Topology of the Dirac equation on spectrally large three-manifolds

The interaction between spin geometry and positive scalar curvature has been extensively explored. In this paper, we instead focus on Dirac operators on Riemannian three-manifolds for which the spectral gap ${\lambda }_1^{\ast }$ of the Hodge Laplacian on coexact $1$-forms is large compared to the curvature. As a concrete application, we show that for any spectrally large metric on the three-torus $T^3$, the locus in the torus of flat $U(1)$-connections where (a small generic pertubation of) the corresponding twisted Dirac operator has kernel is diffeomorphic to a two-sphere. While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin${}^c$ three-manifold $(Y,\mathfrak{s})$ with a large spectral gap ${\lambda }_1^{\ast }$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak{s})$ in terms of the topology of the family of Dirac operators parametrized by the torus of flat $U(1)$-connections on $Y$.