The Moduli Space and Monodromies of N=2 Supersymmetric \(SO(2r+1) \) Yang-Mills Theory

TL;DR

This paper analyzes the moduli space and monodromies of N=2 supersymmetric SO(2r+1) Yang-Mills theory, providing explicit descriptions of weak-coupling monodromies, their relation to braid groups, and the associated complex curve.

Contribution

It introduces a detailed description of weak-coupling monodromies for arbitrary gauge groups and constructs the complex curve for SO(2r+1), verifying its monodromies.

Findings

Weak-coupling monodromies are represented by Sp(2r, Z) matrices.

A one-to-one correspondence between Weyl orbits and generalized braid group elements is established.

The complex curve for SO(2r+1) correctly reproduces the expected monodromies.

Abstract

We write down the weak-coupling limit of N=2 supersymmetric Yang-Mills theory with arbitrary gauge group \( G \). We find the weak-coupling monodromies represented in terms of \( Sp(2r,\bzeta ) \) matrices depending on paths closed up to Weyl transformations in the Cartan space of complex dimension r, the rank of the group. There is a one to one relation between Weyl orbits of these paths and elements of a generalized braid group defined from \( G \). We check that these weak-coupling monodromies behave correctly in limits of the moduli space corresponding to restrictions to subgroups. In the case of $SO(2r+1)$ we write down the complex curve representing the solution of the theory. We show that the curve has the correct monodromies.