Dirac Fermions in Non-trivial Topology Black Hole Backgrounds

We discuss the behaviour of the Dirac fermions in a general spherically symmetric black hole background with a non-trivial topology of the event horizon. Both massive and massless cases are taken into account. The analytical studies of intermediate and late-time behaviour of massive Dirac hair in the background of a black hole with a global monopole and dilaton black hole pierced by a cosmic string will be conducted. It was revealed that in the case of a global monopole swallowed by a static black hole the intermediate late-time behaviour depends on the mass of the Dirac field, the multiple number of the wave mode and the global monopole parameter. The late-time behaviour is quite independent of these factors and has the decay rate proportional to $t^{-5/6}$. As far as the black hole pierced by a cosmic string is concerned the intermediate late-time behaviour depends only on the hair mass and the multipole number of the wave mode while the late-time behaviour dependence is the same as in the previous case. The main modification stems from the topology of the $S^2$ sphere pierced by a cosmic string. This factor modifies the eigenvalues of the Dirac operator acting on the transverse manifold.