Abstract
Supersymmetric partition function of mathcal{N}=1 superconformal theories on Sβ1 × S3 is related to the superconformal index receiving contributions from short representations. The leading coefficients in the small β (high “temperature”) expansion of the index were previously related to the conformal anomaly coefficients of the theory. Assumptions underlying universality of these relations were tested only for simplest low-spin unitary multiplets. Here we consider examples of higher derivative non-unitary mathcal{N}=1 multiplets that naturally appear in the context of extended conformal supergravities and compute their superconformal index. We compare the coefficients in the small β expansion of the index with those proposed earlier for unitary multiplets and suggest some modifications that should apply universally to all types of theories. We also comment on the structure of subleading terms and the case of mathcal{N}=4 conformal supergravity.
Highlights
Unitary representations of N = 1 superconformal algebra play an important role in many aspects of supersymmetric quantum field theory
Supersymmetric partition function of N = 1 superconformal theories on Sβ1 × S3 is related to the superconformal index receiving contributions from short representations
We consider examples of higher derivative non-unitary N = 1 multiplets that naturally appear in the context of extended conformal supergravities and compute their superconformal index
Summary
Applications of non-unitary representations are less studied They appear, in particular, in extended conformal supergravities [1,2,3,4] written in terms of N = 1 multiplets. We shall compute the coefficients in the small β expansion of the index I(β) and compare with their expected expressions in terms of conformal anomaly coefficients proposed earlier on the basis of studies of unitary low-spin examples [19, 20]. In appendix E we shall repeat the computation of the superconformal index and its small β expansion for non-unitary multiplets in case of unequal fugacities which is related to supersymmetric partition function on Sβ1 × Sb3 (with Sb3 being a squashed 3-sphere). A (modified) relation between the superconformal index and the conformal anomaly coefficients may shed light on the values of the latter for superconformal higher spin theories
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