Chern-Simons vortices and holography

In this paper, based on the $AdS_{4}/CFT_{3}$ duality, we have explored the precise connection between the abelian Chern-Simons (CS) Higgs model in ($2+1$) dimensions to that with its dual gravitational counterpart living in one higher dimension. It has been observed that the $U(1)$ current computed at the boundary of the $AdS_{4}$ could be expressed as the local function of the vortex solution that has the remarkable structural similarity to that with the Ginzburg-Landau (GL) type local expression for the current associated with the Maxwell-CS type vortices in ($2+1$) dimensions. In order to explore this duality a bit further we have also computed the coherence length as well as the magnetic penetration depth associated with these vortices. Finally using the knowledge of both the coherence length as well as the magnetic penetration depth we have computed the Ginzburg-Landau coefficient for the Maxwell-CS type vortices in ($2+1$) dimensions.