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

In this article 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 KO${}_{\ast }(\mathbb{R})$ 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 $$\text{spectral flow}=\text{Fredholm index}\mathrm{.}$$ That is, we show how the KO–valued spectral flow relates to a KO-valued index by proving a Robbin-Salamon type result. The Kasparov product is also used to establish a spectral flow $=$ Fredholm index result at the level of bivariant K-theory. We explain how our results incorporate previous applications of $$\mathbb{Z}$/ 2$\mathbb{Z}$$-valued spectral flow in the study of topological phases of matter.