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.
Highlights
In recent years the technique of localization [1] has provided access to a host of exact results in supersymmetric field theories defined on certain curved backgrounds
We study the supersymmetric Casimir energy Esusy of N = 1 field theories with an R-symmetry, defined on rigid supersymmetric backgrounds S1×M3, using a Hamiltonian formalism
In this paper we have shown that the supersymmetric Casimir energy Esusy of fourdimensional N = 1 field theories defined on S1 × M3 is computed by a limit of the index-character counting holomorphic functions on the space R × M3
Summary
In recent years the technique of localization [1] has provided access to a host of exact results in supersymmetric field theories defined on certain curved backgrounds. When all orbits of ξ close, this means that M3 is a Seifert fibred three-manifold, with ξ generating the fibration On such a background, one can consider the partition function of an N = 1 theory with supersymmetric boundary conditions for the fermions. When R × M3 ∼= X \ {o} is the complement of an isolated singularity o in a Gorenstein canonical singularity X, one can elegantly solve for these unpaired modes that contribute to the supersymmetric Casimir energy These include M3 = S3, as well as M3 = L(p, 1) = S3/Zp (i.e. a Lens space), for which X = C2 and X = C2/Zp is an Ap−1 singularity, previously studied in the literature; but this construction includes many other interesting three-manifolds. We have included an appendix A, where we discuss the relation of the index-character to the supersymmetric index [9] and its generalizations
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