We study a class of quantum integrable systems derived from dimer graphs and also described by local toric Calabi-Yau geometries with higher genus mirror curves, generalizing some previous works on genus one mirror curves. We compute the spectra of the quantum systems both by standard perturbation method and by Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way, we obtain some exact analytic results for the classical and quantum periods of the Calabi-Yau geometries. We also determine the differential operators of the quantum periods and compute the topological string free energy in Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other methods such as the topological vertex.

中文翻译：

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二聚体模型和 Calabi-Yau 几何中的量子周期和光谱

我们研究了一类源自二聚体图的量子可积系统，并且还通过具有更高属镜像曲线的局部复曲面 Calabi-Yau 几何描述，概括了以前关于单属镜像曲线的一些工作。我们通过标准微扰方法和 Bohr-Sommerfeld 方法以量子周期作为相体积来计算量子系统的光谱。通过这种方式，我们获得了卡拉比-丘几何的经典和量子周期的一些精确解析结果。我们还确定了量子周期的微分算子并计算了 Nekrasov-Shatashvili (NS) 极限中的拓扑弦自由能。结果与其他方法（如拓扑顶点）的计算结果一致。