Abstract We derive the Thermodynamic Bethe Ansatz (TBA) equations for the exact WKB periods of the Mathieu equation in the weak coupling region. We will use the TBA equations to calculate the quantum corrections to the WKB periods, which are regarded as the quantum periods of N = 2 S U ( 2 ) super Yang-Mills theory at strong coupling. We calculate the effective central charge of the TBA equations, which is found to be proportional to the coefficient of the one-loop beta function of the 4d theory. We also study the spectral problem for the Mathieu equation based on the TBA equations numerically.

中文翻译：

---

Mathieu 方程的精确 WKB 分析和 TBA 方程

摘要 我们推导出了弱耦合区域中 Mathieu 方程的精确 WKB 周期的热力学 Bethe Ansatz (TBA) 方程。我们将使用 TBA 方程来计算 WKB 周期的量子修正，这些周期被视为 N = 2 SU ( 2 ) 强耦合下超杨米尔斯理论的量子周期。我们计算了 TBA 方程的有效中心电荷，发现它与 4d 理论的单环 beta 函数的系数成正比。我们还数值研究了基于 TBA 方程的 Mathieu 方程的谱问题。