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We construct a manifestly supersymmetric and kappa-symmetry invariantworldvolume action describing the coupling of a dynamical membrane to an$\mathcal N=1$, $D=4$ $SU(N)$ super-Yang-Mills multiplet. Worldvolume scalarfields in this action are a Goldstone and a Goldstino associated withspontaneous breaking, by the membrane, of half of $\mathcal N=1$, $D=4$supersymmetry. When the Goldstone fields are set to zero, the model reduces toan $\mathcal N=1$, $d=3$ $SU(N)$ Chern-Simons theory induced by the SYMcoupling. We show that, when the membrane couples to the Veneziano-Yankielowicz(VY) effective theory of the $\mathcal N=1$ SYM, it sources VY bulk fieldequations, separates two distinct SYM vacua and provides the missingcontribution to the tension of BPS saturated domain-wall configurations, forwhich the membrane serves as a core. As a result, we obtain explicit BPSdomain-wall solutions in the Veneziano-Yankielowicz theory. We also brieflydiscuss a supersymmetric system of an open membrane having a string attached toits boundary and coupled to a massive extension of the Veneziano-Yankielowiczmodel.

Supermembranes and domain walls in $\mathcal N=1$, $D=4$ SYM