Coarse Mayer-Vietoris Sequence and Bulk-Edge Correspondence

Abstract: Roe C*-algebras are models of topological insulators. The bulk invariants are given by their K-theory. The bulk-edge correspondence claims that non-trivial bulk invariants lead to the existence of edge states. In a recent preprint, Ludewig and Thiang constructed an integer-valued map to compute the bulk invariants and proved that the spectral gap closes if the map is non-zero. They used a partition of the space, but showed also that the map does not depend much on the partition.

In this talk, I will show that the map defined by Ludewig and Thiang agrees with a composition of boundary maps in coarse Mayer-Vietoris sequences. The insensitivity to partitions is a consequence of the naturality of the coarse Mayer-Vietoris sequence. The boundary maps in coarse Mayer-Vietoris sequences describe the bulk-edge correspondence. These results can be generalised to higher-dimensional spaces.

K-theory and homologyoperator algebras

Audience: researchers in the topic

Göttingen Seminar Noncommutative Geometry

Series comments: This seminar will focus on C*-hulls of *-algebras with respect to a class of integrable representations, roughly following my 2017 paper Representations of *-algebras, C*-hulls, local-global principle, and induction. It takes place in a hybrid format during the winter term 2022-3. The lectures will usually be streamed and recorded. The streams are available at the URL streaming.math.uni-goettingen.de where you have to choose the right room (Sitzungszimmer) I plan that someone in the audience opens the big blue button room on some electronic device in the lecture room to relay questions and comments from outside speakers.