Monopole Floer homology and invariant theta characteristics

Abstract

We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. Consider an automorphism $\varphi$ of a compact Riemann surface $\Sigma$ with quotient $\mathbb{P}^1$. There is a natural correspondence between theta characteristics $L$ on $\Sigma$ which are invariant under $\varphi$ and self-conjugate spin$^c$ structures $\mathfrak{s}_L$ on the mapping torus $M_{\varphi}$ of $\varphi$. We show that the monopole Floer homology groups of $(M_{\varphi},\mathfrak{s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of $\varphi$ on $H^0(L)$, the space of holomorphic sections of $L$. Decategorifying our computation, we also obtain that the dimension of $H^0(L)$ equals the Reidemeister-Turaev torsion of $(M_{\varphi},\mathfrak{s}_L)$. Finally, we combine our description with the Atiyah-Bott $G$-spin theorem to provide explicit computations of the Floer homology groups for all automorphisms $\varphi$ of prime order in terms of ramification data.