• DOI:10.1103/PhysRevD.105.105008
• Corpus ID: 246411202

Gauge anomalies of finite groups

@article{Gripaios2022GaugeAO,
  title={Gauge anomalies of finite groups},
  author={Ben Gripaios},
  journal={Physical Review D},
  year={2022}
}

• B. Gripaios
• Published 27 January 2022
• Mathematics
• Physical Review D

Finite symmetry groups are ubiquitous in physics, e.g. for stabilizing particles such as the proton or dark matter, or for explaining the patterns of masses and mixings of quarks and leptons. The lore of quantum gravity says that they should be gauged and the lore of gauge symmetry says that they should be free of anomalies that can arise when they act on chiral fermions, at least if they are to be linearly realized in vacuo. The study of such anomalies was pioneered by Graham Ross in collaboration with Luis Ibañez [1], who studied cyclic groups by embedding them as subgroups of a spontaneously broken U(1) (see also [2]). The extension to an arbitrary finite abelian group follows immediately, since such a group is isomorphic to a product of cyclic groups, but the general case of groups that are not necessarily abelian is still poorly understood. This is unfortunate, since groups with irreducible representations whose dimensions exceed one, which is a sine qua non for applications to flavour physics, are necessarily non-abelian. Here we show that one can perform a complete analysis of the anomalies of an arbitrary finite group using the theory of characters. To give a flavour of the power of this approach, consider the following situation: given a group and a representation of it, there must exist a unique largest subgroup that is anomaly-free with respect to that representation. By considering all representations, one generates a list of possible anomaly-free subgroups. As the examples considered at the end of this Letter show, this list can be found using the theory of characters with a line or two of menial algebra. (To match representations with subgroups in the list takes another line or two.) Let us begin by recalling the current state of the art, but phrasing things in a way which is both mathematically kosher and as general as possible. Suppose the spacetime dimension is 4 and that the fermions (all with the same chirality) carry the complex representation ρ of G. An argument [3] along the lines of Fujikawa’s [4], shows that the transformation corresponding to g ∈ G is free of a mixed anomaly between G and gravity iff.