Non-factorizable branes on the torus

This work discusses string compactifications on the torus with optional Z4×Z4 or Z2×Z2 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 in H1(T2,Z)3⊂H3(T6,Z), branes studied here can be in generic homology classes, can have arbitrary position and Wilson line, have full complex structure respectively Kähler 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.