Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators

Abstract

We use constructive bounded Kasparov K-theory to investigate the numerical invariants stemming from the internal Kasparov products $K_i(\mathcal{A})\times K{K}^i(\mathcal{A},\mathcal{B})\to K_0(\mathcal{B})\to \mathbb{R}$, $i=0,1$, where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov $(\mathcal{A},\mathcal{B})$-cycles, the invariants are given by the pairing of K-theory of $\mathcal{B}$ with an element of the periodic cyclic cohomology of $\mathcal{B}$, which we call the generalized Connes-Chern character. When $\mathcal{A}$ is a twisted crossed product of $\mathcal{B}$ by ${\mathbb{Z}}^k$, $\mathcal{A}=\mathcal{B}{⋊}_{\xi }^{\theta }{\mathbb{Z}}^k$, we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when $\mathcal{B}=C(\Omega ){⋊}_{{\xi }^{\prime }}^{\varphi }{\mathbb{Z}}^j$, with $C(\Omega )$ the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publication.

To appear in Reviews in Mathematical Physics

References in corpus (5)

• The $K$-theoretic bulk-edge correspondence for topological insulators
• The bulk-edge correspondence for the quantum Hall effect in Kasparov theory
• Effect of Strong Disorder on 3-Dimensional Chiral Topological Insulators: Phase Diagrams, Maps of the Bulk Invariant and Existence of Topological Extended Bulk States
• Factorisation of equivariant spectral triples in unbounded KK-theory
• Intrinsic Chern-Connes Characters for Crossed Products by ${\mathbb{Z}}^d$

Cited by in corpus (10)

• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• Disorder driven phase transitions in weak AIII topological insulators
• Index theory and topological phases of aperiodic lattices
• The FKMM-invariant in low dimension
• Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices
• Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory
• Application of semifinite index theory to weak topological phases
• Spectral decimation of a self-similar version of almost Mathieu-type operators
• The cohomology invariant for class DIII topological insulators
• Mapping Chern numbers in quasi-periodic interacting spin chains