Constraints in supersymmetric gauge theories with hypermultiplets

1/2-BPS vortex strings in $\mathcal{N}=2$ supersymmetric ${\rm U}(1)^N$ gauge theories

We study the string tension in supersymmetric gauge theories with hypermultiplets in the generic setting of an arbitrary fayet-iliopoulos triplet of parameters for each gauge group and an invertible charge matrix.We establish three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in space.It turns out that all existing bogomol'nyi-prasad-sommerfield (bps) solutions have a tension which is linear in the magnetic fluxes, which in turn are linearly related to the winding numbers.We further prove for all cases, that a seemingly vanishing bogomol'nyi bound can not have solutions.Finally, we write down the most general vortex equations in both master form and taubes-like form.