Phases with modular ground states for symmetry breaking by rank 3 and rank 2 antisymmetric tensor scalars

Working with explicit examples given by the 56 representation in SU(8)SU(8), and the 10 representation in SU(5)SU(5), we show that symmetry breaking of a group G⊃G1×G2G⊃G1×G2 by a scalar in a rank three or two antisymmetric tensor representation leads to a number of distinct modular ground states. For these broken symmetry phases, the ground state is periodic in an integer divisor p of N  , where N>0N>0 is the absolute value of the nonzero U(1)U(1) generator of the scalar component Φ   that is a singlet under the simple subgroups G1G1 and G2G2. Ground state expectations of fractional powers Φp/NΦp/N provide order parameters that distinguish the different phases. For the case of period p=1p=1, this reduces to the usual Higgs mechanism, but for divisors N≥p>1N≥p>1 of N it leads to a modular ground state with periodicity p  , implementing a discrete Abelian symmetry group U(1)/ZpU(1)/Zp. This observation may allow new approaches to grand unification and family unification.