Abstract

The modular transformation property of the vacuum amplitudes of the closed bosonic string theories compactified on orbifolds are investigated for both abelian and non-abelian cases. It is shown that the two-dimensional Zn-orbifolds (n=2, 3, 4, 6), which are the typical of abelian orbifolds, give the fully modular invariant vacuum amplitudes. On the other hand, the su(3) and o(6) Weyl orbifolds, which are the simplest examples of non-abelian orbifolds, give only partial modular invariance due to the existence of the subspace which escapes from the action of some non-trivial elements of the Weyl group.

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