Classification of conformality models based on non-Abelian orbifolds

A systematic analysis is presented of compactifications of the type IIB superstring on ${\mathrm{AdS}}_5\times {\mathrm{S}}^5/\Gamma$ where $\Gamma$ is a non-Abelian discrete group. Every possible $\Gamma$ with order $g<~31$ is considered. 45 such groups exist but a majority cannot yield chiral fermions due to a certain theorem that is proved. The lowest order to embrace the non-SUSY standard $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}\times U\left(1\right)\mathrm{}$ model with three chiral families is $\Gamma {=D}_4\times Z_3,$ with $g=24;$ this is the only successful model found in the search. The consequent uniqueness of the successful model arises primarily from the scalar sector, prescribed by the construction, being sufficient to allow the correct symmetry breakdown.