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The statement of the ordinary non-categorical version of geometric Langlands conjecture, which was proven for GL(n) in around 2002 by Frenkel, Gaitsgory and Vilonen, is quite well-known and is easy to find in the literature.

Recently, by talking to some students of Dennis Gaitsgory and postdocs working in this area, I understand that a stronger categorical version of the geometric Langlands conjectures has been in circulation for at least the past few years, and with recent advances Dennis is perhaps even close to proving it (at least in type A). My question is: what is the precise statement of the categorical version of geometric Langlands?

I understand on the left you have something related to the category of $D$-modules on $Bun_G(X)$. After you take the moduli stack of $G^{\vee}$-local systems, on the right you have some category in between the categories of coherent sheaves on this stack, and the category of quasi-coherent sheaves on this stack. I also hear that the categorical version uses ideas from work of Lurie ( $(\infty,1)$-categories, $DG$-categories, etc).

Am I correct that on the left we have the full category of D-modules on $Bun_G(X)$? (Where-as in the simpler non-categorical version we simply request a Hecke eigensheaf for each irreducible local system).

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Minor comment: the 2002 result you mention was for the unramified case. –  S. Carnahan Feb 25 '11 at 1:01
    
I am a non-expert, but my understanding from talking to the experts is that at some point they always slyly say "Now let's restrict to the unramified case". In the local (e.g. p-adic) case (i.e. classical local Langlands) the conjecture is a theorem in full generality, allowing wildly ramified representations, but I half-suspect that no-one can formulate geometric Langlands when you allow nasty ramification. I'd love to be proved wrong though. –  Kevin Buzzard Feb 25 '11 at 11:19
    
@Kevin - I'm not sure that's completely accurate. Certainly the picture is not fully satisfactory with ramification but there are things one can say. I'll try to edit the answer below to reflect that. –  David Ben-Zvi Feb 25 '11 at 16:43
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up vote 22 down vote accepted

For context for Tom's answer, let me state the naive version of the conjecture, which has been around since around 1997 I think (due to Beilinson-Drinfeld). It calls for an equivalence of (dg) categories $$D(Bun_G(X))\simeq QC(Loc_{G^\vee}(X))$$ between (quasi)coherent $D$-modules on the stack of $G$-bundles on a curve $X$, and (quasi)coherent sheaves on the (derived) stack of flat $G^\vee$ connections on the curve. Moreover (and this is where most of the content lies) this equivalence should be an equivalence as module categories for the spherical Hecke category $Rep(G^\vee)$ acting on both sides for every choice of point $x\in X$. The action on the right is given by simple multiplication operators (tensor product with a tautological vector bundle on $Loc_{G^\vee}(X)$ attached to a given representation of $G^\vee$ and point $x$. On the right hand side the action is by convolution (Hecke) functors, associated to modifications of $G$-bundles at $x$ (relative positions at $x$ of $G$-bundles are labeled by the affine Grassmannian, and in order to formulate this statement we use the geometric Satake theorem of Lusztig, Drinfeld, Ginzburg and Mirkovic-Vilonen). The equivalence can be further fixed uniquely by using "Whittaker normalization", a geometric analog of the identification of L-functions in the classical Langlands story.

In this form the conjecture is a theorem for $GL_1$, using the extension of the Fourier-Mukai transform for D-modules. There are no other groups for which it's known (yet - though perhaps Dennis already has $SL_2$), and few curves (one can do $P^1$ for example). Also (as Scott points out) this is only the unramified case, and there are natural conjectures to make with at least tame ramification (a "parabolic" version of the above). There's also a close variant of this conjecture which comes naturally from S-duality for N=4 super-Yang-Mills, thanks to Kapustin-Witten. [Edit: in fact the physics suggests a far more refined version of the whole geometric Langlands program.] It is a theorem of Beilinson-Drinfeld if we restrict on the left hand side to D-modules generated by D itself, and on the right to coherent sheaves living on opers.

