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

Authors

Chris Bourne
Alan L. Carey, University of WollongongFollow
Matthias Lesch
Adam C. Rennie, University of WollongongFollow

RIS ID

146527

Publication Details

Bourne, C., Carey, A., Lesch, M. & Rennie, A. (2020). The KO-valued spectral flow for skew-adjoint Fredholm operators. Journal of Topology and Analysis, 1-52.

Abstract

© 2020 World Scientific Publishing Co. Pte Ltd. All rights reserved. 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.