Quantitative K-Theory Related to Spin Chern Numbers

TL;DR

This paper investigates indices on pairs of almost commuting unitary matrices across different symmetry classes, providing quantitative bounds and validating a computational method for spin Chern numbers.

Contribution

It offers new quantitative results on the indices' behavior and confirms a computational method for spin Chern numbers in specific matrix classes.

Findings

Determines commutator norms guaranteeing index definition

Identifies conditions for indices to be equal

Validates a computational method for spin Chern numbers

Abstract

We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant.