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

areP. van Nieuwenhuizen, Supergravity 273In general we consider algebras with even (E) and odd (0) generators. The commutations relations[E 1.E1] = IkE,, [E,, 0,,] = g,,,”0,~, {0,, O~}= h,,~’E,. (1)In order that this is a consistent system, one must satisfy the Jacobi identities. Those for three evenelements are the usual Jacobi identities for ordinary Lie algebras. In addition there are these Jacobiidentities[E,,[E,,0,,]] — [E,,[E,,0,,]] = [E,, E,], 0,,] (2)[E,,{0,,,0~}]={[E,, 0,,]. 00}+{[E,, On], 0,,} (3)[0,,,{O~,O~}]= [{O,,,O~},O~]+ [{0,,,0~},0~]. (4)The first of these three identities says that the 0,, form a representation of the ordinary Lie algebraspanned by E,. (Consider the 0,, as vectors on which the E act.) The second is equivalent to the first ifthe Killing form is nonsingular in which case g,,,~= h,,~,.* The last identity restricts the possiblerepresentations 0,, of the ordinary Lie algebra. This relation is the reason that not every ordinary Liealgebra can be extended to a superalgebra.At this point we wish to make a distinction between graded algebras and superalgebras (the two areoften confused). A graded algebra is an algebra in which a grading exists, such as for example in theLorentz group, where the rotation generators, denoted by {L0}, and the boosts {L1} define a Z2 grading:[L,,L1] C L1+J,,,0d2.A superalgebra is an algebra with a Z2 grading (“even” and “odd” elements) such that (i) the bracket oftwo generators is always antisymmetric except for two odd elements where it is symmetric and (ii) theJacobi identities (2, 3, 4) are satisfied. The bracket relation between two generators need not always berepresented by a commutator or anticommutator (see (31) for an example), but one can always find arepresentation with this property (namely the regular (adjoint) representation).In order to familiarize the reader a little with super algebras, we first give some explicit representationsof the super Poincaré algebra. The defining representation is a 5 x 5 matrix representation-~yrn(1-y5) 0Pr,, = 0 , =000 00000 0 0000 0(Qa)nl = 0, (Q~)~= [—~(1+ ys)C_u]ma(Qa) = ~(1— y~,,,. (5)The matrix C is the charge conjugation matrix, satisfying Cy,,,C’ = —yT,,.The Killing form has only even—even and odd—odd parts. If it is nonsingular, the superalgebra is nonsingular (the converse is not true).