The character of the supersymmetric Casimir energy

Abstract

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.

47 pages; v2: footnote 6 added, formula (5.29) changed, Section 6 moved to Appendix A

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