Monopole Dynamics and BPS Dyons N=2 Super-Yang-Mills Theories

TL;DR

The paper develops a low-energy framework for monopole and dyon dynamics in pure $N=2$ Yang–Mills with non-aligned Higgs fields, formulating a supersymmetric quantum mechanics on monopole moduli spaces augmented by a potential from tri-holomorphic Killing vectors. BPS states arise as normalizable zero modes of a twisted Dirac operator on the moduli space, with the central charge determining an electric-charge–dependent BPS spectrum that exhibits a sign-driven asymmetry. In the explicit $SU(3)$ case, the relative moduli space is Taub–NUT and the BPS spectrum consists of hypermultiplets with charges tied to the simple roots $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$, including a tower of states with charges proportional to $(\boldsymbol{\alpha}+\boldsymbol{\beta})$. The results are shown to be consistent with Seiberg–Witten weak- and strong-coupling analyses, including monodromy predictions and the spectral curve, thereby linking semi-classical monopole dynamics to the full quantum theory. The work suggests extending the construction to larger gauge groups and to cases with light hypermultiplets, potentially yielding a comprehensive picture of BPS spectra in $N=2$ theories.

Abstract

We determine the low energy dynamics of monopoles in pure N=2 Yang-Mills theories for points in the vacuum moduli space where the two Higgs fields are not aligned. The dynamics is governed by a supersymmetric quantum mechanics with potential terms and four real supercharges. The corresponding superalgebra contains a central charge but nevertheless supersymmetric states preserve all four supercharges. The central charge depends on the sign of the electric charges and consequently so does the BPS spectrum. We focus on the SU(3) case where certain BPS states are realised as zero-modes of a Dirac operator on Taub-NUT space twisted by the tri-holomorphic Killing vector field. We show that the BPS spectrum includes hypermultiplets that are consistent with the strong- and weak-coupling behaviour of the Seiberg-Witten theory.