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

## TL;DR

This paper develops a KO-valued spectral flow for skew-adjoint Fredholm operators, linking it to KO-theory and the Fredholm index, with implications for topological phases of matter.

## Contribution

It introduces a KO-valued spectral flow for skew-adjoint Fredholm operators, providing axiomatic properties and connecting it to the Fredholm index via K-theory.

## Key findings

- Establishes KO-valued spectral flow as a generalization of classical spectral flow.
- Proves spectral flow equals Fredholm index in the KO-theory setting.
- Connects spectral flow to topological phases of matter applications.

## Abstract

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 $\mathbb{Z}/ 2\mathbb{Z}$-valued spectral flow in the study of topological phases of matter.