, in Magnetic Monopoles as Gauge Particles?, conjectured the existence of a dual gauge theory interchanging electric and magnetic charges, and the gauge groups G and ^LG. At the time Witten was a Harvard postdoc, and on a visit to England at the end of 1977 Atiyah told him about this conjecture and first suggested it might have something to do with the Langlands program. Witten met Olive, and they collaborated on the 1978 paper Supersymmetry Algebras That Include Topological Charges where they suggested that Montonen-Olive duality would be most naturally realized in a supersymmetric gauge theory. Later work showed that it is N=4 supersymmetric Yang-Mills that seems to have this duality property, now called S-duality and extended to not just a Z₂ symmetry, but a much larger symmetry under the group SL(2,Z).

Warning: What follows is an absurdly overly simplified discussion that will offend pretty much every mathematician who really knows the subject. Comments correcting anything that isn’t at least in some vague sense more or less morally right are welcome.

From the 1970s on, work on conjectures growing out of the Langlands program has come to be one of the dominant themes of number theory, achieving a fantastic success with the work of Wiles on one such conjecture, the so-called modularity conjecture, that led to the 1995 proof of Fermat’s Last Theorem. Trying to explain the Langlands program in any detail is a huge task, but I’ll try and give a few very vague indications here of what it is about. The field Q of rational numbers can be thought of as “rational functions”, on a “space” called Spec (Z), whose “points” are the prime numbers and a special “point at infinity”. Number fields are extensions of Q, and can be thought of as corresponding to covering spaces of Spec (Z), characterized by Galois groups, in particular the Galois group Gal(Q) of the algebraic closure of Q, which in some sense is the fundamental group of Spec(Z). Many questions in number theory can be expressed as questions about “Galois representations”, representations of Gal(Q) in complex Lie groups such as G=GL(n,C). Thinking of Spec(Z) as a “space”, representations of Gal(Q) correspond to local systems, i.e. flat vector bundles over Spec(Z).

The Langlands program has both a “local” and a “global” aspect. The “local” aspect restricts attention to the neighborhood of a “point” in Spec(Z), and the corresponding “local field” of functions. For the “point at infinity”, the local field is the real number field R, for a “point” corresponding to a prime number p, it is the field called Q_p. The local Langlands conjecture gives a correspondence between representations of Gal(Q_p) into a complex Lie group G and complex representations of the corresponding algebraic group ^LG( Q_p) with Q_p coefficients. This correspondence matches up information on both sides that characterizes the representations, which can be expressed either in terms of L-functions, or in terms of the action of Hecke algebras. One can read this correspondence as possibly giving information in both directions: if you know the Galois representations, a so-called “arithmetic” problem, you get a parametrization of the irreducible representations of a Lie group, a so-called “analytic” problem. If you know about the Lie group representations, you get information about number theory.

In the global version of the Langlands correspondence, on the arithmetic side, the global group in question is just Gal(Q), and its representations in a Lie group G are central objects in number theory that one would like information about. On the analytic side, the global group is much trickier to describe. What one needs is something like a gauge group for bundles over Spec(Z), but remember that each “point” of this “space” has a different nature. One introduces an object called the “adeles” A_Q that puts all the local fields together, and then uses this as the coefficients in an “adelic” group ^LG(A_Q), that perhaps can be thought of as the gauge group of all changes in local trivializations about each “point” in Spec(Z). The representation theory on the analytic side is then harmonic analysis on this adelic group, with irreducible representations characterized by specific functions which are called automorphic forms (so this side of the correspondence is often called the “automorphic” side). Galois representations and automorphic forms are matched up by, equivalently, L-functions or the eigenvalues of the action of a Hecke algebra. For the case of 2d representations, the automorphic forms involved are very classical functions on the upper-half-plane, and readily computable information about the coefficients of their Fourier expansions gives deep information about number theory.

An important idea in number theory/algebraic geometry is that algebraic curves over a finite field F_p have many similar features to the “spaces” like Spec(Z) that characterize number fields. Functions on such curves give so-called “function fields”, which behave very much like number fields, and one can transform number theory questions into analogous questions about these curves. For example, there is an analog of the Riemann hypothesis in the function field case, where it has been proven. One can translate the Langlands program conjectures into the function field setting, and there proofs have been found, for the global case by Drinfeld (rank 2 case) in 1974, and Lafforgue (higher rank) in 1999.

Given the Langlands correspondence for an algebraic curve over a finite field, a natural question is whether there is anything analogous if one replaces the finite field by the complex field, and works with complex algebraic curves, i.e. Riemann surfaces. In 1987 Witten wrote a beautiful paper entitled Quantum Field Theory, Grassmannians, And Algebraic Curves, where he explains how one can think of the holomorphic sector of a conformal field theory on a complex algebraic curve as giving something like an automorphic representation in this context, analogous to the ones studied using adeles for algebraic curves over finite fields. He mentions the Langlands program, but makes no attempt in this paper to describe what would be the analog of the Langlands correspondence.

Several years later, around 1995, Beilinson and Drinfeld formulated what is now known as the geometric Langland correspondence, giving a specific conjectural correspondence that is supposed to be an analog for a complex curve C of what happens in the function field case. On the analog of the arithmetic side, one just has a representation of the fundamental group of C in a Lie group G, i.e. a flat vector bundle. The automorphic side is much trickier, and they define “Hecke eigensheaves” on the moduli space of ^LG bundles that play the role of automorphic forms. In their massive (384 pages at last count), unpublished and still preliminary paper Quantization of Hitchin’s integrable system and Hecke eigensheaves, they write

We would like to mention that E. Witten independently found the main idea [of the construction] and conjectured [the main theorem]. As far as we know he did not publish anything on the subject.

