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Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence. I. (English) Zbl 1442.81059

Summary: We discuss the relation between Liouville theory and the Hitchin integrable system, which can be seen in two ways as a two step process involving quantization and hyper-Kähler rotation. The modular duality of Liouville theory and the relation between Liouville theory and the \(\mathrm{SL}(2)\)-WZNW-model give a new perspective on the geometric Langlands correspondence and on its relation to conformal field theory.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14D24 Geometric Langlands program (algebro-geometric aspects)
32G81 Applications of deformations of analytic structures to the sciences
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53D50 Geometric quantization
81T60 Supersymmetric field theories in quantum mechanics
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