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Domain Lines as Fractional Strings

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We consider N=2 supersymmetric quantum electrodynamics (SQED) with 2 flavors, the Fayet--Iliopoulos parameter, and a mass term $beta$ which breaks the extended supersymmetry down to N=1. The bulk theory has two vacua; at $beta=0$ the BPS-saturated domain wall interpolating between
them has a moduli space parameterized by a U(1) phase $sigma$ which can
be promoted to a scalar field in the effective low-energy theory on the
wall world-volume. At small nonvanishing $beta$ this field gets a
sine-Gordon potential. As a result, only two discrete degenerate BPS
domain walls survive. We find an explicit solitonic solution for domain lines -- string-like objects living on the surface of the domain wall which separate wall I from wall II. The domain line is seen as a BPS kink in the world-volume effective theory. The domain line carries the magnetic flux which is exactly 1/2 of the flux carried by the flux tube living in the bulk on each side of the wall. Thus, the domain lines on the wall confine charges living on the wall, resembling Polyakov's
three-dimensional confinement.