We classify the quantum numbers of the extra $U(1)$ symmetries contained in $E_6$. In particular, we categorize the cases with rational charges and present the full list of models which arise from the chains of the maximal subgroups of $E_6$. As an a
pplication, the classification allows us to determine all embeddings of the Standard Model fermions in all possible decompositions of the fundamental representation of $E_6$ under its maximal subgroups. From this we find alternative chains of subgroups for Grand Unified Theories. We show how many of the known models including some new ones appear in alternative breaking patterns. We also use low energy constraints coming from parity-violating asymmetry measurements and atomic parity non-conservation to set limits on the $E_6$ motivated parameter space for a $Z$ boson mass of~1.2~TeV. We include projected limits for the present and upcoming QWEAK, MOLLER and SOLID experiments.
By comparing successful quark-gluon vertex interaction models with the corresponding interaction extracted from lattice-QCD data on the quarks propagator, we identify common qualitative features which could be important to tune future interaction mod
els beyond the rainbow ladder approximation. Clearly, a quantitative comparison is conceptually not simple, but qualitatively the results suggest that a realistic interaction should be relatively broad with a strong support at about $0.4-0.6$~GeV and infrared-finite.
