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

P. van Nieuwenhuizen, Supergravity 299transformation rules for the dependent fields arise after solving the constraints (see subsection 3.2)= ~Rmn(0)y,. 0 (11)= y” (y5R,,,,,(A) + ~1~,,,,(A)) 0. (12)These results and a similar result for O’f”,. will be derived in subsection 4.We now discuss the 0-invariance of the action. If all fields would transform as given by the structureconstants alone, one would have81 = 8a Jd4x [1~~(0) y5y0,e0R,,,,~(K)+2iR~~(S)(ysR,.,,(A)+~R,.,,(A))eo— ë,.yy . 4’)R,,,,(A)R~~(A) + 2~0y”4’0R,.,,(A)R’”’(A)]. (13)The extra terms O’w,,”’ and 8’4i,. give an extra 8’I, whose general form was derived on page 281.However, since Qi”~ is constant and proportional to “~ for A = M, 5, the Bianchi identity holdsso that the D,,R,,. terms vanishes, and also the 3,,Q terms vanish. One is left withS’I = 8a J d 4x[28~w,.m~1~~”~(K) ~ + 41~”(S)y,~,y~5’4,, + 4ü~,.(y 5R”~(A) + ~1~”’(A))o’q 5,,]. (14)The R(Q) R(K) terms in 81 cancel against the R(K) term in 8’I if and only if 8lw,.mn is as given above.For this to be the case, one needs both 0-constraints; with the dual-chiral constraint alone nocancellation would occur. Furthermore, substituting O’~,,.all other terms in 81+ 8’I cancel at once.We now show that the dilation field cancels from the action. The remaining independent fields areem,,, 4’~,,,A,,, b,. and ftm,. while ffll,. is nonpropagating and is expressed in terms of other fields bysolving its own field equation (just as ~,.mn in ordinary supergravity). Let us consider how these fieldstransform under K-gauges. Only b,. transforms, and since 8Kb,. = 2~K,,. one can gauge away b,.altogether. This one might have expected since there are as many K-gauge degrees of freedom asb-components. However, since the action was K-invariant to begin with, it must therefore beindependent of b,,.The action is thus unique and determines in turn the constraints [523].In tact, as we shall see, thesolution of the ftm,, -field equation is at the same time the solution of an extra constraint [535].At thelinearized level, these results were obtained using superspace methods [191].Let us close this subsection by showing that there are equal numbers of boson and fermion states.Since the graviton and gravitino have higher derivative actions, one has two spin 2 states (El2 counts astwo El) and three spin ~states (Jill counts as three )T~.Indeed, one finds one massless (2,~)multipletand one massive (2, ~,~,1) multiplet [191,526].4.3. Constraints and gauge algebraFor ordinary supergravity, one can formulate the theory in second order formalism either byimposing a constraint on curvatures (R14,,tm (F) = 0) or by solving a field equation belonging to the gauge