Geometric Langlands correspondence

The geometric Langlands correspondence (or geometric Langlands program) in mathematics is a generalization of the Langlands program (of conjectures and theorems). The Langlands program relates number theory and geometry, while the geometric Langlands program relates algebraic geometry and representation theory. In 2007 Kapustin and Witten related the geometric Langlands correspondence to supersymmetric Yang-Mills theory.

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  • Gaitsgory gives an informal introduction to the geometric Langlands program. This is a new and very active area of research which grew out of the theory of automorphic forms and is closely related to it. Roughly speaking, in this theory we everywhere replace functions—like automorphic forms—by sheaves on algebraic varieties; this allows us to use powerful methods of algebraic geometry in order to construcut "automorphic sheaves."
  • Langlands outlined the first version of the programme in 1967, when he was a young mathematician visiting the IAS. His starting point was the theory of algebraic equations (such as the quadratic, or second-degree, equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially. ...
    Inspired by subsequent developments in Galois’s theory, Langlands’ approach allowed researchers to translate algebra problems into the ‘language’ of harmonic analysis, the branch of mathematics that breaks complex waveforms down into simpler, sinusoidal building blocks.
    In the 1980s, Vladimir Drinfel’d, a Ukrainian-born mathematician now at the University of Chicago in Illinois, and others proposed a similar connection between geometry and harmonic analysis. Although this idea seemed to be only loosely inspired by the Langlands programme, mathematicians subsequently found stronger evidence that the two fields are connected. (Drinfel’d received a Fields Medal in 1990.)
  • Hecke transformations are one of the most important ingredients in geometric Langlands. What they mean in terms of physics had bothered me for a long time and eventually had been the last major stumbling block in interpreting geometric Langlands in terms of physics and gauge theory. Finally, while on an airplane flying home from Seattle, it struck me that a Hecke transformation in the context of geometric Langlands is simply an algebraic geometer’s way to describe the effects of a “’t Hooft operator” of quantum gauge theory. I had never worked with ’t Hooft operators, but they were familiar to me, as they had been introduced in the late 1970s as a tool in understanding quantum gauge theory. The basics of how to work with ’t Hooft operators and what happens to them under electric-magnetic duality were well known, so once I could reinterpret Hecke transformations in terms of ’t Hooft operators, many things were clearer to me.
    • Edward Witten, as quoted in Ooguri, Hirosi (May 2015). "Interview with Edward Witten". Notices of the American Mathematical Society 62 (5): 491–506. (quote from p. 505)

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