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

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 KO* (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 spectral flow = 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 Z/2Z-valued spectral flow in the study of topological phases of matter.