The character of the supersymmetric Casimir energy

We study the supersymmetric Casimir energy $E_{\mathrm{s}\mathrm{u}\mathrm{s}\mathrm{y}}$ 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 $E_{\mathrm{s}\mathrm{u}\mathrm{s}\mathrm{y}}$ 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 $E_{\mathrm{s}\mathrm{u}\mathrm{s}\mathrm{y}}$ on $S^1\times S^3$ to the anomaly polynomial. As further applications we compute $E_{\mathrm{s}\mathrm{u}\mathrm{s}\mathrm{y}}$ for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.