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Gauge Theory, Ramification, And The Geometric Langlands Program

Gukov, Sergei and Witten, Edward (2008) Gauge Theory, Ramification, And The Geometric Langlands Program. Current Developments in Mathematics, 2006 . pp. 35-180. ISSN 1089-6384. doi:10.4310/CDM.2006.v2006.n1.a2.

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In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves. We describe the relevant surface operators in N=4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters. Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification. The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Gukov, Sergei0000-0002-9486-1762
Witten, Edward0000-0002-7752-6073
Additional Information:© 2008 International Press. We thank A. Braverman, D. Gaitsgory, E. Frenkel, and D. Kazhdan for patient and extremely helpful explanations. We also thank J. Andersen, P. Aspinwall, M. F. Atiyah, D. Ben-Zvi, R. Bezrukavnikov, R. Bielawski, R. Dijkgraaf, R. Donagi, N. Hitchin, L. Jeffrey, A. Kapustin, P. Kronheimer, Y. Laszlo, H. Nakajima, C. Sorger, and M. Thaddeus, among others, for a wide variety of helpful comments and advice. Research of SG was partly supported by DOE grant DE-FG03-92-ER40701. Research of EW was partly supported by NSF Grant PHY-0503584.
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG03-92-ER40701
Record Number:CaltechAUTHORS:20160511-095533476
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Official Citation:Gukov, Sergei; Witten, Edward. Gauge Theory, Ramification, And The Geometric Langlands Program. Current Developments in Mathematics 2006 (2008), 35--180.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66982
Deposited By: Tony Diaz
Deposited On:11 May 2016 17:06
Last Modified:11 Nov 2021 00:25

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