Quantum curves for Hitchin fibrations and the Eynard-Orantin theory

Abstract

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 $T^{\ast }C$ 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 $\hslash$-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 $SL(2,\mathbb{C})$-character variety of the fundamental group ${\pi }_1(C)$. We show that the semi-classical limit through the WKB approximation of these $\hslash$-deformed $D$-modules recovers the initial family of Hitchin spectral curves.

34 pages

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• 2-parameter $\tau$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-
• Quantum Curve and the First Painlevé Equation
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Quantum curves and conformal field theory
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• Quantum curves
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• Deformation and quantisation condition of the $\mathcal{Q}$-top recursion
• Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies