# On Geometric Langlands Correspondence

The Geometric Langlands correspondence introduced by Drinfeld and Laumon conjectures a 1 to 1 correspondence between (A) local systems on a projective smooth curve over a field and (B) (Hecke eigen-)perverse sheaves on the algebraic stack of vector bundles of rank n on the curve (or more generally G-bundles).

The conjecture seems in good way to be proven (see the recent preprint of Gaitsgory http://arxiv.org/abs/1302.2506). I have the following questions: Is it possible to replace curves by higher dimensional varieties? Is it possible to consider curves (or varieties) minus a finite set of points. Has it been studied? References? Can you formulate (approximatively) what would be the generalization of (B)?

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Rather than working with curves minus a finite set of points, it's better to work with ramified local systems on the whole curve, so you can keep track of the local monodromy at those points. But formulating the conjecture in the ramified case is supposed to be difficult. –  Will Sawin May 9 '13 at 15:59
Regarding status of the conjecture: for $GL_n$ the type of geometric Langlands conjecture you state was proved a while ago by Gaitsgory following joint work with Frenkel and Vilonen --the cited paper concerns a much stronger "harmonic analysis on G-bundles" form of the conjecture. For groups other than $GL_n$ even the form you state is wide open. –  David Ben-Zvi May 9 '13 at 16:44
Dear David, [FGV] prove the case over a finite field and for the whole curve. Thanks to the remark of Will I found the paper of Frenkel-Gaitsgory "Local Geometric Langlands correspondence and affine Kac-Moody algebras". Page 3 of their paper they describe the generalization of (B) to general reductive group and complex projective smooth curves minus some points. This answer pretty much half of my question. I would still be happy to know if one expect a similar conjecture for higher dimensional variety or not... –  user12770 May 9 '13 at 17:30

EDIT Few days ago a survey by A. Parshin appeared in arxiv. I think it is the best place to look on the higher-dimensional Langlands. From the abstract:

A brief survey is given of the classical Langlands correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n). A generalization of the Langlands program to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. We formulate a conjecture on a direct image (=automorphic induction) of automorphic forms which links the Langlands correspondences in dimension 2 and 1. The direct image conjecture implies the classical Hasse-Weil conjecture on the analytical behaviour of the L-functions of curves defined over global fields of dimension 1.

Actually similar question has been asked. Higher-dimensional Langlands is something very intriguing. I did not follow recent advances, but let me mention some part which I know about.

Let us start with NON-geometric local case. Langlands correspondence is roughly speaking "bijection" between representations of Galois group and representations of GL(Local Field).
Abelian case (class field theory) is bijection between characters of Galois and characters Local field, if dualize we get Galois/[Galois,Galois] = (Local Field)^*.

One of the first questions to ask - whether it is possible to generalize local class field theory to higher dimensions ?

The main idea by A. Parshin is that in n-dimensions one should consider Milnor's (n)-th K-group of local field instead of (Local Field)^*. In particular for n=1 K_1^Milnor(Field)=Field^*. (By the way the definition of higher dimensional local field should also be given). Parshin also found higher analogs of various symbols and proved higher analogs of reciprocity laws.

To the best of my knowledge there were no further developments in the field before Kapranov's paper "Analogies between the Langlands correspondence and topological quantum field theory" . In that paper he gave certain vision what higher dimensional Langlands might be.

His idea (quite amazing) that "representations" should be substituted by "k-representations" (i.e. representations in higher categories), (e.g. for surfaces we should consider 2-representations). So in n-th dimensions k-representations of dimension r of Galois group should correspond to (n-k)-representations of GL_r(n-Local Field)

In particular abelian version will correspond to Parshin's higher dimensional class field theory, since Milnor's K-groups correspond higher-representations.

Now about the geometric version. It is quite unclear for me. In 1-dimension, we can think of flat connections as analogs of Galois representations. It seems in higher dimensions we should consider gerbes with flat connections as analogs of higher representations of Galois group, whatever it means...

The geometric substitute for moduli space of vector bundles and Hecke-eigen sheaves is not clear for me. The reason is that in 1-dimension moduli space of vector bundles arise as standard coset G_{out}\G(k((z))/G_{in}, however in higher dimensions I do not see how the group G( k((z,u)) ) may have finite-dimensional quotient. So I do not see something finite-dimensional where "n-Hecke eigensheaf" may live.

Well, it is probably just my own problem. I have speculated around these things in http://arxiv.org/abs/hep-th/0604128, but I am afraid it is very unclear...

Last year there appeared a paper Unramified two-dimensional Langlands correspondence, which probably is the last mile achievement in the question. Unfortunately I have no time to follow these developments.

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There are plenty of contributions to higher local class field theory beyond Parshin's work. See e.g., Kato's 1990 ICM talk: mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0419.0428.ocr.pdf –  S. Carnahan May 27 '13 at 2:03