  Vacuum polarization effects in the cosmic string background are considered.
We find that a current is induced in the vacuum of the quantized massive scalar
field and that the current circulates around the string which is generalized to
a $(d-2)$-brane in locally flat $(d+1)$-dimensional space-time. As a
consequence of the Maxwell equation, a magnetic field strength is also induced
in the vacuum and is directed along the cosmic string. The dependence of the
current and the field strength on the string flux and tension is
comprehensively analyzed. Both the current and the field strength are
holomorphic functions of the space dimension, decreasing exponentially with the
distance from the string. In the case of $d=3$ we show that, due to the vacuum
polarization, the cosmic string is enclosed in a tube of the magnetic flux
lines if the mass of the quantized field is less than the inverse of the
transverse size of the string core.
