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arXiv · 2404.12573

A gerbe-like construction in gauge theory

Mitsuyoshi Adachi
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Abstract

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 $T_B \mathbb{X}$.

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Mitsuyoshi Adachi. 2025-05-22. A gerbe-like construction in gauge theory. https://arxiv.org/abs/2404.12573

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