Mitsuyoshi Adachi | Searcharxiv

arXiv2026

A gerbe-like construction in gauge theory II: the case of homology tori

In the previous paper, the author showed that for a smooth family $X \to \mathbb{X} \to B$ of a homotopy $K3$ surface, the obstruction for the tangent bundle along the fibers $T_B \mathbb{X}$ to have a spin structure is canonically isomorphic to the obstruction for $\mathcal{H}^+(\mathbb{X})$, the vector bundle over $B$ consisting of self-dual harmonic 2-forms, to have a spin structure. In this paper, we show an analogous result for homology tori with odd determinant. The strategy for proof is similar to the case of homotopy $K3$ surfaces: take the determinant line bundle of the $K$-theoretic Seiberg--Witten invariant and construct an anti-linear $\mathbb{Z}/4$-action on it at the representative level. We also see that the anti-linear $\mathbb{Z}/4$-action possesses the information of the ordinary mod 2 Seiberg--Witten invariant. This recovers part of the result by Baraglia(2023) which computes the mod 2 Seiberg--Witten invariants for any closed spin 4-manifold.

math.DG↗

arXiv2025

A gerbe-like construction in gauge theory

In 2022 Baraglia and Konno showed the following: for a smooth family of a homotopy $K3$ surface $X \to \mathbb{X} \stackrelπ{\to} B$, 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 $T_B \mathbb{X}$.

math.DG↗