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

TL;DR

This paper applies the exact WKB method to analyze the quantum Seiberg-Witten curve of 4d N=2 pure SU(3) Yang-Mills theory, revealing new Darboux coordinates and their relation to quantum periods.

Contribution

It introduces a novel application of the exact WKB method to third-order differential equations in the context of SU(3) gauge theory, and proposes new Darboux coordinates related to quantum periods.

Findings

Derived exact quantization conditions for the spectral problem.

Identified higher length-twist Darboux coordinates generalizing Fenchel-Nielsen coordinates.

Numerical evidence supports the conjecture relating Darboux coordinates to quantum periods.

Abstract

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.