Exact WKB and the quantum Seiberg-Witten curve for 4d N=2 pure SU(3) Yang-Mills. 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 CP1 with two irregular singularities. We employ the exact WKB method to study the 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,C)-connections. In particular, in the weak coupling region we encounter coordinates of the higher length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux coordinates are conjectured to admit asymptotic expansions given by the formal quantum periods series and we perform numerical analysis supporting this conjecture.

Automated summary

In this study, 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. We explore the relevant differential equation, its solutions, and associated Stokes phenomena using the exact WKB method. Additionally, we investigate the exact quantization condition for a specific spectral problem. Our analysis also leads us to consider new Darboux coordinates on a moduli space of flat SL(3,C)-connections and numerical analysis supports the conjecture that these coordinates have asymptotic expansions given by the formal quantum periods series.