Symplectic capacities and Weyl laws from Abelian Higgs min max theory
Abstract
For a Hermitian line bundle E→ Y over a Riemannian manifold (Yn,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 Fa 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 {ckAH}k≥1 in dimension four by a min--max construction. In particular, we prove that c1AH 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 CP2 and S2× S2 to prove an upper bound on the higher capacities of balls and the formula ckAH(Z4(a),ω0)=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.
Related
- The ECH and alternative ECH capacities of closed symplectic 4-manifolds
- Fedosov manifolds with parallel symplectic Weyl curvature
- Integral curvature estimates and geodesic limits for stable abelian Higgs fields
- Stable Abelian Yang--Mills--Higgs Fields on the Round Three-Sphere
- Stable solutions in the abelian Higgs model
BibTeX
@misc{shibata2026symplectic,
title = {{Symplectic capacities and Weyl laws from Abelian Higgs min max theory}},
author = {Taisuke Shibata},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34062},
eprint = {2610.34062},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34062},
}
RIS
TY - UNPB
TI - Symplectic capacities and Weyl laws from Abelian Higgs min max theory
AU - Taisuke Shibata
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34062
UR - https://arxiv.org/abs/2610.34062
AB - For a Hermitian line bundle E→ 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≥1 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× 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.
ER -