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"Comments on the classification of the finite subgroups of SU(3)", Patrick Otto Ludl, 2011

Reviewed October 13, 2023

Citation: Ludl, Patrick Otto. "Comments on the classification of the finite subgroups of SU (3)." Journal of Physics A: Mathematical and Theoretical 44.25 (2011): 255204.

Web: https://arxiv.org/abs/1101.2308

Tags: Mathematical

This paper gives a nice exposition of the classification of finite subgroups of SU(3), with an nice new description of the largest families. Many discrete gauge theories arise from symmetry breaking of a larger continuous gauge theory. The standard model of particle physics tells us that the universe is a U(1) x SU(2) x SU(3), corresponding to electromagnetism, the weak force, and the strong force, respectively. Thus, we should expect most discrete gauge theories to arise from finite subgroups of U(1), SU(2), or SU(3). This has led to significat interest in gauge theories with such gauge groups. For SU(3) we have the following references, for instance:

> Bhanot, G., and C. Rebbi. "Monte Carlo simulations of lattice models with finite subgroups of SU (3) as gauge groups." Physical Review D 24.12 (1981): 3319.
> Grosse, Harald, and H. K├╝hnelt. "Phase structure of lattice gauge theories for non-abelian subgroups of SU (3)." Physics Letters B 101.1-2 (1981): 77-81.

The classification of the subgroups of SU(3) helps us understand which gauge theories we should expect to be able to compute in the lab. One thing that's really nice: A5 is a subgroup of SU(3)! Thus, using a non-simple group for topological quantum computation is perhaps not totally out of reach...

There are only finitely many finite subgroups of SU(2). Symmetry breaking of gauge theories in this context is much more well understood. See, for instance, the lovely review article:

> Hartung, Tobias, et al. "Digitizing SU (2) gauge fields and what to look out for when doing so." arXiv preprint arXiv:2212.09496 (2022).