  Using D-brane configurations on the K3 surface, we give six dimensional type
IIA stringy realizations of the Quantum Hall Effect (QHE) in 1+2 dimensions.
Based on the vertical and horizontal lines of the K3 Hodge diamond, we engineer
two different stringy realizations. The vertical line presents a realization in
terms of D2 and D6-branes wrapping the K3 surface. The horizontal one is
associated with hierarchical stringy descriptions obtained from a quiver gauge
theory living on a stack of D4-branes wrapping intersecting 2-spheres embedded
in the K3 surface with deformed singularities. These geometries are classified
by three kinds of the Kac-Moody algebras: ordinary, i.e finite dimensional,
affine and indefinite. We find that no stringy QHE in 1+2 dimensions can occur
in the quiver gauge theory living on intersecting 2-spheres arranged as affine
Dynkin diagrams. Stringy realizations of QHE can be done only for the finite
and indefinite geometries. In particular, the finite Lie algebras give models
with fractional filling fractions, while the indefinite ones classify models
with negative filling fractions which can be associated with the physics of
holes in the graphene.
