Anomalies of non-Abelian finite groups via cobordism

Abstract

Abstract We use cobordism theory to analyse anomalies of finite non-abelian symmetries in 4 spacetime dimensions. By applying the method of ‘anomaly interplay’, which uses functoriality of cobordism and naturality of the η-invariant to relate anomalies in a group of interest to anomalies in other (finite or compact Lie) groups, we derive the anomaly for every representation in many examples motivated by flavour physics, including S 3, A 4, Q 8, and SL(2, 𝔽3). In the case of finite abelian groups, it is well known that anomalies can be ‘truncated’ in a way that has no effect on low-energy physics, by means of a group extension. We extend this idea to non-abelian symmetries. We show, for example, that a system with A 4 symmetry can be rendered anomaly-free, with only one-third as many fermions as naïvely required, by passing to a larger symmetry. As another example, we find that a well-known model of quark and lepton masses utilising the SL(2, 𝔽3) symmetry is anomalous, but that the anomaly can be cancelled by enlarging the symmetry to a ℤ/3 extension of SL(2, 𝔽3).

Keywords

• Anomalies in Field and String Theories
• Discrete Symmetries
• Theories of Flavour
• Topological Field Theories