[Paper Review] Quantization of spectral curves for meromorphic Higgs bundles through topological recursion

Olivia Dumitrescu, Motohico Mulase|arXiv (Cornell University)|Nov 4, 2014

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes a geometric quantization of meromorphic Higgs bundles on a smooth projective curve via topological recursion, constructing a quantum curve as a Rees D-module whose semi-classical limit recovers the Hitchin spectral curve. The method uses topological recursion on the normalized spectral curve to generate an asymptotic WKB expansion of solutions, uniquely determining the quantum curve through canonical normal ordering and resolving singularities via blow-ups.

ABSTRACT

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 spectral curve of a rank $2$ meromorphic Higgs bundle on the base curve corresponds to a quantum curve, which is a Rees $D$-module on the base. The topological recursion then gives an all-order asymptotic expansion of its solution, thus determining a state vector corresponding to the spectral curve as a meromorphic Lagrangian. We establish a generalization of the topological recursion for a singular spectral curve. We show that the partial differential equation version of the topological recursion automatically selects the normal ordering of the canonical coordinates, and determines the unique quantization of the spectral curve. The quantum curve thus constructed has the semi-classical limit that agrees with the original spectral curve. Typical examples of our construction includes classical differential equations, such as Airy, Hermite, and Gauß hypergeometric equations. The topological recursion gives an asymptotic expansion of solutions to these equations at their singular points, relating Higgs bundles and various quantum invariants.

Motivation & Objective

• To establish a geometric quantization of the compactified cotangent bundle of a smooth projective curve using topological recursion.
• To show that the topological recursion on the normalized spectral curve of a rank-2 meromorphic Higgs bundle yields a unique quantum curve as a Rees D-module.
• To resolve singularities in the spectral curve via minimal resolution through successive blow-ups of the compactified cotangent bundle.
• To demonstrate that the resulting quantum curve has a semi-classical limit matching the original Hitchin spectral curve.
• To connect the topological recursion to classical differential equations such as Airy, Hermite, and Gauss hypergeometric equations via asymptotic expansions of their solutions.

Proposed method

• Extends the integral topological recursion to singular spectral curves by constructing the minimal resolution of the divisor Σ ∪ C∞ via blow-ups of the compactified cotangent bundle T*C.
• Applies the topological recursion to the normalized spectral curve ˜Σ, using the Riemann prime form and canonical 1-form W0,1 = ydx to generate correlation forms Wg,n.
• Implements the WKB method to derive an asymptotic expansion of the wave function Ψ(x,ħ) = exp(1/ħ S0(x) + S1(x) + ħ S2(x) + ...), with coefficients Sg(x) determined recursively.
• Uses the consistency condition (6.15) and higher-order recursion (6.16) to uniquely fix the normal ordering of canonical coordinates and ensure quantum curve integrability.
• Applies the construction to classical differential equations by identifying their spectral curves and showing that the topological recursion reproduces the standard hypergeometric and Airy function solutions.
• Verifies that the semi-classical limit of the quantum curve matches the original spectral curve, confirming the geometric quantization correspondence.

Experimental results

Research questions

• RQ1 Can topological recursion be generalized to singular spectral curves arising from meromorphic Higgs bundles on curves of arbitrary genus?
• RQ2 How does the topological recursion uniquely determine the quantization of a spectral curve, and what role does normal ordering play in this selection?
• RQ3 To what extent does the topological recursion on the normalized spectral curve reproduce known solutions of classical differential equations such as the Gauss hypergeometric equation?
• RQ4 What is the precise geometric and algebraic mechanism by which the topological recursion constructs a Rees D-module that serves as a quantum curve?
• RQ5 How do the asymptotic expansions generated by topological recursion relate to the WKB approximation and the semi-classical limit of the quantum curve?

Key findings

• The topological recursion applied to the normalized spectral curve ˜Σ of a rank-2 meromorphic Higgs bundle on a curve of arbitrary genus produces a unique quantum curve as a Rees D-module with the original spectral curve as its semi-classical limit.
• The construction resolves singularities via a minimal resolution of the divisor Σ - 2C∞ through successive blow-ups of the compactified cotangent bundle, enabling the application of topological recursion.
• The wave function Ψ(x,ħ) = exp(1/ħ S0(x) + S1(x) + ħ S2(x) + ...) is generated by the topological recursion, with S0(x) = ∫ y(x) dx and higher Sg(x) determined recursively via (6.15) and (6.16).
• For the Gauss hypergeometric equation, the topological recursion reproduces the standard hypergeometric series up to order ħ³, with coefficients matching the asymptotic expansion of 2F1(a,b;c;x) under appropriate parameter choices.
• The method uniquely selects the normal ordering of canonical coordinates through the PDE version of the topological recursion, ensuring consistency and unambiguity in the quantum curve construction.
• The quantum curve construction is verified to be consistent with known examples: the Airy and Hermite differential equations emerge as special cases of the general framework.