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

P. van Nieuwenhuizen, Supergravity 331to curved space, we consider (7a) as a flat hypersurface in the 4+8 dimensional space spanned by X”L,0L, O~.The global supersymmetry transformations on the coordinates (x”, 0”) in eq. (8) ofX”R,subsection 3 induce the transformationstm L ~m II— LI ~ Pfl _~ ?tlLI j— EL ~ Vt (YUL oX L — 4 ’)’ ~L ~ ~ It nX L= ER+ ~(A ~)*oR, SXinR = ~ + ~ + Am,,X”R. (10)By rewriting these results in two-component notation one finds that SXmR = (5xmL)* and 50R = (SOL)*.To go to the local case, one replaces these combined global transformations by one local function (justas in the case of ordinary gravity) as in (2). One requires also in the local case that right-handed andleft-handed transformations are related by complex conjugation. Finally, Oy~y~0being the flat spacelimit of H”(x, 0) in (1), it is natural to generalize (7a) to (7).Let us now describe how this chiral superspace approach indeed describes ordinary supergravity.First one performs a general super coordinate transformation such that the components of H” (x, 0)with no and one 0 factor are zero. The set of transformations which maintain this gauge haveparameterswhich are field-dependent. The same phenomenon one encounters in the previous approaches tosuperspace. The advantage of this gauge choice for H” (x, 0) is that then the transformation rule,SH”(x, 0)= ~(A”(x+ iH OL)— A*~(x—iH, OR))—~(A”(x+iH, OL)+A (x — iH, OR))8,J-I” (x, 0)— [A”(x +iH, OL)~—+A*~(x—iH, 0R)~~]H”(X~ 0) (11)which is a nonlinear realization of the left- and right-handed supergroup on H” (x, 0), becomespolynomial (though still nonlinear) in H”(x, 0). The parameters are expanded in terms of 0L (or 9~)asA” = —a” +ib” + OL~”L+ OLOL(5” +ip”), with °L°L= OLCOLA” = EL+ WL~O~L+OLOL?1L (12)where a”,. . ., vj’” still depend on X”L = X” + iH”. The bispinor ~L”~ is equal to ~(1+ y5)”PwL~(1+y)U; in other words, viewed as a 4 X 4 matrix it only has nonzero elements in the first two rows andcolumns in a representation in which y~is diagonal.Let us now consider the field-independent terms in 5H”SH” (x, 0) = ~- [2ib” + OL~”— OR~*l~+ OLOL(S” + ip”) — ëROR(s” — ip”)] +~.~, ë~,,.=(13)where all b”,. . . , p” now only depend on x. Clearly there is enough freedom in parameters to choose agauge such that C”(x) and Z”(x) in H”(x, 0) are zero. Let us assume this has been done. Then the full8ff” is polynomial, since third powers of H” vanish.The set of transformations which maintain this gauge have b” = 0 while for the vanishing of the order0 terms in SH”(x, 0) one finds the conditions