Non-Abelian Finite Gauge Theories

TL;DR

This paper explores orbifolds of ${\

Contribution

It constructs a catalogue of finite chiral ${\cal N}=1$ theories from orbifolds and proposes a McKay-type correspondence linking gauge theories to conformal models.

Findings

Matter content aligns with quiver theories for SU(2) orbifolds

Catalogue of finite ${\cal N}=1$ theories for SU(3) orbifolds

Conjectured connection between gauge theories and affine SU(3) modular invariants

Abstract

We study orbifolds of ${\cal N} = 4$ U(n) super-Yang-Mills theory given by discrete subgroups of SU(2) and SU(3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU(2), we have shown how the matter content agrees with current quiver theories and have offered a possible explanation. In the case of SU(3) we have constructed a catalogue of candidates for finite (chiral) ${\cal N}=1$ theories, giving the gauge group and matter content. Finally, we conjecture a McKay-type correspondence for Gorenstein singularities in dimension 3 with modular invariants of WZW conformal models. This implies a connection between a class of finite ${\cal N}=1$ supersymmetric gauge theories in four dimensions and the classification of affine SU(3) modular invariant partition functions in two dimensions.