Quantum Hitchin Systems via beta-deformed Matrix Models

Abstract

We study the quantization of Hitchin systems in terms of beta-deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov-Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the epsilon1-deformation of the chiral observables of the corresponding N=2 four dimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions.

29 pages; v2. refs. added and typos corrected

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• Proving AGT conjecture as HS duality: extension to five dimensions
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• Seiberg-Witten theory as a Fermi gas
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• The Liouville side of the Vortex
• The Stringy Instanton Partition Function
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• Quantum curves and conformal field theory
• Classical limit of irregular blocks and Mathieu functions
• Non-perturbative studies of N=2 conformal quiver gauge theories
• Irregular conformal block, spectral curve and flow equations
• Spectral determinants and quantum theta functions
• On a singular Fredholm-type integral equation arising in N=2 super Yang-Mills theories
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Notes on Mayer Expansions and Matrix Models
• N=2 supersymmetric QCD and elliptic potentials
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach