$\theta$ still is present in the constitutive relations. In fact, the additional contributions of a globally constant $\theta$ to each of the modified Maxwell equations, (\[GaussE\]) and (\[Ampere\]) cancel due to the homogeneous equations.

Now we return to the problem of the allowed values of $
\theta$ to describe topological insulators. $U(1)$ gauge theories with nonzero $\theta$ ($\theta$-ED) exhibit an $SL(2, 
\mathbb{Z})$ duality group which strongly constrains the quantum physics [@Witten3; @Karch]. This group is obtained by repeated applications of the $S$ and $T$ generators of electric-magnetic duality. The $S$ generator is associated with the invariance of classical Maxwell equations in matter (\[MaxEqMatter\]) (supplemented with magnetic charge and current densities) under duality rotations. Only the special case of a duality transformation by $\pi / 2$ is consistent with the requirement that the electric charge and the magnetic charge are quantized.

The aforementioned rescaling symmetry $\theta \to \theta +C$ would allow to set $\theta$ to zero at the classical level. Quantum mechanically, however, given that for properly quantized electric and magnetic fluxes $S _{\theta} / \hbar$ is an integer multiple of $\theta$, only $C = 2 \pi n$ for integer $n$ is an allowed symmetry, otherwise non-trivial contributions to the path 