  The Green function (GF) method is used to analyze the boundary effects
produced by a Chern Simons (CS) extension to electrodynamics. We consider the
electromagnetic field coupled to a $\theta$ term that is piecewise constant in
different regions of space, separated by a common interface $\Sigma$, the
$\theta$ boundary, model which we will refer to as $\theta$ electrodynamics
($\theta$ ED). This model provides a correct low energy effective action for
describing topological insulators (TI). In this work we construct the static GF
in $\theta$ ED for different geometrical configurations of the $\theta$
boundary, namely: planar, spherical and cylindrical $\theta$ interfaces. Also
we adapt the standard Green theorem to include the effects of the $\theta$
boundary. These are the most important results of our work, since they allow to
obtain the corresponding static electric and magnetic fields for arbitrary
sources and arbitrary boundary conditions in the given geometries. Also, the
method provides a well defined starting point for either analytical or
numerical approximations in the cases where the exact analytical calculations
are not possible. Explicit solutions for simple cases in each of the
aforementioned geometries for $\theta$ boundaries are provided. The adapted
Green theorem is illustrated by studying the problem of a point like electric
charge interacting with a planar TI with prescribed boundary conditions. Our
generalization, when particularized to specific cases, is successfully compared
with previously reported results, most of which have been obtained by using the
methods of images.

Mesoscale and Nanoscale Physics;High Energy Physics - Phenomenology;High Energy Physics - Theory;Optics