Question

I'm currently working through Frenkel's beautiful paper: http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf. I'm looking for a good example of a projective curve to get my hands dirty, and go through the general constructions that Frenkel shows there and try to do them manually for this example of a curve. Are there any good instructive examples for doing this? (Or does it always get out of hand very quickly?)

Answer 1

Unfortunately I don't think geometric Langlands is very easy on any curve. The only curve where the objects are readily accessible is $P^1$, but even there the general statement is kind of tricky (see Lafforgue's note here). I would look at Frenkel's writings on the Gaudin model, which is a concrete illustration of the Beilinson-Drinfeld-Feigin-Frenkel approach to geometric Langlands for $P^1$ with several punctures. Also Arinkin and Lysenko worked out explicitly a case of geometric Langlands (in a stronger sense) on $P^1$ minus 4 points -- see the first four papers on a mathscinet search for Arinkin. So the answer is try $P^1$ with some punctures, but don't be surprised if things are rather tricky already there.

(I also think geometric Langlands on an elliptic curve should be accessible, but as far as I know it hasn't been worked out very explicitly.)

Answer 2

This is a kind of question we asked ourselves about 10 years ago :)

Our answer - $P^1$ with nodes and cusps (and more general singularities) are very good examples for doing this. The answer is actually motivated by Serre's "Algebraic groups and algebraic class fields ..." where he works with generalized Jacobions and abelian Langlands (i.e. class field theory).

We concerned the part of the Langlands which treats the Hitchin's D-modules (these NOT all Hecke-eigensheves).

In papers with Dmitry Talalaev we described classical Hitchin system on such curves. http://arxiv.org/abs/hep-th/0303069 Hitchin system on singular curves I

http://arxiv.org/abs/hep-th/0309059 - here are more general singularities

The second step was to quantize Hitchin's hamiltonians. Actually it is the same as to quantize Gaudin's hamiltonian's. Naive recipe - works only for sl(2), sl(3) - http://arxiv.org/abs/hep-th/0404106

The breakthrough that Dmitry Talalaev found http://arxiv.org/abs/hep-th/0404153 Quantization of the Gaudin System

how to do TWO things simultaneously, he proposed a beautiful formula for:

1) All quantum Hitchin (Gaudin) Hamiltonionas (later generalized to whole center of universal enveloping for loop algebra)

2) At the same time it gives the GL-oper explicitly (moreover it gives "universal" GL-oper meaning that its coefficents are quantum Hitchin (Gaudin) hamiltonians, but not complex numbers). Fixing values of Hitchin's hamiltonians we get complex-valued GL-oper, which corresponds by Langlands to these Hitchin's hamiltonians. So the Langlands correspondence: Hitchin D-module -> GL-oper is made very explicit.

3) His formula makes explicit the idea that "GL-oper is quantization of the spectral curve"

To some extent this solves the questions about the Laglands for GL-Hitchin's system. We have not write down the proof of "Hecke-eigenvaluedness" of Hitchin's D-modules. But it seems that is rather clear(may be not the ritht word), if you take appropriate point of view on Hecke's transformations - as in the paper by A. Braverman, R. Bezrukavnikov http://arxiv.org/abs/math/0602255 Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case

One of key ideas - that you can do everything in "classical limit" and than quantize. They worked for finite fields - so they can use some trick to go from classical to quantum, over complex numbers we have explicit formulas by Talalaev so they should do the same.

Let me also mention that Hecke transformations are also known as Backlund transformations in integrability and relevant papers are:

http://arxiv.org/abs/nlin/0004003 Backlund transformations for finite-dimensional integrable systems: a geometric approach V. Kuznetsov, P. Vanhaecke

http://arxiv.org/abs/nlin/0110045 Hitchin Systems - Symplectic Hecke Correspondence and Two-dimensional Version A.M. Levin, M.A. Olshanetsky, A. Zotov

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It would be very nice project to consider from this point of view $P^1$ with cusp, the cotangent to moduli space of vector bundles is $[X,Y]=0/GL(n)$, the same thing which is considered in Etingof's Ginzburg's paper

http://arxiv.org/abs/math/0011114 Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphis

It would be very nice (and should be simple) to explicitly describe the Hecke-Bacclund transformations and their action on CalogeroMoser system and so on...

Answer 3

Mosuli space of SL_2 bundles on any genus 2 curve - is P^3 by Narashimhan and Ramanan. So working with P^3 should be accessible, but may be not so easy as might seems...

Classical Hitchin system has been described by van Geemen, Previato, Gawedzki, et. al.

http://arxiv.org/abs/alg-geom/9410015

http://arxiv.org/abs/solv-int/9710025

Quantum hamiltonians should be contained in Geemen de Jong paper: http://arxiv.org/abs/alg-geom/9701007

So one needs to 1) Quantize (?may be done) it 2) Make Hecke transform and see that result is as predicted by Beilison and Drinfeld - product of initial Hitchin D-module on D-module give by SL_2-oper.

If I would think on this I would try to do the following:

1) find Lax matrix L(z) - it might be done by Gawedzki, et.al.

2) try to guess what is the "qauntum spectral curve" "det"(d/dz - L(z)) - it gives both quantum Hitchin hamiltonians and SL_2-oper (this done by Talalaev for Gaudin-Hitchin for P^1 http://arxiv.org/abs/hep-th/0404153 )

3) try to prove that Hecke transformation agrees with "qauntum spectal curve"

As in previous post I would strongly suggest to use down-to-earth point of view on Hecke transformation.

So in such a way one may prove that the Hitchin's D-module is Hecke eigen-sheave with "eigenvalue" given by the SL_2-oper="quantum spectral curve" (D-module on basic curve).

Considerations of other Hecke eigensheaves (not Hitchin's) is another story...

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In our paper http://arxiv.org/abs/hep-th/0303069 we described classical Hitchin system for the degenerate genus 2 curve y^2 = (x-a)^3 (x-b)^3. However we did not check that our "analogs" of Narasimhan-Ramanan coordinates are indeed limits of true Narashimhan-Ramanan coordinates, so it might not be very helpful. Also Talalaev's formula cannot be applied directly for the Lax matrix in these coordinates. It is coordinate dependent, it is solvable problem but requires some work.