Lower-rank gauge groups in orbifold compactification with background gauge fields

Wilson lines play an important role in the symmetry breaking of the gauge group in orbifold compactification. Previously, Wilson lines in the Cartan subalgebra were introduced as a homo- morphism of the translation group into E8xE8, so that the rank of the gauge group was not changed under the symmetry breakdown. Recently I proposed a new method to reduce the rank of the gauge group with the use of noncommuting automorphisms associated with twisted string boundary conditions. In this paper I give a complete and more general account of our method and give various new examples with lower-rank gauge groups including SU(3)cxSU(2)LxU(1)Y without extra U(1)'s