  Recently, generalizations of quantum Hall effects (QHE) have been made from
2D to 4D and 8D by considering their mathematical frameworks within complex
(C), quaternion (H) and octonion (O) compact (gauge) Lie algebra domains. Just
as QHE in two-dimensional electron gases can be understood in terms of Chern
number topological invariants that belong to the first Chern class, QHE in 4D
and 8D can be understood in terms of Chern number topological invariants that
belong to the 2nd and 4th Chern classes. It has been shown that 2D QHE
phenomena are related to topologically-ordered ground states of Josephson
junction arrays (JJAs), which map onto an Abelian gauge theory with a periodic
topological term that describes charge-vortex coupling. In these 2D JJAs,
magnetic point defects and Cooper pair electric charges are dual to one another
via electric-magnetic duality (Montonen-Olive). This leads to a quantum phase
transition between phase-coherent superconductor and dual phase-incoherent
superinsulator ground states, at a "self-dual" critical point. In this article,
a framework for topological-ordering of Bose-Einstein condensates is extended
to consider four-dimensional quaternion ordered systems that are related to 4D
QHE. This is accomplished with the incorporation of a non-Abelian topological
term that describes coupling between third homotopy group point defects (as
generalized magnetic vortices) and Cooper pair-like charges. Point defects
belonging to the third homotopy group are dual to charge excitations, and this
leads to the manifestation of a quantum phase transition between
orientationally-ordered and orientationally-disordered ground states at a
"self-dual" critical point. The frustrated ground state in the vicinity of this
"self-dual" critical point, are characterized by global topological invariants
belonging to the 2nd Chern class.
