[Paper Review] Interplay between opers, quantum curves, WKB analysis, and Higgs bundles

Olivia Dumitrescu, Motohico Mulase|arXiv (Cornell University)|Feb 2, 2017

Algebraic Geometry and Number Theory45 references3 citations

TL;DR

This paper establishes a geometric quantization framework for Hitchin spectral curves via Rees $π$-modules, showing that two distinct quantization methods—construction of $η$-families of opers and a PDE version of topological recursion—yield equivalent quantum curves for holomorphic and meromorphic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles. The key result is the agreement of these quantization procedures, with the Airy differential equation serving as a canonical example linking quantum curves to Gromov–Witten invariants and mirror symmetry.

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 $SL(2,\mathbb{C})$-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 $SL(2,\mathbb{C})$-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

Motivation & Objective

• To develop a mathematical framework for quantum curves as Rees $\mathcal{D}$-modules whose semi-classical limit recovers Hitchin spectral curves of Higgs bundles.
• To construct a one-parameter family of opers as a geometric quantization of Hitchin spectral curves, using a fixed theta characteristic and projective structure.
• To show that the PDE version of topological recursion provides all-order WKB analysis for these Rees $\mathcal{D}$-modules.
• To prove the equivalence between the oper-based quantization and the topological recursion-based quantization for both holomorphic and meromorphic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles.
• To demonstrate that quantum curves encode quantum invariants such as Gromov–Witten invariants through canonical geometric and analytic procedures.

Proposed method

• Define a quantum curve as a Rees $\mathcal{D}$-module on a smooth projective curve $C$, with $\hbar \in H^1(C, K_C)$ as the deformation parameter.
• Construct $\hbar$-families of $\mathrm{SL}(r,\mathbb{C})$-opers on $C$ using a fixed theta characteristic and projective structure, yielding a canonical quantization of the Hitchin spectral curve.
• Apply a PDE version of topological recursion to compute correlation forms $W_{g,n}$, which are shown to match the WKB expansion of the Rees $\mathcal{D}$-module.
• Use the uniqueness of topological recursion solutions from initial data to prove equivalence between the two quantization methods.
• Verify agreement via explicit computation on the Airy example, where the quantum curve equation recovers free energies via Laplace transform and spectral curve reconstruction.

Experimental results

Research questions

• RQ1 How can one geometrically quantize Hitchin spectral curves using Rees $\mathcal{D}$-modules and what is the role of the Planck constant $\hbar$ in this process?
• RQ2 Can the PDE version of topological recursion provide a complete all-order WKB analysis for the Rees $\mathcal{D}$-modules arising from Higgs bundles?
• RQ3 Do the two distinct quantization procedures—via $\hbar$-families of opers and via topological recursion—yield equivalent quantum curves for $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles?
• RQ4 What is the precise relation between quantum curves and quantum invariants such as Gromov–Witten invariants in this geometric setting?
• RQ5 How does the quantum curve encode the entire free energy data through canonical geometric and analytic operations?

Key findings

• The Rees $\mathcal{D}$-module constructed via $\hbar$-families of opers recovers the original Hitchin spectral curve in the semi-classical limit, confirming the quantization procedure.
• The PDE version of topological recursion produces the same correlation forms $W_{g,n}$ as the WKB expansion of the Rees $\mathcal{D}$-module, establishing equivalence of the two quantization methods.
• For the Airy example, the quantum curve equation (7.9) generates all free energies $F_{g,n}^{\text{Airy}}$ via integration and specialization, with $W_{0,1} = \frac{16}{t^4}$ matching known results.
• The entire function (7.15) encodes rich geometric information through its asymptotic expansion, revealing connections to intersection numbers and quantum invariants.
• The equivalence of the two quantization methods is proven for both holomorphic and meromorphic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles, generalizing Gaiotto's conjecture in the Fuchsian uniformization case.
• The construction realizes mirror symmetry as a Laplace transform between $A$-model intersection numbers and $B$-model spectral data, with the quantum curve as the central object.