  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 of
functionals \begin{align*}
  &E_\epsilon(u,\nabla):=\int_M\Big(|\nabla
u|^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 a
hermitian connection $\nabla$ with curvature $F_\nabla$. While the critical
points of these functionals have been well-studied in dimension two by the
gauge theory community, it was shown in previous work of the second- and
third-named authors that critical points in higher dimension converge as
$\epsilon\to 0$ (in an appropriate sense) to minimal submanifolds of
codimension two, with strong parallels to the correspondence between the
Allen-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, given
a family of couples $(u_\epsilon,\nabla_\epsilon)$ with $\sup_\epsilon
E_\epsilon(u_\epsilon,\nabla_\epsilon)<\infty$, we prove that a suitable gauge
invariant 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 $2 \pi \mathbb{M}(\Gamma)\le\liminf_{\epsilon \rightarrow
0}E_\epsilon(u_\epsilon,\nabla_\epsilon)$. We also obtain a recovery sequence
for 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 the
Yang-Mills-Higgs framework, showing that the former values always provide a
lower bound for the latter. As an ingredient, we also establish a Huisken-type
monotonicity result along the gradient flow of $E_{\epsilon}$.
