Omega vs. pi, and 6d anomaly cancellation

TL;DR

This paper reviews the role of homotopy groups and bordism in understanding global gauge anomalies, showing that non-vanishing homotopy groups are not always indicative of anomalies, especially in 6d theories.

Contribution

It clarifies the relationship between homotopy groups and global anomalies, demonstrating that bordism provides a more accurate criterion for anomaly cancellation in 6d gauge theories.

Findings

Homotopy groups are not necessary or sufficient for global anomalies.

Bordism groups can vanish even when homotopy groups are non-zero.

Conditions for local anomaly cancellation differ from those for global anomalies.

Abstract

In this note we review the role of homotopy groups in determining non-perturbative (henceforth `global') gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of $\pi_d(G)$ is neither a necessary nor a sufficient condition for there being a possible global anomaly in a $d$-dimensional chiral gauge theory with gauge group $G$. To showcase the failure of sufficiency, we revisit `global anomalies' that have been previously studied in 6d gauge theories with $G=SU(2)$, $SU(3)$, or $G_2$. Even though $\pi_6(G) \neq 0$, the bordism groups $\Omega_7^\mathrm{Spin}(BG)$ vanish in all three cases, implying there are no global anomalies. In the case of $G=SU(2)$ we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.