Index theory and topological phases of aperiodic lattices

Abstract

We examine the noncommutative index theory associated to 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.

arXiv comment52 pages, Section 1.6 added and other minor improvements. To appear in Annales Henri Poincaré

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Cited by in corpus (9)

• The Cayley transform in complex, real and graded $K$-theory
• A groupoid approach to interacting fermions
• Localised module frames and Wannier bases from groupoid Morita equivalences
• Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory
• Toeplitz extensions in noncommutative topology and mathematical physics
• Classifying the Dynamics of Architected Materials by Groupoid Methods
• C*-framework for higher-order bulk-boundary correspondences
• Robustness of topological phases on aperiodic lattices
• A Universal Chern Model on Arbitrary Triangulations