It is also well known that this conjecture as stated is too naive, due to bad "functional analysis" of the categories involved (precisely analogous to the analytic issues appearing eg in the Arthur-Selberg trace formula). On the D-module side, the stack $Bun_G$ is not of finite type, and one might want to make more precise "growth conditions" on D-modules along the Harder-Narasimhan strata. On the coherent side, $Loc$ is a derived stack and singular, and one ought to modify the sheaves allowed at the singular points -- in particular at the most singular point, the trivial local system. [Edit:One approach to correcting this involves the "Arthur $SL_2$" -- roughly speaking looking at local systems with an additional flat $SL_2$-action that controls the reducibility..this comes up really beautifully from the physics in work of Gaiotto-Witten, one of the first points where the physics is clearly "smarter" than the math.] These two issues are very neatly paired by the duality. If one restricts to irreducible local systems and "cuspidal" D-modules, these issues are completely avoided (though the conjecture even on this locus remains open except for $GL_n$, where I believe it can be deduced from work of Gaitsgory following his joint work with Frenkel-Vilonen, see his ICM).

In any case, Dennis has now given a precise formulation of a conjecture, which was at some point at least on his website and follows the outline Tom explained. It is very clear that homotopical algebra a la Lurie is crucial to any attempt to prove this, and Dennis and Jacob have made (AFAIK) great progress on this. The basic idea of the approach is the same as that carried out by Beilinson-Drinfeld and suggested by (old) conformal field theory (in particular Feigin-Frenkel) and (new) topological field theory (Kapustin-Witten and Lurie) -- i.e. a local-to-global argument, deducing the result from a local equivalence which comes from representation theory of loop algebras. The necessary machinery for "categorical harmonic analysis" is now available, and I'm looking forward to hearing a solution before very long..

Edit: In response to Kevin's comment I wanted to make some very informal remarks about ramification.

First of all one needs to keep in mind the distinction between reciprocity (which is what the above conjecture captures) and functoriality. In the geometric setting the former is strictly stronger than the latter, while in the arithmetic setting most of the emphasis is on the latter. It is in fact quite easy to formulate a functoriality conjecture in the geometric setting with arbitrary ramification. Namely, fix some ramification and look at the category of D-modules on the stack of G-bundles with corresponding level structure. Then this is a module category over coherent sheaves on the stack of $G^\vee$ connection with poles prescribed by the ramification (eg we can take full level structure and allow arbitrary poles). Then one can conjecture that given a map of L-groups $G^\vee\to H^\vee$ the corresponding automorphic categories are given simply by tensoring the module categories from $QC(Loc_{G^\vee}(X))$ to $QC(Loc_{H^\vee}(X))$ (everything here must be taken on the derived level to make sense). It is not hard to see this follows from any form of reciprocity you can formulate. And there is also a geometric version of the Arthur-Selberg trace formula in the ramified setting (under development).

Second, one can make a reciprocity conjecture with full ramification, though you have todecide to what extent you believe it. In the "completed"/"analytic" form of geometric Langlands that comes out of physics such a conjecture in fact appears in a paper of Witten on Wild Ramification. Roughly speaking in the above reciprocity rather than just looking at the module category structure for D-modules over QC of local systems, you can ask for them to be equivalent... not stated anywhere since maybe I'm too naive and this is known to be too far from the truth, but I think more likely people haven't thought about it very much.

Third, the local story: of course in the p-adic case Kevin discusses one restricts to $GL_n$. There are two things to point out: first of all in the geometric setting one is interested in all groups, and $GL_n$ is not much simpler as far as our understanding of the local story goes. Second, while everything is much harder and deeper in the p-adic setting than the geometric setting, it's worth pointing out that a formulation of a local geometric Langlands conjecture is a much more subtle proposition, involving the representation theory of loop groups on derived categories which is only beginning to be within the reach of modern technology (even at the level of formulation of the objects!)

That being said, there are rough forms of local geometric Langlands conjectures developed by Frenkel, Gaitsgory and Lurie. It is best understood in the so-called "quantum geometric Langlands program", a deformation of the above picture involving the representation theory of quantum groups, where Gaitsgory-Lurie give a precise general local conjecture and make progress on its resolution. The usual case above is a bit degenerate and one needs to be more careful. In any case the rough form of the local conjecture is an equivalence of 2-categories (again everything has to be taken in the appropriate derived sense) between "smooth" LG-actions on categories and quasicoherent sheaves of categories over the stack of connections on the punctured disc..

