We prove the existence of singular harmonic $\mathbf{Z}_2$ spinors on $3$‑manifolds with $b_1 \gt 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg–Witten equation with two spinors. The existence of singular harmonic $\mathbf{Z}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce [Joy17] regarding Donaldson and Segal’s proposal for counting $G_2$-instantons [DS11].

We develop some techniques to study the adiabatic limiting behaviour of Calabi–Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general hypotheses. Assuming a result in pluripotential theory, we prove a uniform fibre diameter bound for a Lefschetz K3 fibred Calabi–Yau $3$‑fold, which reduces the study of the collapsing metric to a locally non-collapsed situation, and we identify the Gromov–Hausdorff limit of the rescaled neighbourhood of the singular fibre.

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and establish a relation with enumerative geometry.