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Lectures on the Langlands program and conformal field theory. (English) Zbl 1196.11091

Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-30307-7/hbk). 389-533 (2007).
Summary: These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. It has long been suspected that the Langlands duality should somehow be related to various dualities observed in quantum field theory and string theory. Indeed, both the Langlands correspondence and the dualities in physics have emerged as some sort of non-abelian Fourier transforms. Moreover, the so-called Langlands dual group introduced by R. P. Langlands in [Lect. Mod. Anal. Appl. III, Lect. Notes Math. 170, 18–61 (1970; Zbl 0225.14022)] that is essential in the formulation of the Langlands correspondence also plays a prominent role in the study of S-dualities in physics and was in fact also introduced by the physicists P. Goddard, J. Nuyts and D. Olive in the framework of four-dimensional gauge theory [Gauge theories and magnetic change, Nuclear Phys., B 125, 1–28 (1977)].
For the entire collection see [Zbl 1104.11003].

MSC:

11G45 Geometric class field theory
19F05 Generalized class field theory (\(K\)-theoretic aspects)
11R39 Langlands-Weil conjectures, nonabelian class field theory
11S37 Langlands-Weil conjectures, nonabelian class field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
11Z05 Miscellaneous applications of number theory

Citations:

Zbl 0225.14022
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