Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization

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.