# Coexisting Vortices and Antivortices Generated by Dually Gauged Harmonic Maps

## TL;DR

This paper introduces a new dually gauged harmonic map model that describes coexisting vortices and antivortices, providing existence, uniqueness, and explicit physical property formulas for solutions on compact surfaces and the plane.

## Contribution

It formulates a novel BPS system for coexisting vortices and antivortices derived from a product Abelian Higgs field theory, with rigorous mathematical results and explicit physical quantities.

## Key findings

- Established existence and uniqueness theorems for vortex solutions.
- Derived necessary and sufficient conditions for solutions on compact surfaces.
- Obtained explicit formulas for physical quantities like magnetic charges and energies.

## Abstract

In this paper we first formulate a dually gauged harmonic map model, suggested from a product Abelian Higgs field theory arising in impurity-inspired field theories, and obtain a new BPS system of equations governing coexisting vortices and antivortices, which are topologically characterized by the first Chern class of the underlying Hermitian bundle and the Thom class of the associated dual bundle. We then establish existence and uniqueness theorems for such vortices. For the equations over a compact surface, we obtain necessary and sufficient conditions for the existence of solutions. For the equations over the full plane, we obtain all finite-energy solutions. Besides, we also present precise expressions giving the values of various physical quantities of the solutions, including magnetic charges and energies, in terms of the total numbers of vortices and antivortices, of two species, and the coupling parameters involved.