The character of the supersymmetric Casimir energy

We study the supersymmetric casimir energy e susy of $$ mathcal{n}=1 $$ field theories with an r-symmetry, defined on rigid supersymmetric backgrounds s 1 ×m 3, 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 × m 3, with respect to both complex structures. the supersymmetric casimir energy 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 s 1 × s 3 to the anomaly polynomial. as further applications we compute e susy for certain secondary hopf surfaces, and discuss how the index-character also be used to compute generalized supersymmetric indices.