Five Vortex Equations

Abstract

The Taubes equation for Abelian Higgs vortices is generalised to five distinct U(1) vortex equations. These include the Popov and Jackiw--Pi vortex equations, and two new equations. The Baptista metric, a conformal rescaling of the background metric by the squared Higgs field, gives insight into these vortices, and shows that vortices can be interpreted as conical singularities superposed on the background geometry. When the background has a constant curvature adapted to the vortex type, then the vortex equation is integrable by a reduction to Liouville's equation, and the Baptista metric has a constant curvature too, apart from its conical singularities. The conical geometry is fairly easy to visualise in some cases.

24 pages, 1 figure, version 2 with additional references

References in corpus (11)

• Vortices in (abelian) Chern-Simons gauge theory
• Analytic vortex solutions on compact hyperbolic surfaces
• Conformal metrics with constant curvature one and finite conical singularities on compact Riemann surfaces
• Integrable vortex-type equations on the two-sphere
• The dynamics of vortices on S^2 near the Bradlow limit
• The Kähler Potential of Abelian Higgs Vortices
• Vortices on Hyperbolic Surfaces
• Vortices as degenerate metrics
• Vortices and Jacobian varieties
• Vortex solutions of the Popov equations
• A Compact Codimension Two Braneworld with Precisely One Brane

Cited by in corpus (21)

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• Integrable Abelian Vortex-like solitons
• Conformal bridge transformation and PT symmetry
• Magnetic Impurities, Integrable Vortices and the Toda Equation
• Nineteen vortex equations and integrability
• The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories
• Cartan Connections and Integrable Vortex Equations
• Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics
• The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory
• Vortex counting and the quantum Hall effect
• Hyperbolic vortices and Dirac fields in 2+1 dimensions
• The volume of a vortex and the Bradlow bound
• Conformal bridge in a cosmic string background
• Nonlinear Problems Inspired by the Born--Infeld Theory of Electrodynamics
• Elizabethan vortices
• Magnetic Zero-Modes, Vortices and Cartan Geometry
• General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations
• Massive Nambu-Goldstone Fermions and Bosons for Non-relativistic Superconformal Symmetry: Jackiw-Pi Vortices in a Trap
• Analytic Approach for Computation of Topological Number of Integrable Vortex on Torus
• Abelian Higgs Vortices and Discrete Conformal Maps
• Vortex Harmonic Spinors on the Nappi-Witten Space