Since the mid-1990s, a lot of mathematical activity has grown up around these ideas, creating a new field that is now generally known as “Geometric Langlands theory”, which connects to a wide range of different kinds of mathematics, and to physics via conformal field theory. With funding from the US Defense Department DARPA program, various workshops were organized that brought physicists and mathematicians together to discuss this subject. One such workshop was held at the IAS in March 2004, and there Witten gave a talk (see the end of these notes) about N=4 supersymmetric Yang-Mills and its dimensional reduction to a non-linear sigma model in two dimensions. He credits David Ben-Zvi with explaining to him crucial facts which made clear to him that what was needed to connect this to geometric Langlands was the introduction of boundary conditions in the sigma model, i.e. branes.

Witten first unveiled his version of geometric Langlands based on N=4 supersymmetric Yang-Mills in a talk at the beach at Stony Brook in August 2005; here are notes and audio from the talk. In April 2006 his 230 page paper with Kapustin, Electric-Magnetic Duality And The Geometric Langlands Program appeared, giving the details of a construction based on a topologically twisted (using the “GL twist”) version of N=4 supersymmetric Yang-Mills, dimensionally reduced to give two topological sigma models with target space the Hitchin moduli space, for group G in one case, ^LG the other. These two models, known as the A and B model, are related by mirror symmetry. They involve boundary conditions and thus branes in two-dimensions, and as a result are related by what mathematicians now refer to as “homological mirror symmetry”. The fact that the Hitchin moduli spaces for G and ^LG could be thought of as mirror partners was shown earlier by my colleague Michael Thaddeus in work with Tamas Hausel.

Late last year Witten and Gukov’s 160 paper Gauge Theory, Ramification and the Geometric Langlands Program appeared, extending the QFT approach to geometric Langlands to the “ramified” case, which is that of a punctured Riemann surface, with non-trivial monodromy about the punctures. This was about the “tamely ramified” case, involving simple pole singularities at the punctures. Last month two new papers totalling 193 pages by Witten on this subject appeared, Gauge Theory And Wild Ramification, which deals with the case of higher order poles, and Geometric Endoscopy and Mirror Symmetry, written with mathematician Edward Frenkel.

The talks by Witten and Frenkel gave very general introductions to the two papers, notes taken by David Ben-Zvi are here and here. Witten mostly just explained the background for the wild ramification problem, not giving any details of how he solved it, so his talk mainly functioned as a good introduction to his recent paper. Frenkel also gave a talk which was more of an introduction to his recent joint paper with Witten. He explained that they were studying a special case of the question of what happens at singularities of the Hitchin fibration, for the simplest kind of singularity (orbifold), and simplest non-trivial case (G=SL(2), ^LG=SO(3), outlining the phenomena that appear. These phenomena are analogous to well-known phenomena in the number field case, where their study goes under the name of “endoscopy”. This part of the Langlands story has recently seen major progress, with the proof by Ngo of what is known in the subject as the fundamental lemma. Ngo is giving a series of talks at the IAS this semester on the subject, and Frenkel promised to give a talk next week about possible relations of what he and Witten have been doing to the work by Ngo.

For the story of a comment by Pierre Deligne during this talk, see this posting by Ben Webster.

To me the most interesting talk was Sergei Gukov’s on D-branes and Representations, in which he described what he is working on with Witten at the moment; no paper has yet appeared. Ben-Zvi’s notes are here, and Gukov gave much the same talk recently at Santa Barbara, notes here, audio here. I’ve been most interested in geometric Langlands because of its relations to 2d QFTs and representation theory, where the simplest story should be seen in the local version of the theory. Also, Gukov’s argument was based upon getting Chern-Simons theory out of the original 4d N=4 GL-twisted SYM theory using boundary conditions (something he didn’t explain other than saying what the boundary conditions are). I’ve always wondered whether it is possible to get Chern-Simons out of some sort of possibly supersymmetric twisted theory involving fermions. Someone in the audience asked if what he was doing gave such a theory, but he somewhat evaded the question, saying he preferred to think of things in 4d with boundary.

Getting down to two dimensions, he said that the Hilbert space of this Chern-Simons theory gave a representation associated to the punctured disk, and mentioned that this was related to local geometric Langlands. Someone asked “what happens on the boundary of the disk?”, and he answered that one only needed to impose boundary conditions at the puncture, not on the boundary. Greg Moore sputtered something like “really, on the boundary of the disk you don’t need boundary conditions??” (for the usual story about this, see this paper, which Greg co-authored), to which Gukov answered something about it being all right since they were only looking for supersymmetric BPS states. He went on, as one can read in the notes, to discuss a way of producing representations of a compact Lie group G (and its complexification and other real forms) that associates Harish-Chandra modules to A-branes on the cotangent space to the flag manifold, working out the details for SL(2, R). At the beginning of the talk, Gukov claimed that this was all leading up to a classification of the admissible representations of a real semi-simple Lie group in terms of D-branes, with the various geometrical constructions (e.g. D-modules) known to mathematicians just different faces of the same physical model. To me, the talk raised all sorts of interesting questions, so I’m looking forward to seeing the details when Gukov and Witten have a paper ready.