  This work discusses string compactifications on the torus with optional Z_4 x
Z_4 or Z_2 x Z_2 orbifold action from the perspective of matrix factorizations.
The method is brought to a level where model building on these backgrounds is
possible. Whereas branes discussed in the literature typically wrap
factorizable cycles, that is cycles which are products of 1-cycles, branes
studied here can be in generic homology classes, can have arbitrary position
and Wilson line, have full complex structure respectively Kahler moduli
dependence and can be subject to any consistent orientifold action. It is shown
how any desired D-brane can be constructed systematically. Three-point
correlators can be computed as is demonstrated at hand of an example. Their
normalization is not discussed.
