We study codimension one branes, i.e., p -branes in (p+2) dimensions, in the superembedding approach for the cases where the worldvolume superspace is embedded in a minimal target superspace with half supersymmetry breaking. This singles out the cases p=1,2,3,5,9 . For p=3,5,9 the superembedding geometry naturally involves a fundamental super 2 -form potential on the worldvolume whose generalised field strength obeys a constraint deducible from considering an open supermembrane ending on the p -brane. This constraint, together with the embedding constraint, puts the system on-shell for p=5 but overconstrains the 9 -brane in D=11 such that the Goldstone superfield is frozen. For p=3 these two constraints give rise to an off-shell linear multiplet on the worldvolume. An alternative formulation of this case is given in which the linear multiplet is dualised to an off-shell scalar multiplet. Actions are constructed for both cases and are shown to give equivalent equations of motion. After gauge fixing a local Sp(1) symmetry associated with shifts in the Sp(1)R Goldstone modes, we find that the auxiliary fields in the scalar multiplet parametrise a two-sphere. For completeness we also discuss briefly the cases p=1,2 where the equations of motion (for off-shell multiplets) are obtained from an action principle.
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