and the constant value $\theta =$ $\theta _{2}$ in region $\mathcal{U}_{2}$. This situation is expressed in the characteristic function $$\theta \left( \textbf{x} \right) =\left\{ 
\begin{array}{c}
\theta _{1}\;\;\;,\;\;\; \textbf{x} \in \mathcal{U}_{1} \\ 
\theta _{2}\;\;\;,\;\;\; \textbf{x} \in \mathcal{U}_{2}%
\end{array}%
\right. .  \label{theta}$$In this scenario the $\theta $-term in the action fails to be a global topological invariant because it is defined over a region with the boundary $%
\Sigma $. Varying the action gives rise to a set of Maxwell equations with an effective additional current density with support at the boundary $$\partial _{\mu }F^{\mu \nu }=\tilde{\theta}\delta \left( \Sigma \right)
n_{\mu }\tilde{F}^{\mu \nu }+4\pi j^{\nu },  \label{FieldEqs}$$where $n_{\mu }$ is the outward normal to $\Sigma $, and $\tilde{\theta}%
= \alpha \left( \theta _{1}-\theta _{2}\right) /\pi $, which enforces the invariance of the classical dynamics under the shifts of $\theta$ by any constant, $\theta \rightarrow \theta + C$. Current conservation can be verified directly by taking the divergence at both sides of Eq. ([FieldEqs]{}) and using symmetry properties.

![[Region over which the electromagnetic field theory is defined.]{}[]{data-label="regions"}](regions.eps)

The set of Eqs. (\[FieldEqs\]) for $\theta $-ED can be written as $$\begin{aligned}
\nabla \cdot \mathbf{E} &=&\tilde{\theta}\delta \left( \Sigma \right) 
\mathbf{B}\cdot \mathbf{n}+4\pi \rho ,  \label{GaussE} \\
