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

TL;DR

This work extends the Seiberg–Witten construction to $SO(2r+1)$ gauge groups by proposing a Weyl-symmetric complex curve that describes the N=2 moduli space and by providing a uniform treatment of weak-coupling monodromies via $Sp(2r,\mathbb{Z})$ actions tied to the Brieskorn braid group. The curve $y^2 = \left(\prod_{i=1}^r (x^2 - b_i^2)\right)^2 - \Lambda^{4r-2} x^2$ encodes moduli through Weyl-invariant polynomials and remains consistent under partial symmetry breaking, yielding factorized sub-curves corresponding to $SO(2r-2k+1)$ and $SU(k)$ factors. In the explicit $SO(5)$ case, the authors compute the Seiberg–Witten differential and monodromies from branch-point braidings, verifying agreement with semi-classical expectations and establishing a Coxeter-braid framework for generating monodromies. They also analyze strong-coupling monodromies using Picard–Lefschetz theory, identifying dyons and showing that the strong- and weak-coupling structures coherently describe the full moduli space. Overall, the paper provides a robust, uniform approach to monodromies for simple groups and sets the stage for extending Seiberg–Witten solutions beyond $SU(N)$ to arbitrary gauge groups.

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.

Figures (3)

• Figure 1: Cycles for the SO(5) curve.
• Figure 2: The action of the braidings on the branch-points for SO(5).
• Figure 3: Vanishing cycles for two strong coupling monodromies for SO(5).