The Z_2-valued spectral flow of a symmetric family of Toeplitz operators
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915133944496128 |
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| author | Braverman, Maxim Sadegh, Ahmad Reza Haj Saeedi |
| author_facet | Braverman, Maxim Sadegh, Ahmad Reza Haj Saeedi |
| contents | 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 analog of the Atiyah-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15534 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Z_2-valued spectral flow of a symmetric family of Toeplitz operators Braverman, Maxim Sadegh, Ahmad Reza Haj Saeedi Differential Geometry 58J20, 58J22, 19K56, 32T15, 58Z05 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 analog of the Atiyah-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. |
| title | The Z_2-valued spectral flow of a symmetric family of Toeplitz operators |
| topic | Differential Geometry 58J20, 58J22, 19K56, 32T15, 58Z05 |
| url | https://arxiv.org/abs/2409.15534 |