  Chiral topological insulator (AIII-class) with Landau levels is constructed
based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the
chiral symmetry of the AIII-class topological insulator in the context of
non-commutative geometry of 4D quantum Hall effect. The many-body groundstate
wavefunction is explicitly derived as a $(l,l,l-1)$ Laughlin-Halperin type
wavefunction with unique $K$-matrix structure. Fundamental excitation is
identified with anyonic string-like object with fractional charge
${1}/({1+2(l-1)^2})$. The Hall effect of the chiral topological insulators
turns out be a color version of Hall effect, which exhibits a dual property of
the Hall and spin-Hall effects.

Strongly Correlated Electrons;High Energy Physics - Theory;Quantum Physics