  In this work, we propose a modern view of the integer spin simple currents
which have played a central role in discrete torsion. We reintroduce them as
nonanomalous composite particles constructed from $Z_{N}$ parafermionic field
theories. These composite particles have an analogy with the Cooper pair in the
Bardeen-Cooper-Schrieffer theory and can be interpreted as a typical example of
anyon condensation. Based on these $Z_{N}$ anomaly free composite particles, we
propose a systematic construction of the cylinder partition function of $Z_{N}$
fractional quantum Hall effects (FQHEs). One can expect realizations of a class
of general topological ordered systems by breaking the bulk-edge correspondence
of the bosonic parts of these FQH models. We also give a brief overview of
various phenomena in contemporary condensed matter physics, such as $SU(N)$
Haldane conjecture, general gapless and gapped topological order with respect
to the quantum anomaly defined by charges of these simple currents and bulk and
boundary renormalization group flow. Moreover, we point out an analogy between
these FQHEs and 2d quantum gravities coupled to matter, and propose a $Z_{N}$
generalization of supersymmetry known as "fractional supersymmetry" in the
composite parafermionic theory and study its analogy with quark confinement.
Our analysis gives a simple but general understanding of the contemporary
physics of topological phases in the form of the partition functions derived
from the operator formalism.
