Supergrand exceptional unification and quark-lepton constituents

Abstract

Supersymmetric gauge multiplets (1,2), whether based on simple (_~ = 1) or extended (N > 1) sypersymmetries, associate fermions in the adjoint representation of the gauge group as supersymmetric partners to the vector gauge fields. Supporting the possibility that matter and gauge fields may be supersymmetrically unified at very high energies, we would like to examine whether the presently known quark-lepton SU 5 or S01o multiplets (3) can be accommodated among these gauge fermions. The S U 5 scheme of unification describes the chiral fields of each generation of quarks and leptons in a reducible representation 15 ~ 10 ~-7, while the S01o scheme includes these in the irreducible spinor representation 16 which decomposes like 16 = 10 § 5 ~ 1 with respect to SUs; the extra singlet being the right-chiral neutrino component. Now it is clear that the adjoint representation of SU~>5 cannot accommodate the antisymmetrie second-rank SU5 tensor 10. The adjoint representation of S0~>1o cannot contain spinor representations in its decomposition with respect to subgroups. Hence it cannot contain the 16 of SOlo or the 10 of SUs. The adjoint representat ion fo SP2~v~SU, v• U1 decomposes with respect to SU~ as (,4-~ 1-}{2} ~{2}*), where ,4 is the adjoint representation of SUN and {2} (and {2}*) is a symmetric secondrank SU~ tensor (and complex conjugate). Hence it cannot accommodate 10 of SU 5. Thus we are left with the five exceptional groups (~). Certainly G~ and F 4 cannot contain SU 5 or S01o. Now we shall present the decompositions of the adjoint representations of E6, E 7 and E s.