Abstract
We present the basic $\mathcal{N} =1$ superconformal field theories in four-dimensional de Sitter space-time, namely the non-abelian super Yang-Mills theory and the chiral multiplet theory with gauge interactions or cubic superpotential. These theories have eight supercharges and are invariant under the full $SO(4,2)$ group of conformal symmetries, which includes the de Sitter isometry group $SO(4,1)$ as a subgroup. The theories are ghost-free and the anti-commutator $\sum_\alpha\{Q_\alpha, Q^{\alpha\dagger}\}$ is positive. SUSY Ward identities uniquely select the Bunch-Davies vacuum state. This vacuum state is invariant under superconformal transformations, despite the fact that de Sitter space has non-zero Hawking temperature. The $\mathcal{N}=1$ theories are classically invariant under the $SU(2,2|1)$ superconformal group, but this symmetry is broken by radiative corrections. However, no such difficulty is expected in the $\mathcal{N}=4$ theory, which is presented in appendix B.
Highlights
In this note we consider a modest exception to the above no-go results: global superconformal theories in dS4
We present the basic N = 1 superconformal field theories in four-dimensional de Sitter space-time, namely the non-abelian super Yang-Mills theory and the chiral multiplet theory with gauge interactions or cubic superpotential
SUSY Ward identities uniquely select the Bunch-Davies vacuum state. This vacuum state is invariant under superconformal transformations, despite the fact that de Sitter space has non-zero Hawking temperature
Summary
In a general supersymmetric field theory, the commutator of two SUSY transformations with spinor parameters , has the structure (on any field Φ(x)):. The spinor bilinears Kμ = ̄ γμ satisfy These bilinears are conformal Killing vectors, since the right hand side is proportional to gμν ; the “weights”. This is one clue that superconformal SUSY is needed in de Sitter space. Spinors that satisfy (2.4) cannot be used, because their bilinears are complex, and their appearance in (2.1) is not consistent with reality properties of the fields This problem can be bypassed by working with conformal Killing spinors (CKS) whose definition, namely γμDν γν Dμ. In which Kν is a future directed time-like CKV Such conformal Killing charges are conserved formally, i.e. if boundary conditions permit, and if the stress tensor is conserved and traceless.
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