Low Energy Dynamics for 1/4 BPS Dyons

TL;DR

This work derives a low-energy effective description for 1/4 BPS dyons in maximally supersymmetric Yang-Mills theories by matching the exact field-theoretic BPS bound $E=|Z|$ with a moduli-space-based dynamics of monopoles. The authors show that the inter-monopole potential is fixed by the geometry of the monopole moduli space and can be expressed as the norm of a triholomorphic Killing vector, enabling a natural supersymmetric extension. In the explicit SU(3) case, they obtain a concrete potential $${\\cal U}(r) = \frac{\\mu_1+\\mu_2}{2g} q_{\\rm tot}^2 + \frac{2\\mu}{g} \frac{(\\Delta q_c)^2}{1+\\frac{1}{2\\mu r}}$$ and demonstrate that the low-energy BPS states reproduce the field-theoretic BPS configurations. The generalization to arbitrary gauge groups and the connection to N=4 supersymmetric quantum mechanics on hyperkähler moduli spaces offer a geometrical framework to study 1/4 BPS dyons and their spectra. Overall, the paper provides a robust, geometry-driven method to capture 1/4 BPS dyons within a low-energy effective theory, including its supersymmetric extension.

Abstract

Classical 1/4 BPS configurations consist of 1/2 BPS dyons which are positioned by competing static forces from electromagnetic and Higgs sectors. These forces do not follow the simple inverse square law, but can be encoded in some low energy effective potential between fundamental monopoles of distinct types. In this paper, we find this potential, by comparing the exact 1/4 BPS bound from a Yang-Mills field theory with its counterpart derived from low energy effective dynamics of monopoles. Our method is generalized to arbitrary gauge groups and to arbitrary BPS monopole/dyon configurations. The resulting effective action for 1/4 BPS states is written explicitly, and shown to be determined entirely by the geometry of multi-monopole moduli spaces. We also explore its natural supersymmetric extension.