arXiv2026
For a Hermitian line bundle $E\to Y$ over a Riemannian manifold $(Y^n,g)$, the self-dual Abelian--Higgs energy is defined by $ \En^Y_{\eps,g}(u,a)=\frac1{2π}\int_Y \left(|\nabla_a u|_g^2+\eps^2|F_a|_g^2+ \frac{(1-|u|^2)^2}{4\eps^2}\right)dV_g, $ where $u$ is a smooth section of $E$, $a$ is a unitary connection, and $F_a$ is its curvature. Pigati and Stern studied the energy as a codimension-two analogue of the Allen--Cahn energy to study minimal submanifolds in a Riemannian manifold. In this paper, we use this functional to define a nondecreasing sequence of symplectic capacities $\{c_k^{\AH}\}_{k\ge1}$ in dimension four by a min--max construction. In particular, we prove that $c_1^{\AH}$ is normalized on the ball and the cylinder. For the upper bound on the cylinder, we combine Gromov's theory of families of $J$-holomorphic spheres with Taubes's planar vortex. For the lower bound on the ball, we use the properties of the Abelian--Higgs theory developed by Pigati--Stern, together with a geometric estimate on smooth approximations to the double of the ball and an equivariant deformation of low energy sublevels. These estimates give the normalization without Seiberg--Witten theory. We then use Seiberg--Witten moduli spaces on $\CP^2$ and $S^2\times S^2$ to prove an upper bound on the higher capacities of balls and the formula $c_k^{\AH}(Z^4(a),ω_0)=ka$. We also prove Euclidean Weyl laws for the capacity sequence and for the min--max sequence with the standard Euclidean metric. For these results, we construct AH sweepouts from moduli spaces of the solutions to the perturbed Seiberg--Witten equations.