# Sheaves on Bun_G

What's the background I need to know to understand the conjectural

 D (Bun_G)  =?=  O(LocSys)


from this question. I know the LHS is about the derived category of D-modules on the space (stack?) of some (stable?) bundles for some preselected group G. What's the RHS? I've seen the physics articles that say something about his topic, but how would you explain it?

I know this goes under the name of Geometric Langlands. Why?

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Don't need stable. It's the stack of all G bundles on a curve. The right hand side is the derived category of local systems, which are vector bundles with flat connection (G-bundles, for the Langlands dual of G in this equation). As for background, that depends on how deep an understanding you want.

For a lot of the positive characteristic stuff, you want to look at stuff by Frenkel and Gaitsgory, is my understanding.

In the complex case, there's a bit of other stuff. I've got to, of course, recommend the work of my advisor, as well as the work of Witten which is closely related. To really get this, you need to have a bit of an understanding of the Hitchin system on Higgs bundles (though not by that name, they're used extensively in this paper).

As for why it's called Geometric Langlands, that'd be because it's, essentially, a geometric formulation related to the classical Langlands conjecture in number theory, which I know fairly little about, except that it involves automorphic forms. I recommend asking your local number theorist for some help, though I do believe that it is explained in the book Introduction to the Langlands Program by quite a few authors, which includes Gaitsgory explaining roughly what Geometric Langlands is and how it all fits together.

Hope that helps.

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The right hand side of the equation describes the derived category of coherent sheaves on the stack of LG torsors with connection. In the traditional Langlands correspondence, one side is described by certain homomorphisms of a Galois group into the complex reductive group that is Langlands dual to G. In the function field setting, the (unramified) Galois group is the etale fundamental group of a proper algebraic curve, and conjugacy classes of homomorphisms to L G are in natural bijection with LG torsors with connection. These torsors are in turn in bijection with certain skyscraper sheaves on the corresponding moduli stack. Passing to coherent sheaves is a natural expansion of the category (but I imagine there are better motivations that escape me at the moment).

There is a more symmetric quantized version of this conjecture, due to Feigin, E. Frenkel, and Stoyanovsky. Coherent sheaves on the right hand side are replaced by twisted D-modules on the stack of LG bundles. There is a discussion of it in the introduction to Gatsgory's paper on the twisted Whittaker model (on the arXiv).

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