Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 077, 25 pages      arXiv:1302.0349
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Quantitative $K$-Theory Related to Spin Chern Numbers

Terry A. Loring
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

Received January 15, 2014, in final form July 13, 2014; Published online July 19, 2014

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.

Key words: $K$-theory; $C^{*}$-algebras; matrices.

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