Atiyah--Singer Index Theorem for Non-Hermitian Dirac Operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911613746937856 |
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| author | da Silva, João Pedro Breveglieri Vassilevich, Dmitri |
| author_facet | da Silva, João Pedro Breveglieri Vassilevich, Dmitri |
| contents | If an operator $H$ anticommutes with a chirality operator $Γ_*$ such that $Γ_*^2=1$, the null space of $H$ can be decomposed in a direct sum of two spaces having positive and negative chiralities, respectively. When both spaces are finite dimensional, one can define an index, $\mathrm{Ind}(Γ_*,H)$, as the difference of dimensions of these two spaces. The key issue is whether $\mathrm{Ind}(Γ_*,H)$ is topologically protected, i.e., whether it remains constant under smooth variations of the parameters and background fields entering $H$. For Hermitian Dirac operators, topological protection of the index is guaranteed by the Atiyah--Singer theorem. In this paper, by using the heat kernel methods, we show that $\mathrm{Ind}(Γ_*,H)$ is topologically protected also for non-hermitian operators $H$ as long as they are diagonalizable and satisfy some ellipticity conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13358 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Atiyah--Singer Index Theorem for Non-Hermitian Dirac Operators da Silva, João Pedro Breveglieri Vassilevich, Dmitri High Energy Physics - Theory Mesoscale and Nanoscale Physics Quantum Physics If an operator $H$ anticommutes with a chirality operator $Γ_*$ such that $Γ_*^2=1$, the null space of $H$ can be decomposed in a direct sum of two spaces having positive and negative chiralities, respectively. When both spaces are finite dimensional, one can define an index, $\mathrm{Ind}(Γ_*,H)$, as the difference of dimensions of these two spaces. The key issue is whether $\mathrm{Ind}(Γ_*,H)$ is topologically protected, i.e., whether it remains constant under smooth variations of the parameters and background fields entering $H$. For Hermitian Dirac operators, topological protection of the index is guaranteed by the Atiyah--Singer theorem. In this paper, by using the heat kernel methods, we show that $\mathrm{Ind}(Γ_*,H)$ is topologically protected also for non-hermitian operators $H$ as long as they are diagonalizable and satisfy some ellipticity conditions. |
| title | Atiyah--Singer Index Theorem for Non-Hermitian Dirac Operators |
| topic | High Energy Physics - Theory Mesoscale and Nanoscale Physics Quantum Physics |
| url | https://arxiv.org/abs/2604.13358 |