Orbifold models and modular transformation

The orbifold models of the heterotic string are constructed on the quotient spaces of generalized tori by translational and rotational discrete symmetries. In order to obtain the consistent orbifold models, the conditions of the modular invariance are derived from a one-loop vacuum amplitude. Z3 orbifold models satisfying such conditions are searched systematically. It is shown that there are infinite possible models with N=2 supersymmetry. Among these models, two examples having E6 and E7 gauge groups are discussed. The orbifold models with N=1 supersymmetry are also discussed in detail. It is shown that there are only five consistent models in the class of these models based on E8 x E8' heterotic string in which the extra six-dimensional torus and the E8 x E8' maximal torus are modded out by the rotational and the translational Z3 symmetries respectively. (orig.)