Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 036, 53 pages      arXiv:1702.00511      https://doi.org/10.3842/SIGMA.2021.036
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles

Olivia Dumitrescu ab and Motohico Mulase cd
a) Department of Mathematics, University of North Carolina at Chapel Hill, 340 Phillips Hall, CB 3250, Chapel Hill, NC 27599-3250 USA
b) Simion Stoilow Institute of Mathematics, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
c) Department of Mathematics, University of California, Davis, CA 95616-8633, USA
d) Kavli Institute for Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa, Japan

Received December 31, 2019, in final form March 12, 2021; Published online April 09, 2021

Abstract
Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic ${\rm SL}(2,\mathbb{C})$-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic ${\rm SL}(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants.

Key words: quantum curve; Hitchin spectral curve; Higgs field; Rees $\mathcal{D}$-module; opers; non-Abelian Hodge correspondence; mirror symmetry; Airy function; quantum invariants; WKB approximation; topological recursion.

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