Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I

Abstract

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 hyperkaehler rotation. The modular duality of Liouville theory and the relation between Liouville theory and the SL(2)-WZNW-model give a new perspective on the geometric Langlands correspondence and on its relation to conformal field theory.

81 pages; V2: small corrections and improvements, references added; V3: Discussion of Yang's potential moved to 5.5 and extended, small additons and corrections otherwise; V4: a reference added, V5: final version to appear in ATMP