Quantum Curves for Hitchin Fibrations and the Eynard-Orantin Theory

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 -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 -character variety of the fundamental group π₁( C). We show that the semi-classical limit through the WKB approximation of these -deformed D modules recovers the initial family of Hitchin spectral curves.