Skew localizer and \(\mathbb{Z}_2\)-flows for real index pairings. (English) Zbl 1477.19006
Index theorems for \(\mathbb{Z}_{2}\)-invariants have been studied by various researchers. There are 16 types secondary \(\mathbb{Z}_{2}\)-valued index pairings. In this paper under review, the authors study a systematic approach to computing \(\mathbb{Z}_{2}\)-valued index pairings and constructs the so-called skew localizer in the all 16 cases. More precisely, the first result in the paper is the index formulas for \(\mathbb{Z}_{2}\)-valued index pairings by using either an orientation flow or a half-spectral flow. The second result is constructing the skew localizer for a pairing stemming from an unbounded Fredholm module and calculating the \(\mathbb{Z}_{2}\)-valued index by using the sign of the Pfaffian and that of the determinant of the skew localizer.
The two main results for one of the 16 cases are provided in Section 1. Section 1 also contains a concrete application to the Kitaev chain, which is an example of a one-dimensional topological insulators.
The two main results for one of the 16 cases are provided in Section 1. Section 1 also contains a concrete application to the Kitaev chain, which is an example of a one-dimensional topological insulators.
Reviewer: Tatsuki Seto (Tokyo)
MSC:
| 19K56 | Index theory |
| 58J30 | Spectral flows |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
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