Bogomol'nyi Equations and Coexistence of Vortices and Antivortices in Generalized Abelian Higgs Theories

TL;DR

This paper derives Bogomol'nyi equations for generalized Abelian Higgs theories, establishing conditions for coexistence of vortices and antivortices on surfaces and the plane, with explicit flux and energy formulas.

Contribution

It introduces new Bogomol'nyi equations allowing vortex-antivortex coexistence and proves existence and uniqueness of solutions with explicit asymptotic behavior.

Findings

Existence of solutions with coexisting vortices and antivortices.

Explicit formulas for magnetic fluxes and energies.

Asymptotic behavior of solutions at infinity.

Abstract

We derive the Bogomol'nyi equations in generalized Abelian Higgs theories which allow the coexistence of vortices and antivortices over a compact Riemann surface or the full plane. In the compact surface situation, we obtain a necessary and sufficient condition for the existence of a unique solution describing a system of coexisting vortices and antivortices. In the full-plane situation, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices and obtain sharp asymptotic behavior of the solution near infinity. These solutions carry quantized magnetic fluxes and energies explicitly expressed in terms of the numbers of vortices and antivortices topologically characterized by the first Chern and Thom classes.