Abstract
We compute the one-loop gauge couplings in six-dimensional non-Abelian gauge theories on the ${T}^{2}/{Z}_{2}$ orbifold with general GUT breaking boundary conditions. For concreteness, we apply the obtained general formulas to the gauge coupling running in a 6D $SO(10)$ orbifold GUT where the GUT group is broken down to the standard model gauge group up to an extra $U(1)$. We find that the one-loop corrections depend on the parity matrices encoding the orbifold boundary conditions as well as the volume and shape moduli of extra dimensions. When the $U(1)$ is broken by the VEV of bulk singlets, the accompanying extra color triplets also affect the unification of the gauge couplings. In this case, the $B\ensuremath{-}L$ breaking scale compatible with the gauge coupling unification is sensitive to the change of the compactification scales.
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