Abstract
Abstract Global gauge anomalies in six dimensions associated with non-trivial homotopy groups $\pi_6(G)$ for $G=SU(2)$, $SU(3)$, and $G_2$ have been computed and utilized in the past. From the modern bordism point of view of anomalies, however, they come from the bordism groups $\Omega^\text{spin}_7(BG)$, which are in fact trivial and therefore preclude their existence. Instead, it was noticed that a proper treatment of the 6D Green–Schwarz mechanism reproduces the same anomaly cancellation conditions derived from $\pi_6(G)$. In this paper, we revisit and clarify the relation between these two different approaches.
Highlights
Introduction and summaryGiven a gauge theory, its gauge anomaly must be canceled as a whole in order for the theory to be consistent
One notable subtlety here is that, even if the perturbative anomaly cancellation is achieved, the theory may still suffer from global anomaly, coming from a global gauge transformation corresponding to non-trivial elements of πd(G), where d is the spacetime dimension and G is the gauge group
This modifies the bordism group to be considered in the classification of anomalies, and there is a global gauge anomaly associated to π6(G), which defines a non-trivial element in the modified bordism group
Summary
Its gauge anomaly must be canceled as a whole in order for the theory to be consistent. One is from the absence of global gauge anomalies associated to non-trivial elements of π6(G), and the other is from the proper consideration of the Green-Schwarz term cancelling the fermion anomaly The aim of this rather technical paper is to reconcile these two points of view. This modifies the bordism group to be considered in the classification of anomalies, and there is a global gauge anomaly associated to π6(G), which defines a non-trivial element in the modified bordism group. Note added: When this paper is almost completed, the authors learned that there is an upcoming work by Davighi and Lohitsiri [DL20c], which has a large overlap with this paper
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