Vortices and Antivortices

Abstract

In this chapter we consider the coexistence of vortices (strings) and antivortices (antistrings) in an Abelian gauge theory. In §11.1, we introduce the gauge field model and state our main existence theorems. In §11.2, we calculate various components of the energy-momentum tensor and reduce the equations of motion into a self-dual system and boil the problem down to a nonlinear elliptic equation with sources through an integration of the Einstein equations. In §11.3, we prove the existence of a unique solution for the elliptic equation governing vortices, in absence of gravity, and, we prove the existence of a solution for the equation governing strings. In §11.4, we calculate the precise value of the quantized energy, and total curvature, of a solution possessing M vortices (strings) and N antivortices (antistrings) and observe the roles played by these two different types of vortices (strings), energetically. In §11.5, we consider the problem over a closed Riemann surface.