Infrared Variables for the $SU(3)$ Yang–Mills Field

We generalize the self-dual parameterization of the $SU(2)$ Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group $SU(3)$. We demonstrate that the duality property intrinsic to the $SU(2)$ gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products $SU(2)^{\otimes3}$.