Threshold Corrections to Gauge Couplings in Orbifold Compactifications

TL;DR

This work shows that threshold corrections to gauge couplings in toroidal orbifolds with generic six-torus lattices are governed by congruence-subgroup automorphic forms rather than the full modular symmetry, a consequence of non-splitting lattice structure. It develops a general formalism focusing on N=2 orbifold sectors and derives explicit Δ_a formulas for Z4, Z6, and Z8 models, each with distinct symmetry groups (e.g., Γ_0(2), Γ^0(3)) and eta-function based moduli dependences. The holomorphic (heavy-string) piece of the threshold, essential for duality anomaly cancellation, cannot be deduced by symmetry arguments alone in this setting, affecting gauge coupling unification through modified boundary conditions and new unification-scale expressions M_X. Overall, the paper links lattice geometry, modular-subgroup automorphy, and spectrum to the precise moduli dependence of thresholds, with implications for phenomenology and model building in string-derived effective theories.

Abstract

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.