Anomalies of non-Abelian finite groups via cobordism

• Published in: The journal of high energy physics Vol. 2022; no. 9; pp. 147 - 45
• Main Authors: Davighi, Joe, Gripaios, Ben, Lohitsiri, Nakarin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 19-09-2022; Springer Nature B.V; SpringerOpen
• Subjects: Anomalies; Anomalies in Field and String Theories; Classical and Quantum Gravitation; Discrete Symmetries; Elementary Particles; Fermions; Flavor (particle physics); Group theory; High energy physics; Leptons; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Symmetry; Theories of Flavour; Topological Field Theories; Discrete Symmetries; Topological Field Theories; Theories of Flavour; Anomalies in Field and String Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP09(2022)147

A
bstract
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
,
𝔽
3
).
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
,
𝔽
3
) symmetry is anomalous, but that the anomaly can be cancelled by enlarging the symmetry to a ℤ
/
3 extension of SL(2
,
𝔽
3
).