A systematic discussion on the cosmic strings and abelian vortex model proposed by Nielsen and Olesen

Due to the wide range of applications and effects of the Abelian vortex model of Nielsen and Olesen in the areas of physics, ranging from condensed matter to astrophysical effects, some work in the literature is necessary to approach this topic in a simple and succinct way that the undergraduate student in both physics and related areas has the possibility to know and understand the mechanisms associated with this vortex model that indicate him as a strong candidate for the source for the topological defects proposed by Vilenkin. Taking into account that this subject is concentrated only in postgraduate studies in physics, and is not so accessible to undergraduate students, we have developed this work by making a small systematic review in the scope of abelian vortices and how these objects influence space-time their when applied to General Relativity.