464 N 8 N 8 Supergravity multiplet � 2 2 3 2 8 1 28 1 2 56 70 1 2 56 1 28 3 2 8

464 n 8 n 8 supergravity multiplet ? 2 2 3 2 8 1 28 1

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4.6.4 N = 8 • N = 8 Supergravity multiplet ( λ = 2) : ( 2 , 3 2 8 , 1 28 , 1 2 56 , 0 70 , 1 2 56 , 1 28 , 3 2 8 , 2 ) . This is TCP self conjugate. Note that each of the supergravity multiplets contains N gravitinos, which are associated to the gauge fields for local supersymmetry. The detailed story of how this works is the subject of supergravity . When N > 8 any supermultiplet necessarily contains states with 26
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helicity greater than two. Hence N > 8 is excluded for an interacting theory. As we will see, the maximum number of conserved supercharges for an interacting theory in any spacetime dimension is 32. N = 8 supergravity in four dimensions is maximally supersymmetric in this sense. The values N = 3 , 5 , 6 are also possible, though they are of less interest. The value N = 7 is equivalent to N = 8 when one forms the supergravity multiplet as the λ = 3 / 2 multiplet plus its TCP conjugate. 4.7 R symmetries R symmetries are transformations that act on the generators of the superalgebra so as to preserve the algebra. In other words, they belong to the automorphism group of the superalgebra. In general, the supercharges Q A α transform nontrivially under R symmetries. Thus, R-symmetry transformations do not commute with susy transformations . They may or may not be true symmetries of a given susy field theory – depending on details. If they aren’t, it would make more sense to call them R transformations rather than R symmetries, but the latter is the more usual usage. Another way to finesse the issue is to speak of R charges without specifying whether or not they are conserved. 4.7.1 Massive R symmetries The N -extended massive D = 4 susy algebra – without central charges – has 4 N super- charges Q A α , ¯ Q ˙ αA , which can be recast as a Clifford Algebra { Γ r , Γ s } = 2 δ rs Γ r = (Γ r ) , (4.38) with 4 N generators ( r, s = 1 , . . . , 4 N ). The way to do this to take the oscillators b αA , b αA α = 1 , 2 , A = 1 , . . . , N introduced earlier and define Γ’s as double the “real” and “imaginary” parts: b + b and ib + ib – giving 4 N Γ’s. As we already saw, there is a representation of dim 2 2 N . This means that the Γ’s can be represented by 2 2 N × 2 2 N matrices. The Clifford algebra has a natural associated SO (4 N ) algebra generated by the matrices Γ rs 1 2 r , Γ s ] r, s = 1 , 2 , . . . , 4 N . While the 2 2 N Fock space states give an irrep of the susy algebra, they give a reducible representation of Spin (4 N ). Specifically, the 2 2 N− 1 bosonic states and the 2 2 N − 1 fermionic states each give spinorial irreps of Spin (4 N ). The SO (4 N ) group has a natural action as an automorphism group of the susy algebra. Specifically, Γ r A r s Γ s , A SO (4 N ) (4.39) preserves the Clifford algebra, and hence there is a corresponding action on the Q A α ’s that preserves the susy algebra. However, this SO (4 N ) cannot be the symmetry of any field 27
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theory – it contains (bosonic) generators that change spin and would violate the CM theorem.
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