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 $\mathbb{Z}\mathrm{/}4$-action on it at the representative level. We also see that the anti-linear $\mathbb{Z}\mathrm{/}4$-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.