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Twisted higher index theory on good orbifolds and fractional quantum numbers

Marcolli, Matilde and Mathai, Varghese (1998) Twisted higher index theory on good orbifolds and fractional quantum numbers. . (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20170713-093723319

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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.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/math/9803051arXivDiscussion Paper
Additional Information:(Submitted on 12 Mar 1998) The second author thanks A. Carey and K. Hannabus for some clarifying comments concerning section 6.
Subject Keywords:Fractional quantum numbers, Quantum Hall Effect, hyperbolic space, orbifolds, C∗- algebras, K-theory, cyclic cohomology, Fuchsian groups, Harper operator, Baum-Connes conjecture.
Classification Code:1991 Mathematics Subject Classification. Primary: 58G11, 58G18 and 58G25
Record Number:CaltechAUTHORS:20170713-093723319
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170713-093723319
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79072
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 Jul 2017 16:47
Last Modified:13 Jul 2017 16:47

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