Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912041270247424 |
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| author | Taubes, Clifford Henry |
| author_facet | Taubes, Clifford Henry |
| contents | This paper constructs sequences of solutions to the Vafa-Witten equations with non-zero (but small) mass term on the product of a 2-dimensional torus with a Riemann surface of genus greater than 1. These are divergent sequences (modulo principle bundle automorphisms) that converge after renormalization to define an 'interesting' $\mathbb{Z}/2\mathbb{Z}$ harmonic 2-form data set. This data set consists of a non-empty, codimension 2 submanifold, a real line bundle defined on the complement of that submanifold with no extension across it, and a self-dual, harmonic 2-form with values in that line bundle that does not extend over the submanifold. Even so, the norm of this 2-form does extend as a Hölder continuous function with that submanifold being its zero locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14959 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits Taubes, Clifford Henry Differential Geometry Analysis of PDEs This paper constructs sequences of solutions to the Vafa-Witten equations with non-zero (but small) mass term on the product of a 2-dimensional torus with a Riemann surface of genus greater than 1. These are divergent sequences (modulo principle bundle automorphisms) that converge after renormalization to define an 'interesting' $\mathbb{Z}/2\mathbb{Z}$ harmonic 2-form data set. This data set consists of a non-empty, codimension 2 submanifold, a real line bundle defined on the complement of that submanifold with no extension across it, and a self-dual, harmonic 2-form with values in that line bundle that does not extend over the submanifold. Even so, the norm of this 2-form does extend as a Hölder continuous function with that submanifold being its zero locus. |
| title | Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2409.14959 |