Some exact Bradlow vortex solutions

Gudnason, Sven Bjarke; Nitta, Muneto

doi:10.1007/jhep05(2017)039

Cited by 5 publications

(4 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[9] on hyperbolic space and further in Ref. [14] on nontrivial geometries with nonconstant curvature. As noticed in Ref.…”

Section: Bradlowmentioning

confidence: 96%

Nineteen vortex equations and integrability

Gudnason

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The class of five integrable vortex equations discussed recently by Manton is extended so it includes the relativistic BPS Chern-Simons vortices, yielding a total of nineteen vortex equations. Not all the nineteen vortex equations are integrable, but four new integrable equations are discovered and we generalize them to infinitely many sets of four integrable vortex equations, with each set denoted by its integer order n. Their integrability is similar to the known cases, but give rise to different (generalized) Baptista geometries, where the Baptista metric is a conformal rescaling of the background metric by the Higgs field. In particular, the Baptista manifolds have conical singularities. Where the Jackiw-Pi, Taubes, Popov and Ambjørn-Olesen vortices have conical deficits of 2π at each vortex zero in their Baptista manifolds, the higher-order generalizations of these equations are also integrable with larger constant curvatures and a 2πn conical deficit at each vortex zero. We then generalize a superposition law, known for Taubes vortices of how to add vortices to a known solution, to all the integrable vortex equations. We find that although the Taubes and the Popov equations relate to themselves, the Ambjørn-Olesen and Jackiw-Pi vortices are added by using the Baptista metric and the Popov equation. Finally, we find many further relations between vortex equations, e.g. we find that the Chern-Simons vortices can be interpreted as Taubes vortices on the Baptista manifold of their own solution.

show abstract

“…[9] on hyperbolic space and further in Ref. [14] on nontrivial geometries with nonconstant curvature. As noticed in Ref.…”

Section: Bradlowmentioning

confidence: 96%

Nineteen vortex equations and integrability

Gudnason

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This solution does not lend an easy choice of gauge that fixes the phase in terms of f (z). Integrability of the Bradlow or type I − 0 equation was discussed in reference [9] on hyperbolic space and further in reference [14] on nontrivial geometries with nonconstant curvature. As noticed in reference [9], it is possible to simply set C 2 := 0 in the solution (58) which yields a class of solutions on the hyperbolic plane.…”

Section: Type II 04mentioning

confidence: 99%

Nineteen vortex equations and integrability

Gudnason¹

2022

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

The class of ﬁve integrable vortex equations discussed recently by Manton is extended so it includes the relativistic BPS Chern-Simons vortices, yielding a total of nineteen vortex equations. Not all the nineteen vortex equations are integrable, but four new integrable equations are discovered and we generalize them to inﬁnitely many sets of four integrable vortex equations, with each set denoted by its integer order n. Their integrability is similar to the known cases, but give rise to diﬀerent (generalized) Baptista geometries, where the Baptista metric is a conformal rescaling of the background metric by the Higgs ﬁeld. In particular, the Baptista manifolds have conical singularities. Where the Jackiw-Pi, Taubes, Popov and Ambjørn-Olesen vortices have conical deﬁcits of 2π at each vortex zero in their Baptista manifolds, the higher-order generalizations of these equations are also integrable with larger constant curvatures and a 2πn conical deﬁcit at each vortex zero. We then generalize a superposition law, known for Taubes vortices of how to add vortices to a known solution, to all the integrable vortex equations. We ﬁnd that although the Taubes and the Popov equations relate to themselves, the Ambjørn-Olesen and Jackiw-Pi vortices are added by using the Baptista metric and the Popov equation. Finally, we ﬁnd many further relations between vortex equations, e.g. we ﬁnd that the Chern-Simons vortices can be interpreted as Taubes vortices on the Baptista manifold of their own solution.

show abstract

“…Bradlow vortex equations, first introduced in Ref. [10], have exact solutions on H 2 , but due to the simplicity of the vortex equation it does in fact have integrable solutions on many compact domains of various curvatures, including vanishing curvature [21].…”

Section: Exotic Vortex Equationsmentioning

confidence: 99%

Magnetic impurities, integrable vortices and the Toda equation

Gudnason

Ross

2021

Lett Math Phys

Self Cite

View full text Add to dashboard Cite

The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong-Wong type. Under certain conditions, these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes equation to the Liouville equation. We further find a family of charge matrices that turn the five vortex equations into either the Toda equation or the Toda equation with the opposite sign. We find exact analytic solutions in all cases and the solution with the opposite sign appears to be new.

show abstract

Some exact Bradlow vortex solutions

Cited by 5 publications

References 15 publications

Nineteen vortex equations and integrability

Nineteen vortex equations and integrability

Nineteen vortex equations and integrability

Magnetic impurities, integrable vortices and the Toda equation

Contact Info

Product

Resources

About