We consider supersymmetric gauge theories coupled to hyper multiplets on five and six dimensional orbifolds and determine the bulk and local fixed point renormalizations of the gauge couplings. We infer from a component analysis that the hyper multiplet does not induce renormalization of the brane gauge couplings on the five dimensional orbifold S 1 /Z 2 . This is not due to supersymmetry, since the bosonic and fermionic contributions cancel separately. We extend this investigation to T 2 /Z N orbifolds using supergraph techniques in six dimensions. On general Z N orbifolds the gauge couplings do renormalize at the fixed points, except for the Z 2 fixed points of an even ordered orbifold. To cancel the bulk one-loop divergences a dimension six higher derivative operator is needed, in addition to the standard bulk gauge kinetic term.

We investigate the structure of local anomalies of heterotic E 8 × E 8 ′ theory on T 6 /Z 4 . We show that the untwisted states lead to anomalies in ten, six and four dimensions. At each of the six dimensional fixed spaces of this orbifold the twisted states ensure, that the anomalies factorize separately. As some of these twisted states live on T 2 /Z 2 , they give rise to four dimensional anomalies as well. At all four dimensional fixed points at worst a single Abelian anomaly can arise. Since the anomalies in all these dimensions factorize in a universal way, they can be canceled simultaneously. In addition, we show that for all U(1) factors at the four dimensional fixed points at least logarithmically divergent Fayet-Ilopoulos tadpoles are generated.

We consider supersymmetric non-Abelian gauge theories coupled to hyper multiplets on five and six dimensional orbifolds, S 1 /Z 2 and T 2 /Z N , respectively. We compute the bulk and local fixed point renormalizations of the gauge couplings. To this end we extend supergraph techniques to these orbifolds by defining orbifold compatible delta functions. We develop their properties in detail. To cancel the bulk one-loop divergences the bulk gauge kinetic terms and dimension six higher derivative operators are required. The gauge couplings renormalize at the Z N fixed points due to vector multiplet self interactions; the hyper multiplet renormalizes only non-Z 2 fixed points. In 6D the Wess-ZuminoWitten term and a higher derivative analogue have to renormalize in the bulk as well to preserve 6D gauge invariance.