Quantization of spectral curves and DQ-modules

Abstract

Given a holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, we prove the existence of a holonomic DQ-module supported by the spectral curve associated to this bundle. Then, we relate quantum curves arising in various situations (quantization of spectral curves of Higgs bundles, quantization of the $A$-polynomial…) to DQ-modules and show that a quantum curve and the DQ-module canonically associated to it have isomorphic sheaves of solutions.