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Conference Paper

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

##### External Resource

https://doi.org/10.1090/pspum/100/01780

(Publisher version)

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##### Fulltext (public)

1411.1023.pdf

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##### Citation

Dumitrescu, O., & Mulase, M. (2018). Quantization of spectral curves for meromorphic
Higgs bundles through topological recursion. In *Topological recursion and its influence in analysis,
geometry and topology* (pp. 179-229). Providence, RI: American Mathematical Society.

Cite as: https://hdl.handle.net/21.11116/0000-0003-A581-0

##### 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 Gau\ss\

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.

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 Gau\ss\

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.