Index of Dirac operators and classification of topological insulators

Abstract

Real and complex Clifford bundles and Dirac operators defined on them are considered. By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard. Through the relations between K-theory groups, Grothendieck groups and symmetric spaces, the periodic table of topological insulators and superconductors is obtained. This gives the result that the periodic table of real and complex topological phases is originated from the Clifford chessboard and index theorems.