Localized anomalies in orbifold gauge theories

We apply the path-integral formalism to compute the anomalies in general orbifold gauge theories (including possible nontrivial Scherk-Schwarz boundary conditions) where a gauge group $\mathcal{G}$ is broken down to subgroups ${\mathcal{H}}_f$ at the fixed points ${y=y}_f.$ Bulk and localized anomalies, proportional to $\delta \left(y-y_f\right),$ do generically appear from matter propagating in the bulk. The anomaly zero mode that survives in the four-dimensional effective theory should be canceled by localized fermions [except possibly for mixed $U\left(1\right)\mathrm{}$ anomalies]. We examine in detail the possibility of canceling localized anomalies by the Green-Schwarz mechanism involving two- and four-forms in the bulk. The four-form can only cancel anomalies which do not survive in the 4D effective theory: they are called globally vanishing anomalies. The two-form may cancel a specific class of mixed $U\left(1\right)\mathrm{}$ anomalies. Only if these anomalies are present in the 4D theory does this mechanism spontaneously break the $U\left(1\right)\mathrm{}$ symmetry. The examples of five- and six-dimensional $Z_N$ orbifolds are considered in great detail. In five dimensions the Green-Schwarz four-form has no physical degrees of freedom and is equivalent to canceling anomalies by a Chern-Simons term. In all other cases, the Green-Schwarz forms have some physical degrees of freedom and leave some nonrenormalizable interactions in the low energy effective theory. In general, localized anomaly cancellation imposes strong constraints on model building.