A gerbe-like construction in gauge theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918029426688000 |
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| author | Adachi, Mitsuyoshi |
| author_facet | Adachi, Mitsuyoshi |
| contents | In 2022 Baraglia and Konno showed the following: for a smooth family of a homotopy $K3$ surface $X \to \mathbb{X} \stackrelπ{\to} B$, if the tangent bundle along the fibers $T_B \mathbb{X}$ admits a spin structure, then $\mathcal{H}^+(\mathbb{X})$ also admits a spin structure, where $\mathcal{H}^+(\mathbb{X})$ is the vector bundle consisting of self-dual harmonic 2-forms. In this paper, we show that $T_B \mathbb{X} \oplus π^\ast \mathcal{H}^+(\mathbb{X})$ admits a canonical spin structure. The proof is carried out by canonically constructing a lifting $O(1)$-gerbe for the spin structure on $\mathcal{H}^+(\mathbb{X})$ using the families Seiberg--Witten equations, starting from a lifting $O(1)$-gerbe for the spin structure on $T_B \mathbb{X}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A gerbe-like construction in gauge theory Adachi, Mitsuyoshi Differential Geometry Geometric Topology In 2022 Baraglia and Konno showed the following: for a smooth family of a homotopy $K3$ surface $X \to \mathbb{X} \stackrelπ{\to} B$, if the tangent bundle along the fibers $T_B \mathbb{X}$ admits a spin structure, then $\mathcal{H}^+(\mathbb{X})$ also admits a spin structure, where $\mathcal{H}^+(\mathbb{X})$ is the vector bundle consisting of self-dual harmonic 2-forms. In this paper, we show that $T_B \mathbb{X} \oplus π^\ast \mathcal{H}^+(\mathbb{X})$ admits a canonical spin structure. The proof is carried out by canonically constructing a lifting $O(1)$-gerbe for the spin structure on $\mathcal{H}^+(\mathbb{X})$ using the families Seiberg--Witten equations, starting from a lifting $O(1)$-gerbe for the spin structure on $T_B \mathbb{X}$. |
| title | A gerbe-like construction in gauge theory |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2404.12573 |