  We investigate the fermion condensate (FC) for a massive spinor field on
background of the 5-dimensional locally anti-de Sitter (AdS) spacetime with a
compact dimension and in the presence of a cosmic string carrying a magnetic
flux. The FC is decomposed into two contributions. The first one corresponds to
the geometry without compactification and the second one is induced by the
compactification. Depending on the values of the parameters, the total FC can
be either positive or negative. As a limiting case, the expression for the FC
on locally Minkowski spacetime is derived. It vanishes for a massless fermion
field and the nonzero FC on the AdS bulk in the massless case is an effect
induced by gravitation. This shows that the gravitational field may essentially
influence the parameters space for phase transitions. For a massive field the
FC diverges on the string as the inverse cube of the proper distance from the
string. In the case of a massless field, depending on the magnetic flux along
the string and planar angle deficit, the limiting value of the FC on the string
can be either finite or infinite. At large distances, the decay of the FC as a
function of the distance from the string is power law for both cases of massive
and massless fields. For a cosmic string on the Minkowski bulk and for a
massive field the decay is exponential. The topological part in the FC vanishes
on the AdS boundary. We show that the FCs coincide for the fields realizing two
inequivalent irreducible representations of the Clifford algebra. In the
special case of the zero planar angle deficit, the results presented in this
paper describe Aharonov-Bohm-type effects induced by magnetic fluxes in curved
spacetime.
