Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles
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DOI：
10.3842/sigma.2021.036
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发表时间：
2017-02
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影响因子：
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通讯作者：
Olivia Dumitrescu;M. Mulase
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文献类型：
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作者：
Olivia Dumitrescu;M. Mulase
量子曲线在物理学文献中被介绍过。我们开发了一个数学框架的情况下与希钦谱曲线。在这种情况下，量子曲线是光滑射影代数曲线上的Rees $\mathcal{D}$-模，其半经典极限产生希格斯束的希钦谱曲线。通过具体构造单参数形变算子族，给出了Hitchin谱曲线的量子化方法。我们提出了一个推广的拓扑递归的Eynard-Orantin和Mirzakhani的背景下奇异Hitchin谱曲线。我们发现了一个令人惊讶的结果，PDE版本的拓扑递归提供了所有阶WKB分析的里斯$\mathcal{D}$-模块，定义为量子化的希钦谱曲线与亚纯$SL（2，\mathbb {C}）$-希格斯丛。因此，拓扑递归被确定为希钦谱曲线的量化过程。我们证明了这两个量子化，一个通过构造opers族，另一个通过PDE拓扑递归，对于全纯和亚纯$SL（2，\mathbb {C}）$-Higgs丛是一致的.经典的微分方程，如艾里微分方程提供了一个典型的例子。通过这些经典的例子，我们看到量子曲线与希格斯束，opers，Gaiotto猜想和量子不变量，如Gromov-Witten不变量有关。
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 the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show a surprising result that a PDE version of the 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 is thus identified 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 topological recursion, 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.