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

290 P. van Nieuwenhuizen, SupergravilyHamiltonian)I =Jd~[4p—A mC(p,q)]q — ~e ~, ~, , W, ~, —— f m , a mnl ~ 0, — 5 m I1eo, ‘PO,a, W0The p are the conjugate momenta, while C(p, q) denote the first class constraints (after eliminating thesecond class constraints by solving the latter) which generate gauge transformationsCm(P, q) = {~m,0”, J,,}. (2)All generators have flat indices. The proof that the action can be written in this way is well-known forpure gravity (due to work by Dirac) but the extension to supergravity which we discuss below is due toTeitelboim and Pilati.m and 8q = {q, Cm} m (where ~m are fieldindependentthe action localisparameters), to be invariant one finds under after 8/Li partially = {p, Cm} integrating (discarding surfaceIfterms)s5(4p) = 48p — (8q)j.5 = tmC. (3)_~mdC0,/dt = éVariation of Cm yields the Poisson bracket of the constraints8C {Cm,Cn}C”. (4)Defining the field-dependent structure constant f,,,,,” by(C, C,,} = fmn”Cp (strongequality) (5)it follows that in order for I to be invariant one must transform the Lagrange multipliers as6A~ ? +fmn”C”Am. (6)From the known transformation rules of em0, 4’o”, ~ as given in section 1 one can thus read off whatf,,P are. This is a very simple and elegant way to find the algebra of constraints, but one can, of course,also obtain the result by directly determining the charges Cm and then compute the anticommutators.In order that 6A has indeed the form as given above, one must combine a general coordinatetransformation with parameter ~ with a local Lorentz transformation with parameter ~m~0,kl and alocal supersymmetry transformation with parameter ~ (the same combination as found byFreedman and the author in the commutator of two local supersymmetry variations). We will call thesetransformations covariant translations.The transformation laws of the fields of supergravity in first order formalism as discussed in section 1read— tmJI—m, ~m ~Lie — 2EV yi,~— ~ ,,e ~84’,,” = ~ me~(D~4’,,”—D~4’,,”)+D~ ”~ . (8)