  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.

High Energy Physics - Theory