标题： 二阶$q$-差分方程的单值群从WKB近似 显示英文标题

标题： Monodromies of Second Order $q$-difference Equations from the WKB Approximation

摘要： 本文通过WKB分析的框架研究了二阶$q$-差分方程的单值数据空间。 我们计算了与WKB波函数的斯托克斯现象相关的连接矩阵，并建立了一个通用框架来参数化$q$-差分方程的单值性。 单值性的计算通过显式例子进行说明，包括一个$q$-马修方程及其退化形式。 在所有例子中，我们证明了$\mathbb{C}^*$原点周围的单值性可以展开为Voros符号或指数化量子周期的组合，其系数为整数。 从物理上讲，这些单值性对应于具有最小超对称性的五维量子场理论中威尔逊线算子的期望值。 在$q$-马修方程的情况下，我们证明了单值性的迹可以与相应$q$-皮卡德方程的哈密顿量相联系。 显示英文摘要

摘要： This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general framework to parameterize monodromies of $q$-difference equations. Computations of monodromies are illustrated with explicit examples, including a $q$-Mathieu equation and its degenerations. In all examples we show that the monodromy around the origin of $\mathbb{C}^*$ admits an expansion in terms of Voros symbols, or exponentiated quantum periods, with integer coefficients. Physically these monodromies correspond to expectation values of Wilson line operators in five dimensional quantum field theories with minimal supersymmetry. In the case of the $q$-Mathieu equation, we show that the trace of the monodromy can be identified with the Hamiltonian of a corresponding $q$-Painlev\'e equation.