Communications in Number Theory and Physics

Volume 1 (2007)

Number 1

Electric-magnetic duality and the geometric Langlands program

Pages: 1 – 236



Anton Kapustin (Department of Physics, California Institute of Technology, Pasadena, Calif.)

Edward Witten (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey)


The geometric Langlands program can be described in a natural way bycompactifying on a Riemann surface $C$ a twisted version of${\EUN}=4$ super Yang-Mills theory in four dimensions. The keyingredients are electric-magnetic duality of gauge theory, mirrorsymmetry of sigma-models, branes, Wilson and ’t Hooft operators,and topological field theory. Seemingly esoteric notions of thegeometric Langlands program, such as Hecke eigensheaves and ${\calD}$-modules, arise naturally from the physics.

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