  The zero modes of the Dirac operator in the background of center vortex gauge
field configurations in $\R^2$ and $\R^4$ are examined. If the net flux in D=2
is larger than 1 we obtain normalizable zero modes which are mainly localized
at the vortices. In D=4 quasi-normalizable zero modes exist for intersecting
flat vortex sheets with the Pontryagin index equal to 2. These zero modes are
mainly localized at the vortex intersection points, which carry a topological
charge of $\pm 1/2$. To circumvent the problem of normalizability the
space-time manifold is chosen to be the (compact) torus $\T^2$ and $\T^4$,
respectively. According to the index theorem there are normalizable zero modes
on $\T^2$ if the net flux is non-zero. These zero modes are localized at the
vortices. On $\T^4$ zero modes exist for a non-vanishing Pontryagin index. As
in $\R^4$ these zero modes are localized at the vortex intersection points.
