We generalize the topological recursion of Eynard-Orantin (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 ℏ-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 ℏ-deformed D-modules recovers the initial family
of Hitchin spectral curves.Comment: 34 page