The unification in an $$ \hat{su}{(8)}_{k_U=1} $$ affine Lie algebra

A bstract A flavor-unified theory based on the simple Lie algebra of $$ \mathfrak{su} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>su</mml:mi> </mml:math> (8) was previously proposed to generate the observed three-generational Standard Model quark/lepton mass hierarchies and the Cabibbo-Kobayashi-Maskawa mixing pattern due to their non-universal symmetry properties. A level-1 affine Lie algebra of $$ \hat{su}{(8)}_{k_U=1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>su</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:msub> <mml:mfenced> <mml:mn>8</mml:mn> </mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>U</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> with the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 supersymmetric extension is found to unify three gauge couplings through the maximally symmetry breaking pattern.