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In the following we will ﬁrst restrict ourselves to helicities that correspond to at most the three-dimensional representation of either one of the SU (2) factors. Hence we have only (3, 3), (3, 2), (2, 3), (3, 1) or (1, 3) representations, as well as the lower-dimensional ones. When a supermultiplet contains (3, 2) or (2, 3) representations, we insist that it will also contain a single (3, 3) representation, because gravitini without a graviton are not expected to give rise to a consistent interacting ﬁeld theory.
21 eψ¯µ Γµνρ (Dν (ω) + 12 gΓν )ψρ . 32 Unity from Duality: Gravity, Gauge Theory and Strings Hence we are dealing with a D-dimensional Einstein space. The maximally symmetric solution of this equation is an anti-de Sitter space, whose Riemann curvature equals Rµνab = 2g 2 eµ[a eν b] . 14) This solution leaves all supersymmetries intact just as ﬂat Minkowski space does. One can verify this directly by considering the supersymmetry variation of the gravitino ﬁeld and by requiring that it vanishes in the bosonic background.
With 8 nonvanishing supercharges we are dealing with an 8-dimensional Cliﬀord algebra, whose irreducible representation space corresponds to the bosonic and fermionic states, each transforming according to a chiral spinor representation. Hence we are dealing with three 8-dimensional representations of SO(8), which are inequivalent. One is the representation to which we assign the supercharges, which we will denote by 8s ; to the other two, denoted as the 8v and 8c representations, we assign the bosonic and fermionic states, respectively.