Application of semifinite index theory to weak topological phases

TL;DR

This paper introduces a semifinite index theory approach to describe weak topological phases in insulators, providing an alternative proof for index formulas and exploring real invariants and bulk-boundary correspondence.

Contribution

It offers a new semifinite index theory framework for weak topological phases, complementing previous KK-theory methods and enhancing understanding of topological invariants.

Findings

Semifinite index theory effectively describes weak topological invariants.

Provides an alternative proof for index formulas in complex topological phases.

Briefly discusses real invariants and bulk-boundary correspondence.

Abstract

Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov's $KK$-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considered.