Fractional Quantum Hall Filling Factors from String Theory using Toric Geometry

Using toric Cartan matrices as abelian gauge charges, we present a class of stringy fractional quantum Hall effect (FQHE) producing some recent experimental observed filling factor values. More precisely, we derive the corresponding Chern-Simons type models from M-theory compactified on four complex dimensional hyper-K\"{a}hler manifolds X^4. These manifolds, which are viewed as target spaces of a particular N=4 sigma model in two dimensions, are identified with the cotangent bundles over intersecting 2-dimensional toric varieties V_i^2 according to toric Cartan matrices. Exploring results of string dualities, the presented FQHE can be obtained from D6-banes wrapping on such intersecting toric varieties interacting with R-R gauge fields. This string theory realization provides a geometric interpretation of the filling factors in terms of toric and Euler characteristic topological data of the compactified geometry. Concretely, explicit bilayer models are worked out in some details.