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 πd(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 G2. Even though π6(G) ≠ 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.
Highlights
Πd(G) fails to detect certain 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
Be it local or global, always corresponds to the non-invariance of the phase of the fermionic partition function, and, a precise formula is known for how that phase varies under an arbitrary gauge transformation A → Ag, g(x) ∈ G [1]
Summary
We have seen that vanishing of the anomaly polynomial Φd+2 only guarantees that Z[A, Md] is invariant under gauge transformations that are connected to the identity. Suppose that spacetime has spherical topology (as is naturally motivated by taking a theory on flat space Rd and requiring the fields “die off” at infinite radius, allowing the point at infinity to be compactified) In this case, non-vanishing of the homotopy group πd(G) means there exists a gauge transformation g : Sd → G, [g] = 0 ∈ πd(G), that cannot be connected to the identity by successive infinitesimal gauge transformations. To work out whether there is really an anomaly, one has to analyze the spectral flow of the Dirac operator coupled to the background gauge field as one interpolates between A and Ag via a gauge field configuration in d + 1 dimensions, for example by considering At = (1 − t)A + tAg, t ∈ I := [0, 1] This spectral flow can be used to deduce whether the partition function, which recall is a regularized product of the eigenvalues of the Dirac operator, is invariant under A → Ag. If not there is a global anomaly. The fermionic partition function varies at most by a phase, as we know on general grounds
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