  The spectrum of the Dirac Hamiltonian in the background of crossing vortices
is studied. To exploit the index theorem, and in analogy to the lattice the
space-time manifold is chosen to be the four-torus $\T^4$. For sake of
simplicity we consider two idealized cases: infinitely fat and thin
transversally intersecting vortices. The time-dependent spectrum of the Dirac
Hamiltonian is calculated and in particular the influence of the vortex
crossing on the quark spectrum is investigated. For the infinitely fat
intersecting vortices it is found that zero modes of the four-dimensional Dirac
operator can be expressed in terms of those eigenspinors of the Euclidean
time-dependent Dirac Hamiltonian, which cross zero energy. For thin
intersecting vortices the time gradient of the spectral flow of the Dirac
Hamiltonian is steepest at the time at which the vortices cross each other.
