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

We investigate more generally the possible uni cation Yang-Mills groups GYM and representations with CP as a gauge symmetry. Besides the possible Yang-Mills groups E8, E7, SO(2n + 1), SO(4n), SP (2n), G2 or F4 (or a product of them) which only allow selfcontragredient representations, we present other uni cation groups GYM 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 E8 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 uni cation group. We have also generalized the possible groups as uni cation GYM to e-mail: [email protected], Present address: P.O.Box 17660, Jackson State University, Jackson, MS 39217 Where work supported by a DOE fellowship include non-semisimple Lie groups with CP arising as a gauge symmetry. In this case with U(1) ideals in the GYM , we found that the UY (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 brie y discussed. We expect that our results may open new alternatives for uni ed model building, especially with deeper or more generalized understanding of anomaly-free theories.