Anomalies on Orbifolds

TL;DR

This work analyzes chiral anomalies in a 5D theory on the S1/Z2 orbifold with chiral boundary conditions, showing that the full 5D current divergence is localized on the orbifold fixed planes and is independent of the detailed bulk KK-mode wavefunctions. The authors derive the key result $\partial_C J^C(x,x_4) = \frac{1}{2}[\delta(x_4) + \delta(x_4-L)] {\cal Q}$ with ${\cal Q} = \frac{1}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}$, tying the 5D anomaly to the familiar 4D chiral anomaly while demonstrating that 4D anomaly cancellation suffices for 5D consistency. The analysis combines a KK decomposition, a matrix formulation of the 5D current, and a careful evaluation of the chiral trace, connecting the bulk result to a boundary (Callan–Harvey) Chern–Simons mechanism in the appropriate limits. Overall, the paper clarifies how orbifold-induced anomalies factorize between bulk and fixed points, providing a practical criterion for anomaly cancellation in higher-dimensional orbifold theories.

Abstract

We discuss the form of the chiral anomaly on an S1/Z2 orbifold with chiral boundary conditions. We find that the 4-divergence of the higher-dimensional current evaluated at a given point in the extra dimension is proportional to the probability of finding the chiral zero mode there. Nevertheless the anomaly, appropriately defined as the five dimensional divergence of the current, lives entirely on the orbifold fixed planes and is independent of the shape of the zero mode. Therefore long distance four dimensional anomaly cancellation ensures the consistency of the higher dimensional orbifold theory.