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arXiv · 1310.6022

Quantum curves for Hitchin fibrations and the Eynard-Orantin theory

Olivia Dumitrescu·Motohico Mulase
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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 $T^*C$ 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 $\hbar$-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 $π_1(C)$. We show that the semi-classical limit through the WKB approximation of these $\hbar$-deformed $D$-modules recovers the initial family of Hitchin spectral curves.

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BibTeXRIS

Olivia Dumitrescu, Motohico Mulase. 2014-01-14. Quantum curves for Hitchin fibrations and the Eynard-Orantin theory. https://doi.org/10.1007/s11005-014-0679-0

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