Given a Hermitian line bundle Lrightarrow M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as epsilon rightarrow 0, of couples (u_epsilon ,nabla _epsilon ) critical for the rescalings Eϵ(u,∇)=∫M(|∇u|2+ϵ2|F∇|2+14ϵ2(1-|u|2)2)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} E_\\epsilon (u,\\nabla )=\\int _M\\Big (|\\nabla u|^2+\\epsilon ^2|F_\\nabla |^2+\\frac{1}{4\\epsilon ^2}(1-|u|^2)^2\\Big ) \\end{aligned}$$\\end{document}of the self-dual Yang–Mills–Higgs energy, where u is a section of L and nabla is a Hermitian connection on L with curvature F_{nabla }. Under the natural assumption limsup _{epsilon rightarrow 0}E_epsilon (u_epsilon ,nabla _epsilon )<infty , we show that the energy measures converge subsequentially to (the weight measure mu of) a stationary integral (n-2)-varifold. Also, we show that the (n-2)-currents dual to the curvature forms converge subsequentially to 2pi Gamma , for an integral (n-2)-cycle Gamma with |Gamma |le mu . Finally, we provide a variational construction of nontrivial critical points (u_epsilon ,nabla _epsilon ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral (n-2)-varifolds in an arbitrary closed Riemannian manifold.