Notes on topological insulators

Abstract

This paper is a survey of the [Formula: see text]-valued invariant of topological insulators used in condensed matter physics. The [Formula: see text]-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The [Formula: see text] invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.