Strings on an orbifold with background gauge fields

I study the symmetry breaking of string theories by the introduction of abackground gauge field in the framework of orbifold compactification based on aLorentzian even self-dual lattice. The condition of modular invariance in the presence of Wilson lines is derived by constructing one-loop partition functions explicitly. The extended gauge symmetry obtained by the Lorentzian toroidal compactification is broken by the Wilson lines consistently with modular invariance. I give the example of the three-generation model with the ''observable'' gauge symmetry of SU(3) x SU(3) x U(1)'s