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 $SU(8)$, and the 10 representation in $SU(5)$, we show that symmetry breaking of a group ${\cal G}\supset {\cal G}_1 \times {\cal G}_2$ 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 $N>0$ is the absolute value of the nonzero $U(1)$ generator of the scalar component $\Phi$ that is a singlet under the simple subgroups ${\cal G}_1$ and ${\cal G}_2$. Ground state expectations of fractional powers $\Phi^{p/N}$ provide order parameters that distinguish the different phases. For the case of period $p=1$, this reduces to the usual Higgs mechanism, but for divisors $N\geq p>1$ of $N$ it leads to a modular ground state with periodicity $p$, implementing a discrete Abelian symmetry group $U(1)/Z_p$. This observation may allow new approaches to grand unification and family unification. 
Stephen L. Adler
hep-th hep-ph math-ph math.MP