Skew localizer and $\mathbb{Z}_2$-flows for real index pairings

Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real structure, namely projections can be real, quaternionic, even or odd Lagrangian and unitaries can be real, quaternionic, symmetric or anti-symmetric. There are $64$ such real index pairings of real $K$-theory with real $K$-homology. For $16$ of them, the Noether index of the pairing vanishes, but there is a secondary $\mathbb{Z}_2$-valued invariant. The first set of results provides index formulas expressing each of these $16$ $\mathbb{Z}_2$-valued pairings as either an orientation flow or a half-spectral flow. The second and main set of results constructs the skew localizer for a pairing stemming from a Fredholm module and shows that the $\mathbb{Z}_2$-invariant can be computed as the sign of its Pfaffian and in $8$ of the cases as the sign of the determinant of its off-diagonal entry. This is of relevance for the numerical computation of invariants of topological insulators.