Alan Carey (ANU, Canberra)
Witten introduced the notion of real spectral flow in the late 1980s in connection with anomalies.
His idea is quite correct but there is a gap in the argument that may be filled in using an extension of
the analytic approach of John Phillips from the case of spectral flow along paths in the self adjoint Fredholm operators in the complex case to flow along paths in the space of skew adjoint Fredholm operators on a real Hilbert space.
The recent examples motivating a revival of interest in real spectral flow come from condensed matter physics. The intersection of the mathematics with the physics occurs due to the way the classifying spaces for real K theory were introduced by Atiyah-Singer in the late 1960s.