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Quantum Curves for Hitchin Fibrations and the Eynard–Orantin Theory

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Abstract

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 \({\hbar}\)-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 π1(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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Authors and Affiliations

  1. Fakultät für Mathematik und Physik, Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Olivia Dumitrescu

  2. Department of Mathematics, University of California, Davis, CA, 95616–8633, USA

    Motohico Mulase

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  1. Olivia Dumitrescu
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  2. Motohico Mulase
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Correspondence to Motohico Mulase.

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O. Dumitrescu is a member of the Simion Stoilow Institute of Mathematics of the Romanian Academy.

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Dumitrescu, O., Mulase, M. Quantum Curves for Hitchin Fibrations and the Eynard–Orantin Theory. Lett Math Phys 104, 635–671 (2014). https://doi.org/10.1007/s11005-014-0679-0

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