# Jacobi Identity in de Sitter Superalgebra

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In the book "Supergravity" (by D. Freedman and A. Van Proeyen) they talk about why $\mathcal{N}=1$ de-sitter superalgebra is impossible to construct (Section 12.6.1).

Basically de-Sitter algebra is a modification of flat space symmetry algebra with: $[P_μ,P_ν]=−\frac{1}{4L^2}M_{μν}$

Then if we try to embed this into a superalgebra with say: $[P_μ,Q_α]=\frac{a}{4L}(γ_μQ)_α$ with arbitrary "a" we see that the Jacobi identity involving $[P_μ,P_ν,Q_α]$is not satisfied.

Whereas in the paper "Construction of the de Sitter supergravity"(arXiv:1602.01678v2) the authors say:

Super-dS algebras do exist for even $\mathcal{N}=1$ in 4 dimensions (and for other D ≤ 6).

Which implies that having an R symmetry, somehow makes the above problem go away. My question is, how does that work out? Any reference is welcome.

Note: I tried to read W. Nahm's paper in which he classifies all the SUSY algebras. But that paper just has the list of algebras. If there's a reference that tackles the above problem in detail, it would be really appreciated.

recategorized Apr 22, 2020

arXiv:1602.01678v2 states $\mathcal{N}$ must be even.  In particular $\mathcal{N} \neq 1.$

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