Cyclic cohomology for graded $C^{\ast ,r}$-algebras and its pairings with van Daele $K$-theory

Abstract

We consider cycles for graded $C^{\ast ,r}$-algebras (Real $C^{\ast }$-algebras) which are compatible with the $\ast$-structure and the real structure. Their characters are cyclic cocycles. We define a Connes type pairing between such characters and elements of the van Daele $K$-groups of the $C^{\ast ,r}$-algebra and its real subalgebra. This pairing vanishes on elements of finite order. We define a second type of pairing between characters and $K$-group elements which is derived from a unital inclusion of $C^{\ast }$-algebras. It is potentially non-trivial on elements of order two and torsion valued. Such torsion valued pairings yield topological invariants for insulators. The two-dimensional Kane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are special cases of torsion valued pairings. We compute the pairings for a simple class of periodic models and establish structural results for two dimensional aperiodic models with odd time reversal invariance.

57 pages

References in corpus (4)

• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological index for periodically driven time-reversal invariant 2D systems
• The $K$-theoretic bulk-edge correspondence for topological insulators
• Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals

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• The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors
• The KO-valued spectral flow for skew-adjoint Fredholm operators
• The Cayley transform in complex, real and graded $K$-theory
• Application of semifinite index theory to weak topological phases
• Index theory and noncommutative geometry of topological insulator
• Exact sequence between real and complex bivariant K theories and application to the Z2 pairing