The character of the supersymmetric Casimir energy

Journal of High Energy Physics, Aug 2016

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 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 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 may also be used to compute generalized supersymmetric indices.

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Dario Martelli, James Sparks. The character of the supersymmetric Casimir energy, Journal of High Energy Physics, 2016, 117, DOI: 10.1007/JHEP08(2016)117