The character of the supersymmetric Casimir energy

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