Spectral Flow of Toeplitz operators and bulk-edge correspondence

Abstract: We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain in $\mathbb{C}^n$. This result is similar to the Boutet de Monvel's computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. Finally, we show that the bulk-boundary correspondence in a tight-binding model of topological insulators is a special case of our results. At the end I will explain KK-theoretical extension of the main theaorem to families of Toeplitz operators parametrized by an arbitrary compact manifold, obtained by Koen van den Dungen.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( video )

Noncommutative Geometry in NYC

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