Quantum Hitchin Systems via beta-deformed Matrix Models

TL;DR

This work demonstrates that β-deformed matrix models capture the quantization of Hitchin systems in the Nekrasov-Shatashvili limit, reproducing quantum Hamiltonians on the sphere and torus and yielding exact wave-functions via degenerate Liouville block insertions. By deriving loop equations at genus zero and one and translating them into Schrödinger-type equations with potentials built from Penner-type (sphere) and elliptic (torus) data, the authors connect matrix-model resolvents to the quantum Seiberg-Witten differential and Gaudin-type Hamiltonians. The approach unifies conformal blocks, matrix models, and Hitchin integrable systems within the AGT framework, providing a concrete pathway to compute ε1-deformed chiral observables and prepotentials. The results point to a broader quantization program for integrable systems using β-deformed matrix models, with potential extensions to higher genus, higher rank, and q-deformations. The work has implications for Langlands duality and the geometric understanding of 4d N=2 gauge theories via integrable systems.

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.