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.

Toeplitz Extensions in Noncommutative Topology and Mathematical Physics

“…In [11], the authors replaced crossed products C * -algebras by groupoid C *algebras. While crossed products of commutative C * -algebras are naturally an example of groupoid C * -algebras, the advantage of this more general setting lies in the possibility of studying systems without translational symmetries, like those resulting from non-periodic R d -actions and the above mentioned Delone sets.…”

“…Quite remarkably, in the one-dimensional case, the groupoid C * -algebra admits an alternative description as Cuntz-Pimsner algebra of a self-Morita equivalence bimodule (cf. [11,Subsection 2.3]). The map implementing the bulk-edge correspondence is realised as a Kasparov product with the unbounded representative for the class of the extension (23), as constructed in [26] (see also [2]).…”

Untwisting twisted spectral triples

“…The context is typically that of a dynamical system on a metric measure space, for which the dynamics does not preserve the metric nor the measure. The phenomenon has been examined in detail for Cuntz-Krieger algebras [26, Section 5 and 6], Cuntz-Pimsner algebras of vector bundles [28,Section 4], group C * -algebras and boundary crossed products of groups of Möbius transformations [44,Section 4] and Delone sets with finite local complexity [7,Section 5]. For later reference in this context, we include the following general construction.…”

“…Proof. That (A, ℓ 2 (V t ), D af,t , σ) is a twisted spectral triple follows from noting that for a ∈ A 7) which is bounded because for any a ∈ A 0 and k ∈ Z,…”

“…Our first general result states that functional calculus using the C 1 -function sgnlog : R → R, x → sgn(x) log(1 + |x|), can be used to turn both twisted and higher order spectral triples into ordinary spectral triples without changing the K-homology class (for details on the twisted case, see Theorem 1 below). This logarithmic dampening has been used before in specific examples [7,24,26,30,44,46], and in Section 1 of the present paper we formalise the procedure.…”