  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.
