Higgs bundles, Mirror symmetry, and Geometric Langlands
Summary:
This talk gives a summary of the Kapustin-Witten appraoch to the geometric langlands correspondence. This realizes geometric langlands as a duality between A-branes (langrangian submanifolds) on the moduli space of Higgs bundles with gauge group G, and B-branes (coherent sheaves) on the moduli space of Higgs bundles with the langlands dual gauge group G^L. This flavor of mirror symmetry arises from S-duality of an N=4 supersymmetric 4D gauge theory. On the Higgs bundles side, it comes from T-duality of the Hitchin integrable system.
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