  We perform a detail study of higher dimensional quantum Hall effects and
A-class topological insulators with emphasis on their relations to
non-commutative geometry. There are two different formulations of
non-commutative geometry for higher dimensional fuzzy spheres; the ordinary
commutator formulation and quantum Nambu bracket formulation. Corresponding to
these formulations, we introduce two kinds of monopole gauge fields;
non-abelian gauge field and antisymmetric tensor gauge field, which
respectively realize the non-commutative geometry of fuzzy sphere in the lowest
Landau level. We establish connection between the two types of monopole gauge
fields through Chern-Simons term, and derive explicit form of tensor monopole
gauge fields with higher string-like singularity. The connection between two
types of monopole is applied to generalize the concept of flux attachment in
quantum Hall effect to A-class topological insulator. We propose tensor type
Chern-Simons theory as the effective field theory for membranes in A-class
topological insulators. Membranes turn out to be fractionally charged objects
and the phase entanglement mediated by tensor gauge field transforms the
membrane statistics to be anyonic. The index theorem supports the dimensional
hierarchy of A-class topological insulator. Analogies to D-brane physics of
string theory are discussed too.
