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Mirror symmetry, Hitchin’s equations, and Langlands duality. (English) Zbl 1298.81151
García-Prada, Oscar (ed.) et al., The many facets of geometry. A tribute to Nigel Hitchin. Oxford: Oxford University Press (ISBN 978-0-19-953492-0/hbk). 113-128 (2010).
The author and A. Kapustin [Commun. Number Theory Phys. 1, No. 1, 1–236 (2007; Zbl 1128.22013)] have established a grand description of the geometric Langlands program for complex Riemann surfaces, as is usual with Witten, within the framework of quantum field theory. The crucial step was based upon the reduction of Wilson and t’Hooft line operators to topological operators acting naturally on the branes of the two-dimensional sigma-model. The idea was generalized [in: Current developments in mathematics, 2006. Jerison, David (ed.) et al., Somerville, MA: International Press. 35–180 (2008; Zbl 1237.14024)] by the author and S. Gukov to tame ramification, while its generalization to wild ramification was discussed in [Anal. Appl., Singap. 6, No. 4, 429–501 (2008; Zbl 1177.81101)] by the author. This paper is a good introduction to these developments.
For the entire collection see [Zbl 1192.00076].

MSC:
81T10 Model quantum field theories
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
20F36 Braid groups; Artin groups
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