--THAT being said we don't have a proof of this for $GL_n$ so again the number theorists win! just wanted to give some sense that there is a reasonable understanding of full ramification.

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Thanks for the comments on ramification issues. On the arithmetic side there are two stories---one local and one global. In the local case the unramified story is understood completely. But the ramified story is not, except for some groups ($GL_n$, $GSp_4$...). I had initially thought that one might be able to formulate a local conjecture via global methods, but this also seems hard because globally you can have two $\pi$s that agree everywhere other than at one place, where they can be rather different. In particular even if the number theorists had functoriality in some great generality... –  Kevin Buzzard Feb 25 '11 at 20:06
    
...it is not clear to me that they would necessarily have a local conjecture, let alone a theorem, for general $G$. Maybe my comment that the geometric guys restrict to the unramified case when the going gets tough is unfair, but on the other hand it seems to me that David makes a precise global conjecture in the unramified case above, and in the ramified case it looks to my untrained eyes that his comments indicate that things are much vaguer---which in my mind is in perfect analogy with the arithmetic setting, where things are also vague. –  Kevin Buzzard Feb 25 '11 at 20:09
    
In particular I can not give a precise statement, which incorporates ramification, of any of the below things in the arithmetic setting: (1) functoriality (2) local Langlands (3) global Langlands, for a general $G$, even though I could talk all day about how things might work, and could make some more precise statements for $GL(n)$. I am just trying to make a "counter-point" to David's point: the OP asked for a precise conjecture, and I am suggesting that if you allow ramification then precise conjectures incorporating these phenomena are perhaps thinner on the ground! –  Kevin Buzzard Feb 25 '11 at 20:12
    
Thanks! One question: Is there a formulation of the quantum local geometric langlands conjecture, which you mention, anywhere? I think I once saw a talk Dennis gave on the subject, and Dennis's paper "Twisted Whittaker models & Factorizable Sheaves" seems to be on the "global" quantum geometric Langlands (unless I'm mistaken..). –  Vinoth Feb 25 '11 at 22:17
    
@Kevin - thank you for the informative comments. Of course I agree that things are much vaguer in the ramified setting, except for functoriality, which can be formulated precisely with arbitrary ramification (it's a straightforward Tannakian exercise to formulate in the geometric setting). –  David Ben-Zvi Feb 26 '11 at 15:32
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My (incomplete) understanding is that you are supposed to modify the "naive" attempt at the spectral side (i.e. coherent sheaves on $G^\vee$-local systems) as follows. There is a (recent) notion of microsupport for coherent complexes on local complete intersection stacks (and the stack $\operatorname{Loc}_{G^\vee}$ of $G^\vee$-local systems is such a stack). Now, in the corrected conjecture one should take only the part of the coherent derived category of $\operatorname{Loc}_{G^\vee}$ consisting of complexes with microsupport in the global nilpotent cone.

I should say that I think there are lots of details in the various documents on Gaitsgory's web page: http://www.math.harvard.edu/~gaitsgde/GL/. You are better off looking there than reading what I wrote above!

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Thanks! I'll try to make sense of this ... is there a reference which describes the microsupport condition for coherent complexes on (suitably nice) stacks? Also, can you tell me which of Dennis's papers on that webpage are most relevant to this? Dennis seems to have a key paper formulating the categorical correspondence that he hasn't posted called "Formulation of the GL conjecture"... –  Vinoth Feb 25 '11 at 0:58
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Vinoth, if I remember correctly that's the one that describes the microsupport condition. I first heard about it from David Ben-Zvi, who heard about it from Tony Pantev...I think maybe Tony and Dima Arinkin (or maybe just Dima??) worked it out. But I think it's explained in that paper of Dennis. Which appears not to exist on his web page any longer...though I think I have the file somewhere on my laptop... –  Thomas Nevins Feb 25 '11 at 1:46
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p.s. You might ask a new question, which asks for a description of the microsupport condition for coherent complexes on an LCI scheme/stack. Then you might get Tony (or Dima? is he an MO user?) to give a detailed answer, which would be a great community service since this is surely a fundamental new notion. –  Thomas Nevins Feb 25 '11 at 1:53
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