  We study a model whose dynamics is determined by a Maxwell Lagrangian coupled
to a complex scalar and a Dirac fermion field, in an $AdS_3$ black hole
background. Our study is performed within the context of the Euclidean
formalism, in terms of an effective action $S^{eff}$ that results from
integrating out the fermion field. In particular, $S^{eff}$ includes an induced
parity breaking part which reduces, in the weak coupling limit, to Chern-Simons
terms for both the gauge and spin connections, with temperature dependent
coefficients. We find numerically the effective action minimum and, applying
the AdS/CFT correspondence, we discuss the properties of the dual quantum field
theory defined on the boundary. We show that, in contrast with what happens in
the absence of fermions, the system does not undergo a phase transition at any
finite temperature.
