Index theorem for domain walls in supersymmetric gauge theories

The supersymmetric Abelian Higgs model with N scalar fields admits multiple domain wall solutions. We perform a Callias-type index calculation to determine the number of zero modes of this soliton. We confirm that the most general domain wall has $2(N\ensuremath{-}1)$ zero modes, which can be interpreted as the positions and phases of $(N\ensuremath{-}1)$ constituent domain walls. This implies the existence of moduli for a D-string interpolating between N $D5$-branes in type IIB string theory.