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 → X π→ B, if the tangent bundle along the fibers TB X admits a spin structure, then H⁺(X) also admits a spin structure, where H⁺(X) is the vector bundle consisting of self-dual harmonic 2-forms. In this paper, we show that TB X ⊕ π^ H⁺(X) admits a canonical spin structure. The proof is carried out by canonically constructing a lifting $O(1)$-gerbe for the spin structure on H⁺(X) using the families Seiberg--Witten equations, starting from a lifting $O(1)$-gerbe for the spin structure on TB X.