We generalize the topological recursion of Eynard-Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347-452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T*C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal h{stroke}-deformation family of D modules over an arbitrary projective algebraic curve C of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the SL(2, ℂ)-character variety of the fundamental group π1(C). We show that the semi-classical limit through the WKB approximation of these h{stroke}-deformed D modules recovers the initial family of Hitchin spectral curves. © 2014 Springer Science+Business Media Dordrecht.

© 2018 World Scientific Publishing Co. Pte. Ltd. This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed.

We derive the spectral curves for q-part double Hurwitz numbers, r-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)-geometry. We quantize this family of spectral curves and obtain the Schrödinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.

We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in [Eynard-Orantin]. As an application, we prove the conjecture proposed by Bouchard and Marino, relating Hurwitz numbers to the spectral invariants of the Lambert curve exp(x)=y exp(-y).