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

200 P. van Nieuwenhuizen. Supergravityderivatives in their action while fermion fields have only one (see eq. (1)). Hence the dimension of is—~ according to eq. (5). However, this means that in the inverse transformation oF — Be there is a gap ofone unit of dimension. The only dimensionful object available in flat space with massless fields to fill thisgap is a derivative. Thus, purely on dimensional grounds (omitting indices)oF=(8B) . (8)The reader may check this relation for the special case of eq. (2).(5) Commutator: Consider now two consecutive infinitesimal global supersymmetry transformationsof a boson field B (the same arguments hold for F). The first rotation transforms B into F, but thesecond rotates F back into 3MB. Thus two internalsupersymmetry transformations have led to a space-timetranslation. In any globally supersymmetric model one always finds the same relation[8(e1),8(e 2)]B = ~(i2y~Lei)8,~B. (9)For F the same result holds, see subsection 8. We leave it as an exercise to verify this relation for eq. (2)using the appendix. Hence, global supersymmetry is the square root of the translation operator, and oneexpects that local supersymmetry is the square root of general relativity, just as the Dirac equation is thesquare root of the Klein—Gordon equation. This we will show in more detail in subsection 3.5.We derived this extremely interesting result from the simple fact that bosons and fermions havedifferentdimensions. For a discussion of the complete algebra of symmetries of globally supersymmetricmodels, see below. For further details of global supersymmetry, see the Physics Report by P. Fayet andS. Ferrara.1.2.2. SupergravityWe now turn to local supersymmetry and ask what would happen if we make “For example, in eq. (2) we definelocal, hence “ (x).OA = ~(/(A — iy5B)) (x) without ~terms in OAsince A is considered as a matter field. (Quite generally, only gauge fields have in their transformationrules derivatives of parameters (in this case (x))and only on the parameters to which they belong. Forexample J~ewill only occur in the supersymmetry gauge field (the gravitino) but not in, say, theYang—Mills field transformation law.) Since for constant the action in (1) is invariant under (2), itfollows that for local e the variation of the action must be proportional to 3,~e(x)4x (3~,,i(x))j~. (10)SI = J dHerej~is the Noether current, and can be obtained from (2) and (4) as follows-. o.~’ 1-A. 1Indeed, for constant one has O5~’= ~ + (8./8~)5~= 3L~Ke~so that on shell (where