We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations of the spectral curve that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to order hbar^5, that the quantum curve satisfies the properties expected from matrix models. As a side result, we obtain an infinite sequence of identities relating A-cycle integrals of elliptic functions and quasi-modular forms.

中文翻译：

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量化魏尔斯特拉斯

我们研究了 Eynard-Orantin 拓扑递归与由 Weierstrass 方程给出的属 1 光谱曲线家族的量子曲线之间的联系。我们构建消除微扰和非微扰波函数的光谱曲线的量化。特别是，对于非微扰波函数，我们证明，直到 hbar^5 阶，量子曲线满足矩阵模型预期的特性。作为附带的结果，我们获得了与椭圆函数的 A 循环积分和拟模形式相关的无穷恒等式。