A gerbe-like construction in gauge theory

In 2022 Baraglia and Konno showed the following: for a smooth family of a homotopy $K3$ surface $X \to \mathbb{X} \stackrel{\pi}{\to} B$, 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 $T_B \mathbb{X} \oplus \pi^\ast \mathcal{H}^+(\mathbb{X})$ 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 $T_B \mathbb{X}$.