Geometric construction of classes in van Daele's K-theory

Abstract

We describe explicit generators for the "real" K-theory of "real" spheres in van Daele's picture. Pulling these generators back along suitable maps from tori to spheres produces a family of Hamiltonians used in the physics literature on topological insulators. We compute their K-theory classes geometrically, based on wrong-way functoriality of K-theory and the geometric version of bivariant K-theory, which we extend to the "real" case.

References in corpus (8)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Chern-Simons Currents and Chiral Fermions on the Lattice
• Index pairings in presence of symmetries with applications to topological insulators
• Coarse geometry and topological phases
• Equivariant representable K-theory
• Bivariant K-theory via correspondences
• Equivariant embedding theorems and topological index maps
• The Cayley transform in complex, real and graded $K$-theory