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

J. Teschner

TL;DR

This paper develops a unifying framework linking the Hitchin moduli space, Liouville conformal field theory, and the geometric Langlands correspondence. It posits a two-step mechanism—hyperkähler rotation and quantization—relating the classical Hitchin system to Liouville theory, with isomonodromic deformations acting as a bridge. A central theme is the use of opers, conformal blocks, and the separation of variables to map local system data to D-modules on Bun_G, and to reinterpret Langlands duality through Liouville/WZNW dualities and modular structures. The work also proposes a quantum geometric Langlands program, where degenerate fields, Hecke functors, and KZ/KZB equations generate a rich algebraic structure connecting global local systems with conformal blocks. Overall, the paper offers a comprehensive, symmetry-rich landscape suggesting deep ties between gauge theory, integrable systems, and conformal field theory with potential for broad generalizations to higher rank theories.

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.

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

TL;DR

This paper develops a unifying framework linking the Hitchin moduli space, Liouville conformal field theory, and the geometric Langlands correspondence. It posits a two-step mechanism—hyperkähler rotation and quantization—relating the classical Hitchin system to Liouville theory, with isomonodromic deformations acting as a bridge. A central theme is the use of opers, conformal blocks, and the separation of variables to map local system data to D-modules on Bun_G, and to reinterpret Langlands duality through Liouville/WZNW dualities and modular structures. The work also proposes a quantum geometric Langlands program, where degenerate fields, Hecke functors, and KZ/KZB equations generate a rich algebraic structure connecting global local systems with conformal blocks. Overall, the paper offers a comprehensive, symmetry-rich landscape suggesting deep ties between gauge theory, integrable systems, and conformal field theory with potential for broad generalizations to higher rank theories.

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.

Paper Structure

This paper contains 116 sections, 251 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. From the Hitchin integrable system to Liouville theory
  4. Separation of Variables
  5. Geometric Langlands correspondence
  6. Modular duality vs. Langlands duality
  7. The classical Hitchin system
  8. Self-duality equations vs. Higgs pairs
  9. The Hitchin integrable system
  10. Separation of variables
  11. Special geometry of the base of the Hitchin fibration
  12. Isomonodromic deformations as a deformation of the Hitchin system
  13. Hitchin moduli space as space flat connections
  14. Flat connection vs. local systems
  15. Hyperkähler structure
  16. ...and 101 more sections

Theorems & Definitions (2)

  • Remark 1
  • Remark 2