The character of the supersymmetric Casimir energy

We study the supersymmetric Casimir energy Esusy of N = 1 field theories with an R-symmetry, defined on rigid supersymmetric backgrounds S1 x M3, using a Hamiltonian formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to Esusy arise from certain twisted holo- morphic modes on R x M3, with respect to both complex structures. The super- symmetric 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 Esusy on S1 x S3 to the anomaly polynomial. As further applications we compute Esusy for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.