The ${\mathbb{Z}}_2$-valued spectral flow of a symmetric family of Toeplitz operators

Abstract

We consider families $A(t)$ of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary ${\mathbb{Z}}_2$-valued spectral flow of such families. We prove an analogue of the Atiyah–Patodi–Singer–Robbin–Salamon theorem, showing that this secondary spectral flow of $A(t)$ is equal to the secondary ${\mathbb{Z}}_2$-valued index of the suspension operator $\frac{d}{dt}+A(t)$.
Applying this result, we show that the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold equals the secondary index of a certain Callias-type operator. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel’s index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf–Porta module for 2D topological insulators of type AII.