Euclidean dilaton black hole vortex and Dirac fermions

We considered the behavior of Dirac fermion modes in the background of a Euclidean dilaton black hole with an Abelian Higgs vortex passing through it. Fermions were coupled to the fields due to the superconducting string model. The case of nonextremal and extremal charged black holes in the theory with an arbitrary coupling constant between the dilaton field and U(1)-gauge field was considered. We elaborated the cases of zero and nonzero Dirac fermion modes. One finds evidence that the system under consideration can support fermion fields acting like a superconducting hair on a black hole in the sense that a nontrivial spinor field configuration can be carried by a Euclidean spherically symmetric charged dilaton black hole. It was revealed that the localization of Dirac fermion modes depended on the cosmic string winding number and the value of black hole surface gravity.