Résume | For a complex curve C and reductive group G, the space of G-bundles on C has been of much interest to many mathematicians. For the purposes of the geometric Langlands correspondence, one wishes to construct certain `Hecke eigensheaves' over this space. It has long been expected (and in some cases known) that these should arise from quantization of fibers of Hitchin's integrable system, this being the map h: T*Bun(C, G) --> A which, for G = GL(n), records the spectral curve of a Higgs bundle. Historically this means that one tries to associate a D-module on Bun(C, G) to each fiber of h.
More recently, the fact that Langlands dual groups give rise to dual Hitchin fibrations has led to the expectation that geometric Langlands duality should be some sort of homological mirror symmetry. In this talk we will take a step towards making this precise: recent results on the localization of wrapped Fukaya categories allow us to use Floer theory to associate a constructible sheaf on Bun(C, G) to a fiber of the Hitchin fibration. (More precisely, we may do for smooth fibers, in components of Bun(C, G) where there are no strictly semistable Higgs bundles, and should assume G connected center). We don't yet know how to check that we have eigensheaves, but can check some expected properties: our sheaves have the expected endomorphisms, rank, microstalks on certain components, and sheaves from different fibers are orthogonal. |