integral would result. Furthermore, since $\mathbf{E}\cdot \mathbf{B}$ is odd under $t \to -t$, only $\theta = 0$ and $\theta = \pi$ give a time-reversal symmetric theory. Thus, time reversal takes $\theta$ into $- \theta$, so $\theta = 0$ is time reversal invariant per se, whereas $\theta = \pi$ is invariant after the shift $\theta \rightarrow \theta + 2 \pi$. This is typically referred as the $T$ generator of the electric-magnetic duality. The two transformations $S$ and $T$ generate the $SL(2, \mathbb{Z})$ symmetry group acting on the fields [@Karch].

Next we study the effects of a $\theta$-interface in the propagation of the fields. Assuming that the time derivatives of the fields are finite in the vicinity of the surface $\Sigma $, the field equations imply that the normal component of $\mathbf{E}$, and the tangential components of $\mathbf{B}$, acquire discontinuities additional to those produced by superficial free charges and currents, while the normal component of $\mathbf{B}$, and the tangential components of $\mathbf{E}$, are continuous. For vanishing free sources on the surface the boundary conditions read $$\begin{aligned}
 \mathbf{E}_{z}\big|^{z=a^+}_{z=a^-} &=&\tilde{\theta}\mathbf{B}_{z}\big|%
_{z=a},\;\;\;\;\;\;\;\;\; \mathbf{B}_{\parallel }\big|^{z=a^+}_{z=a^-}=-\tilde{%
\theta}\mathbf{E}_{\parallel }\big|_{z=a},  \label{Ampere-BC} \\
 \mathbf{B}_{z}\big|^{z=a^+}_{z=a^-} &=&0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\; \mathbf{E}%
_{\parallel }\big|^{z=a^+}_{z=a^-}=0.  \label{Faraday-BC}\end{aligned}$$ The continuity conditions (\[Faraday-BC\]) imply that the right hand sides 