Electric-Magnetic Duality And The Geometric Langlands Program
The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.
A.K. would like to thank D. Arinkin, R. Bezrukavnikov, and D. Orlov for useful conversations. In particular, Orlov's explanations in 2002 about the abelian case of the geometric Langlands program partially motivated the paper , which will enter our story in section 11. E.W. would like to thank the many mathematicians who over the years have explained matters relevant to the Langlands program, including A. Beilinson, P. Deligne, V. Drinfeld, and K. Vilonen, and especially M. F. Atiyah, D. Ben-Zvi, R. Donagi, E. Frenkel, and D. Kazhdan, and most recently M. Goresky and R. MacPherson. In addition, among others, T. Hausel, N. Hitchin, M. Hopkins, P. Kronheimer, L. Jeffrey, J. Morgan, G. Moore, D. Morrison, N. Nekrasov, M. Thaddeus, C. Vafa, and E. J. Weinberg clarified some points relevant to the present paper, and many of the physicists at the IAS, including S. Hellerman, K. Intriligator, J. Maldacena, N. Seiberg, and J. Walcher, made helpful comments.
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