  We study the supersymmetric Casimir energy $E_\mathrm{susy}$ of
$\mathcal{N}=1$ field theories with an R-symmetry, defined on rigid
supersymmetric backgrounds $S^1\times M_3$, using a Hamiltonian formalism.
These backgrounds admit an ambi-Hermitian geometry, and we show that the net
contributions to $E_\mathrm{susy}$ arise from certain twisted holomorphic modes
on $\mathbb{R}\times M_3$, with respect to both complex structures. The
supersymmetric Casimir energy may then be identified as a limit of an
index-character that counts these modes. In particular this explains a recent
observation relating $E_\mathrm{susy}$ on $S^1\times S^3$ to the anomaly
polynomial. As further applications we compute $E_\mathrm{susy}$ for certain
secondary Hopf surfaces, and discuss how the index-character may also be used
to compute generalized supersymmetric indices.
