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Given a hermitian line bundle $L\to M$ on a closed Riemannian manifold$(M^n,g)$, the self-dual Yang-Mills-Higgs energies are a natural family offunctionals \begin{align*}  &E_\epsilon(u,\nabla):=\int_M\Big(|\nablau|^2+\epsilon^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4\epsilon^2}\Big) \end{align*}defined for couples $(u,\nabla)$ consisting of a section $u\in\Gamma(L)$ and ahermitian connection $\nabla$ with curvature $F_\nabla$. While the criticalpoints of these functionals have been well-studied in dimension two by thegauge theory community, it was shown in previous work of the second- andthird-named authors that critical points in higher dimension converge as$\epsilon\to 0$ (in an appropriate sense) to minimal submanifolds ofcodimension two, with strong parallels to the correspondence between theAllen-Cahn equations and minimal hypersurfaces.  In this paper, we complement this idea by showing the $\Gamma$-convergence of$E_\epsilon$ to ($2\pi$ times) the codimension two area: more precisely, givena family of couples $(u_\epsilon,\nabla_\epsilon)$ with $\sup_\epsilonE_\epsilon(u_\epsilon,\nabla_\epsilon)<\infty$, we prove that a suitable gaugeinvariant Jacobian $J(u_\epsilon,\nabla_\epsilon)$ converges to an integral$(n-2)$-cycle $\Gamma$, in the homology class dual to the Euler class $c_1(L)$,with mass $2 \pi \mathbb{M}(\Gamma)\le\liminf_{\epsilon \rightarrow0}E_\epsilon(u_\epsilon,\nabla_\epsilon)$. We also obtain a recovery sequencefor any integral cycle in this homology class.  Finally, we apply these techniques to compare min-max values for the$(n-2)$-area from the Almgren-Pitts theory with those obtained from theYang-Mills-Higgs framework, showing that the former values always provide alower bound for the latter. As an ingredient, we also establish a Huisken-typemonotonicity result along the gradient flow of $E_{\epsilon}$.

Convergence of the self-dual $U(1)$-Yang-Mills-Higgs energies to the $(n-2)$-area functional