  Superstring models describing curved 4-dimensional magnetic flux tube
backgrounds are exactly solvable in terms of free fields. We first consider the
simplest model of this type (corresponding to `Kaluza-Klein' Melvin
background). Its 2d action has a flat but topologically non-trivial
10-dimensional target space (there is a mixing of angular coordinate of the
2-plane with an internal compact coordinate). We demonstrate that this theory
has broken supersymmetry but is perturbatively stable if the radius R of the
internal coordinate is larger than R_0=\sqrt{2\a'}. In the Green-Schwarz
formulation the supersymmetry breaking is a consequence of the presence of a
flat but non-trivial connection in the fermionic terms in the action. For R <
R_0 and the magnetic field strength parameter q > R/2\a' there appear
instabilities corresponding to tachyonic winding states. The torus partition
function Z(q,R) is finite for R > R_0 (and vanishes for qR=2n, n=integer). At
the special points qR=2n (2n+1) the model is equivalent to the free superstring
theory compactified on a circle with periodic (antiperiodic) boundary condition
for space-time fermions. Analogous results are obtained for a more general
class of static magnetic flux tube geometries including the a=1 Melvin model.
