Fermionic Zero Modes in Self-dual Vortex Background on a Torus

We study fermionic zero modes in the self-dual vortex backgroundon an extra two-dimensional Riemann surface in (5+1) dimensions.Using the generalized Abelian-Higgs model, we obtain the innertopological structure of the self-dual vortex and establish theexact self-duality equation with topological term. Then we analyzethe Dirac operator on an extra torus and the effective Lagrangianof four-dimensional fermions with the self-dual vortex background.Solving the Dirac equation, the fermionic zero modes on a toruswith the self-dual vortex background in two simple cases are obtained.