Twisted higher index theory on good orbifolds and fractional quantum numbers

Abstract

In this paper, we study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, and we apply these results to obtain qualitative results, related to generalizations of the Bethe-Sommerfeld conjecture, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher traces on K-theory, which we then apply to compute the range of values of the Hall conductance in the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in this case is that the Hall conductance again has plateaus at all energy levels belonging to any gap in the spectrum of the Hamiltonian, where it is now shown to be equal to an integral multiple of a fractional valued invariant. Moreover the set of possible denominators is finite and has been explicitly determined. It is plausible that this might shed light on the mathematical mechanism responsible for fractional quantum numbers.

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