Bosonic and Fermionic Singularities in Diffeology
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918360852201472 |
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| author | Iglesias-Zemmour, Patrick |
| author_facet | Iglesias-Zemmour, Patrick |
| contents | We explore the differential geometry of the quadrant $C_2 = [0,\infty[^2$, equipped with the subset diffeology of $\mathbf{R}^2$. We show a striking dichotomy between differential forms and symmetric tensors. While differential forms on $C_2$ are simply restrictions of smooth forms on $\mathbf{R}^2$ (a "Fermionic" behavior where singularities are hidden), symmetric tensors exhibit a "Bosonic" behavior where singularities accumulate. We prove a decomposition theorem identifying exactly the singular parts: they are purely axial. Surprisingly, the mixed interaction term is forced to be regular by the symmetries of the corner. Finally, we introduce the notion of \emph{singular capacity} to quantify the order of singularity a tensor can support. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00032 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bosonic and Fermionic Singularities in Diffeology Iglesias-Zemmour, Patrick Differential Geometry Primary 58A40, Secondary 58A10, 58A32, 53A45, 58K70 We explore the differential geometry of the quadrant $C_2 = [0,\infty[^2$, equipped with the subset diffeology of $\mathbf{R}^2$. We show a striking dichotomy between differential forms and symmetric tensors. While differential forms on $C_2$ are simply restrictions of smooth forms on $\mathbf{R}^2$ (a "Fermionic" behavior where singularities are hidden), symmetric tensors exhibit a "Bosonic" behavior where singularities accumulate. We prove a decomposition theorem identifying exactly the singular parts: they are purely axial. Surprisingly, the mixed interaction term is forced to be regular by the symmetries of the corner. Finally, we introduce the notion of \emph{singular capacity} to quantify the order of singularity a tensor can support. |
| title | Bosonic and Fermionic Singularities in Diffeology |
| topic | Differential Geometry Primary 58A40, Secondary 58A10, 58A32, 53A45, 58K70 |
| url | https://arxiv.org/abs/2603.00032 |