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${}_{*}(mathbb{R})$ 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}. ] 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 $mathbb{Z}/ 2mathbb{Z}$-valued spectral flow in the study of topological phases of matter.