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Using D-brane configurations on the K3 surface, we give six dimensional typeIIA 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 engineertwo different stringy realizations. The vertical line presents a realization interms of D2 and D6-branes wrapping the K3 surface. The horizontal one isassociated with hierarchical stringy descriptions obtained from a quiver gaugetheory living on a stack of D4-branes wrapping intersecting 2-spheres embeddedin the K3 surface with deformed singularities. These geometries are classifiedby 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 occurin the quiver gauge theory living on intersecting 2-spheres arranged as affineDynkin diagrams. Stringy realizations of QHE can be done only for the finiteand indefinite geometries. In particular, the finite Lie algebras give modelswith fractional filling fractions, while the indefinite ones classify modelswith negative filling fractions which can be associated with the physics ofholes in the graphene.

Engineering of Quantum Hall Effect from Type IIA String Theory on The K3 Surface