not include a coupling to the YM Pontryagin invariant, so its Abelian limit does not reproduce $\theta $-ED. The inclusion of the topological coupling $\theta (z)$ in addition to the YM coupling $g(z)$ is developed in Refs. [@Witten2; @Kim2], where 1/2 BPS vacuum configurations are studied in particular. As shown in Ref. [@Witten2] half of the original supersymmetries can be maintained provided such couplings are constrained by the relations $1/g^{2}(z)=D\sin 2\psi (z)$ and $\theta (z)=\theta_{0}+8\pi ^{2}D\cos 2\psi (z)$. The case of a sharp interface respecting the above constraints is also considered in Ref. [@Kim2] and it is studied in the abelian Coulomb phase, by setting two different constant values $\psi _{1}$ and $\psi _{2}$ at each side of a planar boundary. However, in the case of $\theta $-ED, supersymmetry does not enter and we are choosing the electromagnetic coupling to be globally constant, *i.e.* $g_{1}=g_{2}=e$, while only the topological coupling $\theta (z)$ becomes discontinuous at the sharp boundary, with constant values $\theta_{1}\neq \theta _{2}$ in each side. As can be seen already, these two systems are not equivalent and we will later discuss this issue in more detail.

The paper is organized as follows. In section \[model\] we review the 