1/2, 1/4 and 1/8 BPS equations in SUSY Yang-Mills-Higgs systems: Field theoretical brane configurations

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 many new 1/4 and 1/8 BPS equations are found in dimension less than six by dimensional reductions.