Monopole Floer homology and invariant theta characteristics

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

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