TITLE: Sheaves on Bun_G
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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).