Chern-Simons deformations of the gauged O(3) Sigma model on compact surfaces

TL;DR

The work develops a rigorous framework to analyze Chern-Simons deformations of the gauged O(3) sigma model on compact surfaces. By converting distributional Bogomol’nyi equations into a regular elliptic system via Green’s functions and applying Leray-Schauder continuation, it proves existence of solutions for small deformation and extends this to global κ under a Bradlow-type constraint, with enhanced results when the vortex and antivortex counts match. The paper also characterizes the asymptotic behavior as κ grows, revealing three distinct large-κ regimes and Maxwell/Chern-Simons limiting profiles, and corroborates the theory with sphere-scale numerical experiments using symmetry reductions and continuation methods. Overall, the results illuminate the persistence and multiplicity of moduli spaces under Chern-Simons deformations and delineate the asymptotic landscape of solutions on compact surfaces.

Abstract

Existence of solutions to the field equations of the gauged Chern-Simons-O(3)-Sigma model on a compact Riemann surface is proved by a topological method. Existence of a minimal deformation constant $κ_{*} > 0$ is proved, such that for any prescribed configuration of vortices and antivortices, at least one solution exists for $|κ| \leq κ_{*}$. For small values of the Chern-Simons deformation parameter $κ$, it is proved that the field equations admit multiple solutions, provided the total number of vortices and antivortices are different. The Maxwell limit is computed for solutions of the field equations. In contrast, if the number of vortices equals the number of antivortices, it is proved that the field equations admit at least one solution for any value of $κ$ and the limit $κ\to \infty$ is proved. dependence of the fields on the deformation parameter is investigated numerically on the sphere.

Figures (2)

• Figure 1: Family of solutions to the Bogomol'nyi equations on a sphere of radius 2 along declination angle $\theta$. Two vortices are located at north pole with no antivortices. Top row. Higgs field north pole projection $\phi_{3}$ and magnetic field $B$. Bottom row. Scalar field rescaling by the Chern-Simons constant and electric field in the declination angle direction. The asymmetry parameter was set to $\tau = 0$ and $q = 1$. Pseudo-arclength continuation advances towards lighter colour.
• Figure 2: Family of solutions to the Bogomol'nyi equations on a sphere of radius $1.5$ along declination angle $\theta$. A vortex is located at north pole and an antivortex at south pole. Top row. Higgs field north pole projection $\phi_{3}$ and magnetic field $B$. Bottom row. Scalar field and electric field in the declination angle direction, both rescaled by the Chern-Simons deformation constant $\kappa$. The asymmetry parameter was set to $\tau = 0$ and $q = 1$. Dashed lines represent the non-trivial limits as $\kappa \to \infty$. Pseudo-arclength continuation advances towards lighter colour.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof