structure constant and $j^{\mu }\;$is a conserved external current. The coupling constant for the $\theta$-term, $\alpha / 4 \pi ^{2}$, is chosen in such a way that the total electric charge $q 
_{e} = \frac{1}{4 \pi} \int d\textbf{S} \cdot \textbf{D}$ has to be an integer multiple of the electron charge $e$, whereas the magnetic charge $q _{m} = \frac{1}{4 \pi} 
\int d\textbf{S} \cdot \textbf{B}$ should be an integer multiple of $g 
= e / 2 \alpha$ by the Dirac quantization condition [@Wilczek:1987mv]. Later we will recall the reasoning which shows that, quantum mechanically, the allowed values of $\theta$ are $0$ or $\pi$ (mod $2 \pi$).

The $(3+1)$-dimensional spacetime is $\mathcal{M}=\mathcal{U}\times \mathbb{R%
}$, where $\mathcal{U}$ is a three-dimensional manifold and $\mathbb{R}$ corresponds to the temporal axis. We make a partition of space in two regions: $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ in such a way that manifolds $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ intersect along a common two-dimensional boundary $\Sigma $, to be called the $\theta $-boundary, so that $\mathcal{U}=\mathcal{U}_{1}\cup \mathcal{U}_{2}$ and $\Sigma =\mathcal{%
U}_{1}\cap \mathcal{U}_{2}$, as shown in Fig. \[regions\]. We also assume that the field $\theta $ is piecewise constant in such way that it takes the constant value $\theta =\theta _{1}$ in region $\mathcal{U}_{1}$ 