AI 中文总结
该研究探究$(r,s)$-Airy结构的q-形变，构造q-WKB解，分类容许$(r,s,q)$对及q-卡西米尔构型，建立量子谱曲线与可积系统的关联，为镜像曲线q-量子化提供新见解。
AI 中文摘要
本研究探讨了$(r,s)$-Airy结构的q-形变及其通过q-差分算子的实现，建立了量子谱曲线与可积系统之间的桥梁。我们为与q-量子化曲线$E_q(x,y)=0$相关的矩阵系统构造了全阶q-WKB解，证明由此得到的非微扰连通q-振幅满足一组移位q-环方程，该方程可解释为q-形变$\boldsymbol{\textrm{W}}(\boldsymbol{\frak{gl}}_r)$代数的Ward恒等式。主要结果是对满足q-拓扑型性质的容许$(r,s,q)$对及q-卡西米尔构型进行严格分类，确保半经典展开由q-拓扑递归唯一支配，为镜像曲线的q-量子化及其底层代数结构提供了新见解。
英文摘要
This work investigates the $q$-deformation of $(r,s)$-Airy structures and their realization via $q$-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order $q$-WKB solution for the matrix systems associated with the $q$-quantized curve $E_q(x,y)=0$. We demonstrate that the resulting non-perturbative connected $q$-amplitudes satisfy a set of shifted $q$-loop equations, which can be interpreted as the Ward identities of a $q$-deformed $\mathcal{W}(\mathfrak{gl}_r)$ algebra. Our main result provides a rigorous classification of admissible $(r,s,q)$ pairs and $q$-Casimir configurations that satisfy the $q$-topological type property. This ensures that the semi-classical expansion is uniquely governed by the $q$-topological recursion, offering new insights into the $q$-quantization of mirror curves and their underlying algebraic structures.
