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

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.

💡 Research Summary

The paper investigates the gauged O(3) sigma model in 2+1 dimensions with an added Chern‑Simons term, focusing on the existence and multiplicity of vortex‑antivortex solutions on a compact Riemann surface Σ. Starting from the Lagrangian L_O(3) and introducing a neutral scalar N together with a Chern‑Simons deformation parameter κ, the authors derive the self‑dual Bogomol’nyi equations (1.5)–(1.9). By defining the scalar field u = log