1/2, 1/4 and 1/8 BPS Equations in SUSY Yang-Mills-Higgs Systems -- Field Theoretical Brane Configurations --

TL;DR

This work systematically classifies 1/2,1/4, and 1/8 BPS equations in SUSY gauge theories with eight supercharges across dimensions $d=6$ to $d=2$. By establishing projection operators in six dimensions, the authors show that the fundamental 1/2 BPS objects are instantons (strings) and vortices (3-branes), and they derive two distinct 1/4 BPS equation families and a unique 1/8 BPS equation in $d=6$, with all lower-dimensional cases obtained via Scherk–Schwarz and ordinary dimensional reductions. The paper then organizes a comprehensive zoo of BPS composites—including instanton–vortex, monopole–vortex–wall, Hitchin vortex–wall junctions, and Nahm-wall constructions—articulating their charges, energies, and extension to wavy and dyonic solitons. A key outcome is the unification of field-theory branes with D/M-brane pictures, providing a Mother 1/8 BPS system from which all others descend, and outlining rich avenues for exact solutions, moduli spaces, and higher-dimensional generalizations. Overall, the work significantly expands the landscape of BPS configurations in eight-supercharge gauge theories and sets a framework for exploring brane-like solitons in field theory and their string/M-theory connections.

Abstract

We systematically classify 1/2, 1/4 and 1/8 BPS equations in SUSY gauge theories in d=6, 5, 4, 3 and 2 with eight supercharges, with gauge groups and matter contents being arbitrary. Instantons (strings) and vortices (3-branes) are only allowed 1/2 BPS solitons in d=6 with N=1 SUSY. We find two 1/4 BPS equations and the unique 1/8 BPS equation in d=6 by considering configurations made of these field theory branes. All known BPS equations are rederived while several new 1/4 and 1/8 BPS equations are found in dimension less than six by dimensional reductions.