Quark zero modes in intersecting center vortex gauge fields ∗

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 ± 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.