Spectral flow measures the number of eigenvalue crossings of continuous paths of Fredholm operators (which have finite-dimensional kernels). Atiyah, Patodi and Singer first defined spectral flow as a way to study index theory for odd dimensional manifolds.

Recent applications of index theory to topological phases, where anti-linear symmetries may occur, have motivated the study of spectral flow on real Hilbert spaces. The paper develops a theory of spectral flow for skew-adjoint Fredholm operators on real Hilbert spaces, which are a classifying space for KO-theory. Our construction applies to bounded and unbounded Fredholm operators, possibly with additional Clifford symmetries, and generalises all previous notions of analytic spectral flow. The results are also relevant to free-fermionic topological phases, where the topological obstruction to two symmetric Hamiltonians having the same strong topological phase can be exactly measured by the KO-valued spectral flow.

Chris Bourne, Alan L. Carey, Matthias Lesch, Adam Rennie
"The KO-valued spectral flow for skew-adjoint Fredholm operators"
doi: 10.1142/S1793525320500557
arXiv: 1907.04981