A Monograph on the Classification of the Discrete Subgroups of SU(4)

TL;DR

This monograph resolves the problem of classifying the discrete finite subgroups of $SU(4)$ by presenting a complete classification of the discrete finite subgroups of $SL(4,\mathbb{C})$, organized into primitive, imprimitive, and intransitive families. It provides explicit matrix generators and subgroup data, and computes the character tables for the 30 exceptional primitive cases (with online availability), enabling construction of quivers and analysis of orbifold theories. The work connects to string theory, WZW models, and Gorenstein resolutions by supplying a ready-to-use catalogue of finite subgroups and their representations, including detailed lifting data to $SU(4)$. It also discusses how these groups arise from various product and tensor constructions and outlines directions for further exploration in AdS/CFT, M-theory, and higher-dimensional singularities. Overall, the paper delivers a modern, explicit framework for applying finite subgroups of $SU(4)$ to high-energy theory, geometry, and mathematical physics.

Abstract

We here present, in modern notation, the classification of the discrete finite subgroups of SU(4) as well as the character tables for the exceptional cases thereof (Cf. https://github.com/yanghuihe/SU4Subgroups). We hope this catalogue will be useful to works on string orbifold theories, quiver theories, WZW modular invariants, Gorenstein resolutions, nonlinear sigma-models as well as some recently proposed inter-connections among them.