Self-Dual Chern--Simons Vortices in Higgs Field

By decomposing the Bogomol’nyi self-dual equation in the Abelian Chern--Simons Higgs model, we find a self-dual topological term that was ignored all the time in the Bogomol’nyi self-duality equation due to the improper decomposition of the complex Higgs field. We also present a new self-dual equation that includes the topological term. It is shown that the self-dual vortex just arises from the symmetric phase of the Higgs field Ф=0. Using our Ф-mapping theory, the inner topological structure of the vortex and double vortex is given.