标题： 狄拉克方程在谱大三维流形上的拓扑结构 显示英文标题

标题： Topology of the Dirac equation on spectrally large three-manifolds

摘要： 自旋几何与正标量曲率之间的相互作用已经被广泛探讨。 在本文中，我们则关注黎曼三流形上的狄拉克算子，其中与曲率相比，上同调闭合的$1$形式上的霍奇拉普拉斯算子的谱间隙$\lambda_1^*$很大。 作为一个具体应用，我们证明了对于三环面$T^3$上任何谱大的度量，环面上平坦的$U(1)$连接的区域中（对应扭曲狄拉克算子的小的通用扰动）有核的情况，微分同胚于一个二维球面。 虽然该结果仅涉及线性算子，但其证明依赖于塞伯格-威滕方程的非线性分析。 这是在扭量自旋$^c$三流形$(Y,\mathfrak{s})$上具有大谱间隙$\lambda_1^*$的单极 Floer 同调背景下的横截性更一般理解的结果。 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 $Y$. 显示英文摘要

摘要： 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 $\lambda_1^*$ 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 $\lambda_1^*$. 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 $Y$.