SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

3.3. The Haag—Lopuszanski—Sohnius theoremP. van Nieuwenhuizen, Supergravity 283One can, to a large extent, determine the possible extra symmetries in supersymmetric theories byanalyzing the commutator algebra. We stress that in what follows we do not analyze the gauge algebra,but only field-independent super algebras. Thus, for example, central charges which vanish on shell arenot discovered in this way. Nevertheless, it is amazing how much can be deduced from (seemingly)rather weak assumptions.Consider a super algebra which is an extension of the super Poincaré algebra. It contains thePoincaré algebra itself, and N conserved spin ~charges (i = 1, N)[Q”j, Fm] = 0, [O”i, Mmn] = ~ (1)In addition, the following (anti) commutators are present, using two-component notation (see appendix)(QA. QB} = ~I1~Z, (2){QA. ~l} = o~ Om)F ~ (QAEAB)* (3)[QA. B,] — (b,)/0A 1. (4)The Z, are a subset of the internal charges B, and, according to (2) these Z, commute with P0,. Wechoose these Z, to be a basis for the Z,1 = [1,’Z,,hence there are in general fewer Z, than Z,1. In fact, itfollows quite generally from the Coleman—Mandula theorem that all B, are Lorentz scalars whichcommute with F,,,. This theorem states that the most general bosonic extension of the Poincaré algebrais a direct sum of the Poincaré algebra, a semisimple Lie algebra and an Abelian algebra. Since theColeman—Mandula theorem holds only for symmetries of the S-matrix, the only symmetries discussed inthis subsection are those of the S-matrix. Since the Z, carry no Lorentz indices, the invariant tensor ~ARmust appear in (2) (in four-component notation this means that central charges appear as U, + iy5V,)and it follows that Ii’,, is antisymmetric in (ii). We choose all B, anti-Hermitean in order to preserveunitarity.From now on we will use the Jacobi identities to analyze the algebra further. We will denote theseidentities for three generators A, B and C by (ABC). Let us consider first (0, 0, F) and note that in (3)one cannot add a term with Mmn on the right-hand side. (In a de-Sitter space, [0”, P] is non-vanishingand one does indeed find terms with Mmn. Here we consider non-Sitter spaces where [Fm, F,,] = 0 andhence [QaP}0) -From (OBB) it follows that the b, are a representation of B,. From (BOO) it follows that(b,),” = _((b,)ki)* ~ b, anti-Hermitean (5)assuming that F,,, and B, (and hence Z,) are anti-Hermitean.The notation (Q~~,)* = ~ will now be justified. If a tensor T, transforms under a given group asMITJ let us define a tensor T’ to transform as (M.~.T)i~T1.(Note that M_l.T is again a representation ofthe same algebra of which M is a representation.) One could define two more kinds of tensors,transforming under M* and under Mlt. (These four sets of tensors correspond in two-componentformalism to XA,f, XA and x’~.)Since according to (5) the group elements generated by b, are unitary,