Quantum Hitchin Systems via beta-deformed Matrix Models

TL;DR

This paper connects beta-deformed matrix models to the quantization of Hitchin systems, revealing how their loop equations reproduce quantum Hamiltonians and eigenvalues relate to deformed gauge theory observables.

Contribution

It demonstrates that beta-deformed matrix models encode the quantum Hitchin systems and provides explicit wave-functions via conformal blocks with degenerate fields.

Findings

Loop equations match quantum Hitchin Hamiltonians.

Eigenvalues correspond to epsilon1-deformed gauge theory observables.

Exact wave-functions derived from matrix model conformal blocks.

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.