Zero Modes of Fermions Trapped by Giant Vortices

Zero-energy solutions of the Dirac equation for the fermions bound to giant vortices of large winding number \(n\) are studied in the abelian Higgs and Chern-Simons Higgs models. The case of Jackiw-Rossi theory of the Majorana states in topological superconductors is discussed in detail. By expanding in inverse powers of \(n\) we find an analytic result for asymptotically all \(n\) solutions required by the index theorem. In the abelian Higgs model the zero modes fill the vortex core and reveal a universal structure independent of fine details of the gauge and scalar field interactions which, in particular, determines the general properties of the large-\(n\) superconducting cosmic strings. On the contrary, for the Chern-Simons Higgs vortices the zero modes are localized on the core boundary and the explicit solution is obtained for the supersymmetric couplings in a self-dual background.