Towards the fractional Quantum Hall Effect: a noncommutative geometry perspective

In this joint work with Varghese Mathai we propose an approach to the fractional Quantum Hall Effect within the framework of noncommutative geometry, using hyperbolic geometry to simulate electron-electron interactions. By computing the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, we determine the range of values of the Connes-Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in our case is that the Connes-Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. The set of possible fractions obtained in this model can be compared with recently available experimental data.