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Threshold Corrections to Gauge Couplings in Orbifold Compactifications

We derive the moduli dependent threshold corrections to gauge couplings in toroidal orbifold compactifications. The underlying six dimensional torus lattice of the heterotic string theory is not assumed ---as in previous calculations--- to decompose into a direct sum of a four--dimensional and a two--dimensional sublattice, with the latter lying in a plane left fixed by a set of orbifold twists. In this more general case the threshold corrections are no longer automorphic functions of the modular group, but of certain congruence subgroups of the modular group. These groups can also be obtained by studying the massless spectrum; moreover they have larger classes of automorphic functions. As a consequence the threshold corrections cannot be uniquely determined by symmetry considerations and certain boundary conditions at special points in the moduli space, as was claimed in previous publications.