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Unification Yang-Mills Groups and Representations with CP as a Gauge Symmetry

We investigate more generally the possible unification Yang-Mills groups $G_{YM}$ and representations with CP as a gauge symmetry. Besides the possible Yang-Mills groups $E_8$, $E_7$, $SO(2n+1)$, $SO(4n)$, $SP(2n)$, $G_2$ or $F_4$ (or a product of them) which only allow self-contragredient representations, we present other unification groups $G_{YM}$ and representations which may allow CP as a gauge symmetry. These include especially $SU(N)$ containing Weyl fermions and their CP-conjugates from low-energy spectra in a basic irreducible representation (IR). Such an example is the 496-dimensional basic IR (on antisymmetric tensors of rank two) of $SU(32)$ containing $SO(32)$ as a subgroup in the adjoint IR, or $SU(248)$ in a fundamental IR containing $E_8$ as a subgroup in the adjoint IR. Our consideration also leads to the construction of a physical operator (CP) intrincially as an inner automorphism of order higher than two for the unification group. We have also generalized the possible groups as unification $G_{YM}$ to include non-semisimple Lie groups with CP arising as a gauge symmetry. In this case with $U(1)$ ideals in the $G_{YM}$, we found that the $U_Y(1)$ for weak hypercharge in the standard model or a $U(1)$ gauge symmetry at low energies in general is traceless. Possible relevance to superstring theory is also briefly discussed. We expect that our results may open new alternatives for unified model building, especially with deeper or more generalized understanding of anomaly-free theories.