Seiberg-Witten theory and the geometric structure $\mathbf{R} \times H^2$

Abstract

The moduli space of the solutions to the monopole equations over an oriented closed 3-manifold M carrying the geometric structure $\mathbf{R} \times H^2$ is studied. Solving the parallel spinor equation, we obtain an explicit solution to the monopole equations. The moduli space consists of a single point with the Seiberg-Witten invariant $\pm 1$. Further, the (anti-)canonical line bundle $K_M^\pm 1$ gives a monopole class of M.