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Index Theory and Topological Phases of Aperiodic Lattices

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Abstract

We examine the non-commutative index theory associated with the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

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Acknowledgements

We thank Jean Bellissard, Magnus Goffeng, Johannes Kellendonk, Aidan Sims and Makoto Yamashita for helpful discussions. We thank the anonymous referees for their careful reading of the manuscript and valuable feedback. CB was supported by a postdoctoral fellowship for overseas researchers from The Japan Society for the Promotion of Science (No. P16728), and both authors were supported by a KAKENHI Grant-in-Aid for JSPS fellows (No. 16F16728). This work is also supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. BM gratefully acknowledges support from the Hausdorff Center for Mathematics and the Max Planck Institute for Mathematics in Bonn, Germany, as well as Tohoku University, Sendai, Japan, for its hospitality. Part of this work was carried out during the Lorentz Center programme KK-theory, Gauge Theory and Topological Phases held in Leiden, Netherlands, in March 2017. We also thank the Leibniz Universität Hannover, Germany, the Radboud University Nijmegen, Netherlands, and the Erwin Schrödinger Institute, University of Vienna, Austria, for hospitality.

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  1. WPI-Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

    C. Bourne

  2. RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

    C. Bourne

  3. Max Planck Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    B. Mesland

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Bourne, C., Mesland, B. Index Theory and Topological Phases of Aperiodic Lattices. Ann. Henri Poincaré 20, 1969–2038 (2019). https://doi.org/10.1007/s00023-019-00764-9

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