Bogomol'nyi Equations and Coexistence of Vortices and Antivortices in Generalized Abelian Higgs Theories
Aonan Xu, Yisong Yang
TL;DR
This work generalizes the Abelian Higgs framework to allow coexisting vortices and antivortices on both compact Riemann surfaces and the full plane by deriving Bogomol'nyi equations tied to topological invariants $c_1$ (first Chern class) and $\tau$ (Thom class). It reduces the problem to a nonlinear elliptic PDE with delta-function sources and proves sharp existence/uniqueness results: on compact surfaces a necessary and sufficient bound $|M-N| < |S|/(2\pi)$, with energy $E = 2\pi(M+N)$ and topological charges, and on $\mathbb{R}^2$ a full existence/uniqueness theory with exponential decay and flux quantization under structural conditions (C1)–(C2). The approach combines Bogomol’nyi decompositions, Leray–Schauder fixed-point arguments, and sub-/supersolution methods, enriching the landscape of vortex moduli and linking to geometric topology via $c_1$ and $\tau$. The results extend single-species vortex analyses to arbitrary distributions of vortices and antivortices, with potential applications in duality, symmetry breaking, and cosmic-string or condensed-matter contexts.
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.