Quantum Hitchin Systems via $${\beta}$$ β -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 $${\epsilon_1}$$ -deformation of the chiral observables of the corresponding $${{\mathcal{N}=2}}$$ 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.