Question

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki because there isn't really a "right" answer: what does GLC imply? Lots of big conjectures have well known consequences (Riemann Hypothesis and distribution of primes) but what about GLC? Are there any nice things that are known to follow from this equivalence of derived categories?

EDIT: The Geometric Langlands Conjecture says the following: Let C be an algebraic curve (any field, though I think the formulation I know is only good in characteristic 0), G a reductive algebraic group, its Langlands dual (the characters of G are cocharacters of , if I recall correctly). Then there's a natural equivalence of categories from the derived category of coherent sheaves on the stack of G-local systems to the derived category of coherent D-modules on the moduli stack of principal -bundles, such that the structure sheaf of a point is sent to a Hecke Eigensheaf (and I'm not going to sit down and define that on top of the rest here...the idea is that G-local systems on the curve are equivalent to eigensheaves for some collection of operators, but actually making it precise and having a hope of being true gets technical)

Edit 2: This paper states one version of the conjecture (for GL(n) only) as 1.3, after defining the Hecke operators.

Answer 1

OK, this is a very broad question so I'll be telegraphic. There is a sequence of increasingly detailed conjectures going by the name GL -- it's really a "program" (harmonic analysis of D-modules on moduli of bundles) rather than a conjecture -- and only the first of this sequence has been proved (and only for GL_n), but I don't want to get into this.

There are several kinds of reasons you might want to study geometric Langlands:

1. direct consequences. One application is Gaitsgory's proof of de Jong's conjecture (arXiv:math/0402184). If you prove the ramified geometric Langlands for GL_n, you will recover L. Lafforgue's results (Langlands for function fields), which have lots of consequences (enumerated eg I think in his Fields medal description), which I won't enumerate. (well really you'd need to prove them "well" to get the motivic consequences..) In fact you'll recover much more (like independence of l results). To me though this is the least convincing motivation..

2. Original motivation: by understanding the function field version of Langlands you can hope to learn a lot about the Langlands program, working in a much easier setting where you have a chance to go much further. In particular the GLP (the version over C) has a LOT more structure than the Langlands program -- ie things are MUCH nicer, there are much stronger and cleaner results you can hope to prove, and hope to use this to gain insight into underlying patterns.

By far the greatest example of this is Ngo's proof of the Fundamental Lemma --- he doesn't use GLP per se, but rather the geometry of the Hitchin system, which is one of the key geometric ingredients discovered through the GLP. To me this already makes the whole endeavor worthwhile..

1. Relations with physics. Once you're over C, you (by which I mean Beilinson-Drinfeld and Kapustin-Witten) discover lots of deep relations with (at least seemingly) different problems in physics.

a. The first is the theory of integrable systems -- many classical integrable systems fit into the Hitchin system framework, and geometric Langlands gives you a very powerful tool to study the corresponding quantum integrable systems. In fact you (namely BD) can motivate the entire GLP as a way to fully solve a collection of quantum integrable systems. This has has lots of applications in the subject (eg see Frenkel's reviews on the Gaudin system, papers on Calogero-Moser systems etc).

b. The second is conformal field theory (again BD) --- they develop CFT (conformal, not class, field theory!) very far towards the goal of understanding GLP, leading to deep insights in both directions (and a strategy now by Gaitsgory-Lurie to solve the strongest form of GLP).

c. The third is four-dimensional gauge theory (KW). To me the best way to motivate geometric Langlands is as an aspect of electric-magnetic duality in 4d SUSY gauge theory. This ties in GLP to many of the hottest current topics in string theory/gauge theory (including Dijkgraaf-Vafa theory, wall crossing/Donaldson-Thomas theory, study of M5 branes, yadda yadda yadda)...

1. Finally GLP is deeply tied to a host of questions in representation theory, of loop algebras, quantum groups, algebraic groups over finite fields etc. The amazing work of Bezrukavnikov proving a host of fundamental conjectures of Lusztig is based on GLP ideas (and can be thought of as part of the local GLP). (my personal research program with Nadler is to use the same ideas to understand reps of real semisimple Lie groups). This kind of motivation is secretly behind much of the work of BD --- the starting point for all of it is the Beilinson-Bernstein description of reps as D-modules.

There's more but this is already turning into a blog post so I should stop.

Answer 2

I once typed lecture notes from a talk by Tony Pantev on Geometric Langlands, which I still find a good quick while precise resource for the basic statement:

http://ncatlab.org/nlab/show/geometric+Langlands.

(If David B.-Z. ever feels the need for an online space to talk in more detail about Geometric Langlands, this would be one, by the way...)

Somehow I keep having the feeling that with the right language used the statement might become pretty much a tautology. Well, probably I am wrong, but with suitable flexible interpretation of the terms, the geom. Langlands statement seems to be saying:

"functions" on the "space" of flat G-bundles

correspond to

"flat functions" on the "space" of G^L-bundles .

Answer 3

The DARPA program funding a lot of the early work suggested that it could actually have real-world implication for materials: DARPA said that "seemingly esoteric ideas from [geometric Langlands] will be used to design new materials with novel properties that have the long-range potential to produce nano-level computing devices." See about a third of the way through this post:

http://www.math.columbia.edu/~woit/wordpress/?p=633

I'm not sure what the actual connection is though, and it's possible that it was oversold.