The Moduli Space and Monodromies of the $N=2$ Supersymmetric Yang-Mills Theory with any Lie Gauge Groups

TL;DR

This paper develops a unified method to determine the hyperelliptic curves, monodromies, and dyon spectra for $N=2$ supersymmetric Yang-Mills theories with any Lie gauge group, extending known results to exceptional groups.

Contribution

It introduces a general scheme for deriving hyperelliptic curves for all Lie gauge groups, including exceptional ones, and verifies their consistency and monodromies.

Findings

Derived hyperelliptic curves for $F_4, E_{6,7,8}$ groups.

Determined exact monodromies and dyon spectra for these theories.

Showed that monodromies depend solely on the Cartan matrix.

Abstract

We propose a unified scheme for finding the hyperelliptic curve of $N=2$ SUSY YM theory with any Lie gauge groups. Our general scheme gives the well known results for classical gauge groups and exceptional $G_2$ group. In particular, we present the curve for the exceptional gauge groups $F_4, E_{6,7,8}$ and check consistency condition for them. The exact monodromies and the dyon spectrum of these theories are determined. We note that for any Lie gauge groups, the exact monodromies could be obtained only from the Cartan matrix.