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

References in corpus (8)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• The quantum dilogarithm and representations quantum cluster varieties
• Darboux coordinates, Yang-Yang functional, and gauge theory
• On AGT conjecture
• A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections

Cited by in corpus (15)

• The Omega Deformation, Branes, Integrability, and Liouville Theory
• Darboux coordinates, Yang-Yang functional, and gauge theory
• Wild Quiver Gauge Theories
• Vertices, Vortices & Interacting Surface Operators
• The Liouville side of the Vortex
• Quantum Hitchin Systems via beta-deformed Matrix Models
• 1+1 Gaudin Model
• Dimer Models, Integrable Systems and Quantum Teichmuller Space
• Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function
• Baxter's T-Q equation, SU(N)/SU(2)^{N-3} correspondence and Ω-deformed Seiberg-Witten prepotential
• On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories
• Symmetries of quantum Lax equations for the Painlevé equations
• SQCD, Superconducting Gaps and Cyclic RG Flows
• Angular Momentum and Gravimagnetization of the $\mathcal{N}=2$ SYM vacuum
• Les Houches lectures on non-perturbative Seiberg-Witten geometry