{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34062",
  "author_total": 1,
  "authors": [
    {
      "name": "Taisuke Shibata"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55gdgqhfsgm2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34062",
  "fulltext_url": null,
  "id": "55gdgqhfsgm2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34062",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7223907208",
  "pdf_url": "https://arxiv.org/pdf/2610.34062",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55gdgqhfsgm2r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34062",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34062"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Symplectic capacities and Weyl laws from Abelian Higgs min max theory",
  "title_text": "Symplectic capacities and Weyl laws from Abelian Higgs min max theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/55gdgqhfsgm2r",
  "venue": "arXiv",
  "versions": []
}