Vacuum polarization in $d+1\mathrm{/}2$ dimensions

We study the main properties of the one-loop vacuum polarization function (${\mathrm{\Pi }}_{\alpha \beta }$) for spinor $QED$ in `$d+1\mathrm{/}2$ dimensions’, i.e., with fields defined on $\mathcal{M}\subset {\mathbb{R}}^{d+1}$ such that ${\mathcal M} = {(x_0,…,x_d) | x_{d}\geq 0 }$, with bag-like boundary conditions on the boundary $\mathrm{\partial }\mathcal{M}=\{(x_0,\mathrm{.}\mathrm{.}\mathrm{.},x_d)\mathrm{\mid }{x}_d=0\}$. We obtain an exact expression for the induced current due to an external constant electric field normal to the boundary. We show that, for the particular case of 2+1 dimensions, there is a transverse component for the induced current, which is localized on a region close to $\mathrm{\partial }\mathcal{M}$. This current is a parity breaking effect purely due to the boundary.