Signatures for $J$-hermitians and $J$-unitaries on Krein spaces with Real structures

TL;DR

This paper studies the topological invariants of $J$-hermitian and $J$-unitary operators on Krein spaces, introducing a global signature and secondary invariants, especially in the presence of Real structures, linking to classifying spaces.

Contribution

It introduces a homotopy-invariant global Krein signature for $J$-hermitian operators and extends the analysis to operators with Real symmetry, connecting to Atiyah and Singer classifying spaces.

Findings

Global Krein signature is a homotopy invariant.

Sets of $J$-hermitian Fredholm operators with Real symmetry retract to classifying spaces.

Secondary $ extbf{Z}_2$-invariants label connected components.

Abstract

For $J$-hermitian operators on a Krein space $(\mathcal{K},J)$ satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of $J$-hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary $\mathbb{Z}_2$-invariants are introduced to label their connected components. Related invariants are also analyzed for $J$-unitary operators.