TITLE: Langlands conjectures in higher dimensions
QUESTION [5 upvotes]: Geometric class field theory (curves over a finite field) has been generalized
to higher dimensional varieties over a finite field (and other arithmetical fields). Some of the key names here are Lang, Serre, Bloch, Kato-Saito...
Question: Does there exist a precise formulation of Langlands's conjectures which constitutes a non-abelian generalization of these results? 
Probably relevant: Kazhdan's talk here at the Panel on Open Questions held at Gross's 60th Birthday Conference.

REPLY [6 votes]: From what I remember, Kapranov's paper "Analogies between the Langlands correspondence and topological quantum field theory" has some speculations about this. A wise old friend once called it the most speculative paper he'd ever seen in print.