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arXiv · 1907.04981

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

Chris Bourne·Alan L. Carey·Matthias Lesch·Adam Rennie
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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.

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Chris Bourne, Alan L. Carey, Matthias Lesch, Adam Rennie. 2019-07-11. The KO-valued spectral flow for skew-adjoint Fredholm operators. https://doi.org/10.1142/s1793525320500557

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