The KO-valued spectral flow for skew-adjoint Fredholm operators
| Autor: | Alan L. Carey, Adam Rennie, Matthias Lesch, Chris Bourne |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
010102 general mathematics Clifford algebra Hilbert space Skew Spectral flow FOS: Physical sciences K-Theory and Homology (math.KT) Mathematical Physics (math-ph) Clifford module 01 natural sciences Functional Analysis (math.FA) Mathematics - Functional Analysis symbols.namesake Flow (mathematics) 0103 physical sciences Mathematics - K-Theory and Homology symbols FOS: Mathematics 010307 mathematical physics Geometry and Topology 0101 mathematics Analysis Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1907.04981 |
| Popis: | 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. Comment: v2: 47 pages, applications to physics expanded |
| Databáze: | OpenAIRE |
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