Removal of singularities and Gromov compactness for symplectic vortices

Ott, Andreas

doi:10.4310/jsg.2014.v12.n2.a3

Cited by 13 publications

(34 citation statements)

References 24 publications

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“…If we assume the connection is continuous across the singular point, then the singularity is removable [9]. Similar results were also abtained for minimal YMH fields in the symplectic setting, which are often referred as symplectic vortices [3]. However, in general, one can not expect that the limit holonomy around the singular point to be trivial.…”

Section: Introductionsupporting

confidence: 62%

Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

Chen

Song

2020

International Mathematics Research Notices

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We study isolated singularities of two dimensional Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general the singularity can not be removed due to possibly non-vanishing limit holonomy around the singular points. We establish a sharp asymptotic decay estimate of the Yang-Mills-Higgs field near a singular point, where the decay rate is precisely determined by the limit holonomy. Our result can be viewed as a generalization of the classical removable singularity theorem of two dimensional harmonic maps.

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Section: Introductionsupporting

confidence: 62%

Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

Chen

Song

2020

International Mathematics Research Notices

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“…The proof of this theorem has been given by Ott [30] in the case that z ∈ IntH and by the second named author [40] in the case G = S 1 . The detail of a general proof is left to the reader because the proof has no essential difference from the proof of the standard Gromov compactness for holomorphic disks or spheres, and because our main concern is the compactness with respect to the adiabatic limit.…”

Section: 4mentioning

confidence: 99%

“…Remark 2.16. Besides [30] and [40], there are other works treating compactness of vortex equations in various settings, such as [27], [8], [29], [33].…”

Section: 4mentioning

confidence: 99%

Compactness in the adiabatic limit of disk vortices

Wang

2016

Math. Z.

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This paper is the first input towards an open analogue of the quantum Kirwan map. We consider the adiabatic limit of the symplectic vortex equation over the unit disk for a Hamiltonian G-manifold with Lagrangian boundary condition, by blowing up the metric on the disk. We define an appropriate notion of stable solutions in the limit, and prove that any sequence of disk vortices with energy uniformly bounded has a subsequence converging to such a stable object. We also proved several analytical properties of vortices over the upper half plane, which are new type of bubbles appearing in our compactification.

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“…This would lead to the definition of a virtual moduli cycle and a new quantum product in equivariant cohomology. For more related results in this direction, we refer to [2,16,31,34] and references therein. It is also worthy mentioning that, by the Hitchin-Kobayashi correspondence, the moduli space of vortices corresponds to the moduli space of stable holomorphic bundles [1,13,27].…”

Section: Definition and Motivationmentioning

confidence: 99%

Convergence of Yang–Mills–Higgs fields

Song

2015

Math. Ann.

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In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces, where the fiber is a compact symplectic manifold and the conformal structure of the underlying surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth YMH field modulo finitely many harmonic spheres, while near the nodes where the conformal structure degenerates, the YMH fields converges to a pair consisting of a flat connection and a twisted geodesic (with potential). In particular, we generalize the recent compactness results on both harmonic maps from surfaces and twisted holomorphic curves to general YMH fields.

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Removal of singularities and Gromov compactness for symplectic vortices

Cited by 13 publications

References 24 publications

Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

Compactness in the adiabatic limit of disk vortices

Convergence of Yang–Mills–Higgs fields

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