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Spectral flows of Toeplitz operators and bulk-edge correspondence. (English) Zbl 1426.58005

Summary: 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 the tight-binding model of topological insulators is a special case of our result. In “Appendix,” Koen van den Dungen reviewed the main result in the context of (unbounded) KK-theory.

MSC:

58J30 Spectral flows
32T15 Strongly pseudoconvex domains
19K56 Index theory
58J32 Boundary value problems on manifolds
58J22 Exotic index theories on manifolds
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