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Manton's Vortex Equations and Anti-Self-Dual Connections
Abstract
We study four of Manton's five vortex equations on a Riemann surface X in relation to anti-self-dual connections. From a solution of each vortex equation, we construct an equivariant holomorphic bundle over X×ℙ1, X×Δ, or X×ℂ, together with an invariant Hermitian or pseudo-Hermitian metric whose associated connection is anti-self-dual. The equivariant structures are given by the actions of SU(2), SU(1,1), and SE(2), respectively. Conversely, we show that an equivariant holomorphic bundle equipped with an invariant (pseudo-)Hermitian metric gives rise to a solution of the corresponding vortex equation when the associated connection is anti-self-dual.
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BibTeX
@misc{ono2026mantons,
title = {{Manton's Vortex Equations and Anti-Self-Dual Connections}},
author = {Takashi Ono},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05962},
eprint = {2610.05962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05962},
}
RIS
TY - UNPB
TI - Manton's Vortex Equations and Anti-Self-Dual Connections
AU - Takashi Ono
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05962
UR - https://arxiv.org/abs/2610.05962
AB - We study four of Manton's five vortex equations on a Riemann surface X in relation to anti-self-dual connections. From a solution of each vortex equation, we construct an equivariant holomorphic bundle over X×ℙ^1, X×Δ, or X×ℂ, together with an invariant Hermitian or pseudo-Hermitian metric whose associated connection is anti-self-dual. The equivariant structures are given by the actions of SU(2), SU(1,1), and SE(2), respectively. Conversely, we show that an equivariant holomorphic bundle equipped with an invariant (pseudo-)Hermitian metric gives rise to a solution of the corresponding vortex equation when the associated connection is anti-self-dual.
ER -