Classical conformal solutions of two-dimensional Euclidean scalar electrodynamics with topological coupling

Classical conformal solutions of two-dimensional (2D) Euclidean scalar electrodynamics with topological coupling are investigated. For suitable choices of the scalar potential and of the topological coupling, the determining equations become well-known nonlinear equations of 2D physics. A particularly interesting case is when the matter density solves the Liouville equation. This system describes magnetic vortices, and the model is related to the nonlinear O(3) model with local symmetry and to the CP1 model. The non-Abelian generalization of the model leading to Liouville vortices has also been provided.