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

TL;DR

This paper explores the deep connections between Liouville theory, Hitchin moduli spaces, and the geometric Langlands correspondence, emphasizing the roles of quantization and dualities in understanding these mathematical physics structures.

Contribution

It introduces a novel perspective linking Liouville theory and Hitchin systems through quantization and hyperkähler rotation, shedding light on the geometric Langlands program.

Findings

Establishes a relation between Liouville theory and Hitchin integrable systems.

Highlights the role of modular duality in Liouville theory.

Provides new insights into the geometric Langlands correspondence.

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.