What becomes of vortices when they grow giant

We discuss vortex solutions of the abelian Higgs model in the limit of large winding number $n$. We suggest a framework where a topological quantum number $n$ is associated with a ratio of dynamical scales and a systematic expansion in inverse powers of $n$ is then derived in the spirit of effective field theory. The general asymptotic form of ${\it giant}$ vortices is obtained. For critical coupling the axially symmetric vortices become ${\it integrable}$ in the large-$n$ limit and we present the corresponding analytic solution. The method provides simple asymptotic formulae for the vortex shape and parameters with accuracy that can be systematically improved, and can be applied to topological solitons of other models. After including the next-to-leading terms the approximation works remarkably well down to $n=1$.