The KO-valued spectral flow for skew-adjoint Fredholm operators

Abstract

In this paper, we give a comprehensive treatment of a “Clifford module flow” along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in [Formula: see text] via the Clifford index of Atiyah–Bott–Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that [Formula: see text] That is, we show how the [Formula: see text]-valued spectral flow relates to a [Formula: see text]-valued index by proving a Robbin–Salamon type result. The Kasparov product is also used to establish a [Formula: see text] result at the level of bivariant [Formula: see text]-theory. We explain how our results incorporate previous applications of [Formula: see text]-valued spectral flow in the study of topological phases of matter.