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

@article{Dumitrescu2017InterplayBO, title={Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles}, author={Olivia Dumitrescu and Motohico Mulase}, journal={arXiv: Algebraic Geometry}, year={2017} }

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 generalization of…

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## References

SHOWING 1-10 OF 84 REFERENCES

### Quantization of spectral curves for meromorphic Higgs bundles through topological recursion

- MathematicsProceedings of Symposia in Pure Mathematics
- 2018

A geometric quantization using the topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin…

### Quantum Curves for Hitchin Fibrations and the Eynard–Orantin Theory

- Mathematics
- 2014

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…

### A journey from the Hitchin section to the oper moduli

- MathematicsProceedings of Symposia in Pure Mathematics
- 2018

This paper provides an introduction to the mathematical notion of \emph{quantum curves}. We start with a concrete example arising from a graph enumeration problem. We then develop a theory of quantum…

### Topological Strings from Quantum Mechanics

- Mathematics
- 2014

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant…

### Hitchin integrable systems, deformations of spectral curves, and KP-type equations

- Mathematics
- 2007

An effective family of spectral curves appearing in Hitchin fibrations is determined. Using this family the moduli spaces of stable Higgs bundles on an algebraic curve are embedded into the Sato…

### Spectral theory and mirror symmetry

- MathematicsProceedings of Symposia in Pure Mathematics
- 2018

Recent developments in string theory have revealed a surprising connection between spectral theory and local mirror symmetry: it has been found that the quantization of mirror curves to toric…

### Invariants of spectral curves and intersection theory of moduli spaces of complex curves

- Mathematics
- 2011

To any spectral curve S, we associate a topological class {\Lambda}(S) in a moduli space M^b_{g,n} of "b-colored" stable Riemann surfaces of given topology (genus g, n boundaries), whose integral…

### Quantum Curves and D-Modules

- Mathematics, Physics
- 2008

In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an I-brane configuration, which consists of D4 and D6-branes intersecting…

### Intersection numbers of spectral curves

- Mathematics
- 2011

We compute the symplectic invariants of an arbitrary spectral curve with only 1 branchpoint in terms of integrals of characteristic classes in the moduli space of curves. Our formula associates to…