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

P. van Nieuwenhuizen, Supergrauity 347transformation rules are always supercovariant. This means that there must be extra terms in thecommutator of two local supersymmetry transformations. These terms are: an extra local Lorentztransformation with parameterI’)~/i\1 ilk f—i k ‘j .Le —i k I ‘11~ ~E 2Ei ma 2E275 1 ma)and an extra local supersymmetry transformation with parameter(2V2)- ~ (E’2e~A — E’275 i y5A) (12)and an 0(3) Yang—Mills gauge transformation with parameterA’ = (2V2)~ ”(E’~e~) —SA,. = 9,~A’+ ge’~”~A~,.A”, &m,. = 0 (13)5i/J~,.= gekikçfri,.A’c, SA = 0.Note that the parameter composition rules do not depend on g (or only through F’,,,,, as in (11))although the transformation rules depend on g explicitly.There is a global U(3) invariance for g = 0 on-shell. Off-shell only the 0(3) invariance remains, sinceall other symmetries involve not only chiral transformations but also duality transformations. Theprecise rules follow by truncation from the U(4) group of the S0(4) version of the N = 4 model (seebelow). For g 0, the 0(3) invariance remains valid off-shell, but since the Yang—Mills coupling involvesbare A,, fields and the mass term breaks chiral invariance, the U(3) invariance is again lost for g~0 onand off-shell.The gauge parameter in (13) contains a nonvanishing term in the global limit (when fields are setequal to zero). Hence, there is a central charge in the super Poincaré algebra (i.e. for g = 0), but thischarge does not act on physical states. Also in the super de-Sitter algebra (with g 0) there is an“electric” charge in the (0, Q} anticommutator, but now this charge is no longer a central charge, sinceit does not commute with supersymmetry. As a result S(gauge) i/i,. 0, so that for g 0 the electriccharge does act on physical states.From the N = 3 theory (or any other theory) one can obtain other theories by consistent truncation.A truncation means putting certain fields equal to zero, and 2, consistency = i/is,. = A = 0requires which yields that the alsoN their = 2variations then vanish. Two consistent truncations are: A~= model [511’],while A2,. A= A3,, = i/it = i/r~.= 0 yields the (2, ~)+ (1, ~) Maxwell—Einstein systems [507].We now discuss the N = 4 extended supergravity. New here is the occurrence of scalars, for exampleA, which appear in a nonpolynomial way. This is possible since KA is dimensionless. For photons, KA,.is also dimensionless, but Maxwell gauge invariance forbids an infinite series in photons [202]while onecan also show that fermion fields cannot appear nonpolynomially [202].There are actually two versions of the N = 4 model known, the S0(4) model [109,90, 91] withfields (e, i/i”, V,”, A”, A, B’) and the SU4 model [190]with fields (e, i/i’, A,”, B,.”, A’, 4, B). The sixfields V,” are all vector fields, but the three A,.” are vector fields while the B,” are axial vector fields.One can obtain the SU(4) model by reduction_of the N = 1 model in d = 10 dimensions. Spinors ind = 10 can satisfy both the Majorana condition A’~= AM and the Weyl condition A = ~(1± y11)A. Thus,