Pontryagin invariant has different constant values at each side of a plane interface and suffers a jump across such boundary. In relation to $\theta $-ED, it is interesting to recall that the authors of Ref. [@CFKS] proposed a model for the dual theory arising from the Janus solution, where the $\mathcal{N}=4$ SYM coupling $g(z)$ affects only the kinetic term of the non-Abelian gauge field, together with the interaction terms in the original Lagrangian for the standard $\mathcal{N}=4$ SYM theory. The model completely breaks the $16$ original supersymmetries of the $\mathcal{N}=4$ SYM theory. Moreover, $g(z)$ is taken as constant on each side of a planar interface $(z=0)$, with a sharp jump across it. In this way, the gauge field part of the action is the non-abelian generalization of the Maxwell action in an inhomogeneous medium with permitivity $\epsilon $ and permeability $\mu $ related by $\epsilon (z)=1/\mu (z)=1/g^{2}(z)$. The YM fields satisfy boundary conditions at the interface, which are derived by integrating the equations of motion over the standard infinitesimal pill-shaped regions across the boundary, in a way similar to standard electrodynamics. The YM Green’s function is also obtained by using image methods. Nevertheless, let us emphasize that this model does 