An Index Theorem for Domain Walls in Supersymmetric Gauge Theories

TL;DR

This work counts the zero modes of domain walls in ${\cal N}=2$ supersymmetric QED with $N$ hypermultiplets and a Fayet–Iliopoulos parameter. It adapts a Callias-type index theorem to the linearized BPS equations and shows the continuum spectrum is gapped, allowing a precise count of zero modes. The analysis yields ${\cal I}=N-1$, implying the most general domain wall carries $2(N-1)$ real moduli, interpreted as the positions and phases of $(N-1)$ constituent walls. In the IIB string theory context, this corresponds to moduli for a D-string interpolating between $N$ D5-branes, linking gauge-theory soliton moduli to brane dynamics. The regulator-dependent nature of the index $I(M^2)$ is discussed, analogous to mass renormalization effects in related soliton problems.

Abstract

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-1) zero modes, which can be interpreted as the positions and phases of (N-1) constituent domain walls. This implies the existence of moduli for a D-string interpolating between N D5-branes in IIB string theory.