  We consider a vertex model in d dimensions characterized by lines which run
in a preferred direction. We show that this vertex model is soluble if the
weights of vertices with intersecting lines are given by a free-fermion
condition, and that a fugacity -1 is associated to each loop of lines. The
solution is obtained by mapping the model into a dimer problem and by
evaluating a Pfaffian. We also determine the critical point and the singular
behavior of the free energy.
