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

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

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1-52
Number of pages52
JournalJournal of Topology and Analysis
Publication statusAccepted/In press - 2020


  • Clifford algebra
  • Fredholm index
  • K- and KK-theory
  • Spectral flow

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology


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