2d Fu-Kane-Mele invariant as Wess-Zumino action of the sewing matrix

Abstract

We show that the Fu-Kane-Mele invariant of the 2d time-reversal-invariant crystalline insulators is equal to the properly normalized Wess-Zumino action of the so-called sewing matrix field defined on the Brillouin torus. Applied to 3d, the result permits a direct proof of the known relation between the strong Fu-Kane-Mele invariant and the Chern-Simons action of the non-Abelian Berry connection on the bundle of valence states.