A gerbe-like construction in gauge theory
Mitsuyoshi Adachi
TL;DR
The paper shows that for a smooth family of homotopy K3 surfaces, one can canonically lift the spin geometry from the fiber tangent data to the bundle of self-dual harmonic 2-forms, via an O(1) gerbe-theoretic framework and a finite-dimensional approximation to the Seiberg–Witten map. It constructs a canonical spin structure on $T_B\mathbb{X} \oplus \pi^*\mathcal{H}^+(\mathbb{X})$ and proves the obstruction class $\alpha$ for lifting to $\mathrm{Diff}^+(X,\mathfrak{s})$ equals the second Stiefel–Whitney class of $\mathcal{H}^+(\mathbb{X})$, while ensuring functoriality under base change and stabilization. The approach unites gauge-theoretic finite-dimensional models with a geometric interpretation via Spin(3) triples and $O(1)$-gerbes, yielding a canonical, globally defined spin structure on $\mathcal{H}^+(\mathbb{X})$ even when $T_B\mathbb{X}$ lacks a spin or spin^c structure. This advances the understanding of families Seiberg–Witten theory by providing explicit, canonical geometric objects associated to the self-dual form bundle and its interaction with the base geometry.
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 $T_B \mathbb{X}$.