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Twisted Index Theory on Good Orbifolds, II: Fractional Quantum Numbers

Marcolli, Matilde and Mathai, Varghese (2001) Twisted Index Theory on Good Orbifolds, II: Fractional Quantum Numbers. Communications in Mathematical Physics, 217 (1). pp. 55-87. ISSN 0010-3616. doi:10.1007/s002200000351.

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This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to 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, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. 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. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers.

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Alternate Title:Twisted higher index theory on good orbifolds, II: fractional quantum numbers
Additional Information:© 2001 Springer-Verlag. Received: 4 November 1999; Accepted: 22 September 2000. Communicated by A. Connes. We thank J. Bellissard for his encouragement and for some useful comments. The second author thanks A. Carey and K. Hannabuss for some helpful comments concerning the Sect. 4. The first author is partially supported by NSF grant DMS-9802480. Research by the second author is supported by the Australian Research Council. The second author acknowledges that this work was completed in part for the Clay Mathematical Institute.
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Record Number:CaltechAUTHORS:20111104-093842954
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Official Citation:Marcolli, M. & Mathai, V. Commun. Math. Phys. (2001) 217: 55.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27625
Deposited By: Tony Diaz
Deposited On:04 Nov 2011 19:02
Last Modified:09 Nov 2021 16:50

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