2021/01/22 by Nora Doll, Doll, Nora, Hermann Schulz‐Baldes +1
Computer Science · Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Mathematical Physics (math-ph) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2101.09226
openalex publication_date 2021/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℤ2-valued invariant. The first set of results provides index formulas expressing each of these 16 ℤ2-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 ℤ2-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.