We investigate the exact WKB method for the quantum Seiberg-Witten curve of 4d $N=2$ pure $SU(3)$ Yang-Mills, in the language of abelianization. The relevant differential equation is a third-order equation on $\mathbb{CP}^1$ with two irregular singularities. We employ the exact WKB method to study solutions to such a third-order equation and the associated Stokes phenomena. We also investigate the exact quantization condition for a certain spectral problem. Moreover, exact WKB analysis leads us to consider new Darboux coordinates on a moduli space of flat SL(3,$\mathbb{C}$)-connections. In particular, in the weak coupling region we encounter coordinates of higher length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux coordinates are conjectured to admit asymptotic expansions given by the formal quantum periods series; we perform numerical analysis supporting this conjecture.
TBA equations for $SU(r+1)$ quantum Seiberg-Witten curve: higher-order Mathieu equation
Institute for Astrophysics, School of Physics, Zhengzhou University（郑州大学物理学院天体物理研究所）；Beijing Institute of Mathematical Sciences and Applications（北京数学与应用研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Analytic approaches to perturbations of strongly coupled Yang-Mills plasma
School Mathematical Sciences and STAG Research Centre, University of Southampton（南安普顿大学数学科学学院和STAG研究中心）；National Centre for Nuclear Research（国家核研究中心）；Physics Department, University of Białystok（比亚韦斯托克大学物理系）
