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

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334 P. van Nieuwenhuizen, Supergravityand47’(x, 0)= exp(A 118 11) 47(x, 0)= 47 +A 118 1147 ~ (21)11(x, 0) A *(x 0)11AChiral superfields are defined by1 8and depend thus on x and0A only (not yet on XLThus, with (21),(22)11 which maintain chirality.and OL). Consider those A8A(S47) = 8B(A’~8A+ AA8A + A”8,.)çb = 0. (23)It follows that AA and A” are chiral but AA is unrestricted since 8A4., = 0 from (22).Similarly, 47 ~ is antichiral, ~(1+ y~)(8/80)47 * = 8A47 * = 0, and requiring that also antichirality ismaintained, one finds 8BA *A = 8BA*F’ = 0. Again A *A is unrestricted. (The star operation is in a generalrepresentation not equal to complex conjugation C, but rather x” = C~y~{.In a Majorana representation* = C.)The chiral scalar superfields of Ogievetski and Sokatchev are obtained as follows47(XL, OL) = exp(iU) ~(x, gA)47*(xR OR) = exp(—iU) 47(x, OA)(24)where U = U’~(x, 0)8~with (U’~)* = UA. This field U transforms per definition as(exp iU)’ = (exp A *fI~~) (exp iU) (exp — A”811). (25)To lowest order in A and U one hasso that one can gauge away the spinor superfields UA and UA by fixing AA and A *A appropriately.Clearly, (25) preserves the reality (Majorana character) of U’5. In this gauge (24) yields= (exp iU)x”, X”R = (exp — iU)x” (27)as one may easily check. It follows that X”L — X”g = 2i U” +~... Hence, to lowest order, U” = H” isthe axial vector superfield discussed before. This is superconformal superspace supergravity.To obtain Poincaré superspace supergravity, consider a chiral scalar density VSV= A 118 11V+(8,.A” — 8AA’~)V. (28)(26)
The gauge choice V = constant leads then to further constraints on A 11P. van Nieuwenhuizen. Supergravity 3358,.A”~8AA”=0. (29)Clearly, 1/47 is again chiral, and f d~xd20 V47 is an invariant action.Siegel and Gates construct explicit constraint-free supervielbeins, and, after a conventional choice ofthe superconnection, they find that the Wess—Zumino constraints are satisfied identically.The motivation for writing the fields such that they appear in an exponential is to have a formalismwhich is analogous to the super Yang—Mills case. For the extended supergravities it seems unavoidablethat one needs constraints on H” itself. Thus here the distinction between the chiral and non-chiralapproach seems less clear even at the starting point.5.7. Gauge supersymmetryThe first approach to local supersymmetry is the theory of Arnowitt and Nath [17—43],called gaugesupersymmetry. Its tangent group is not, as in supergravity, 0(3, 1) x 0(N), but the larger supergroupOsp(3, 1/4N). The dynamical equations of motion are the Einstein equations with a cosmologicalconstant, generalized to a space with four boscnic and 4N fermionic coordinates. Due to thecosmological constant, the metric g,.~= 8,.,. is not a solution of the field equations, while globalsupersymmetry (see subsection 3) is only a solution at the tree level when N = 2. When quantumcorrections are taken into account, global supersymmetry could also be a solution for other values of N.The main problem of gauge supersymmetry is that it contains higher spin fields and ghosts. As such it isnot a good particle theory. Nevertheless, we feel it important to discuss also this theory here because itis different from supergravity in its geometrical structure.*We will begin by defining tensors. This is more complicated than in supergravity, since the tangentgroup has Bose—Fermi parts in gauge supersymmetry, but not in supergravity.The base manifold is z’~= x”, 0”’ where i = 1, N and N is arbitrary. The basic field is the metrictensor g~n(z)where the caret indicates that the two indices transform differently as we shall see.Coordinates transform as= — ~eA di’s = dz’~— ~ dz’ (1)where in this subsection we follow the literature and always use right derivatives. Requiring that the lineelement(ds)2 = dz’tgAn dz” with g~i~ = (~)“ ~“~‘5”gIIA (2)is invariant, one finds that the metric transforms under infinitesimal general supercoordinate transformationsas— Z \(Z+1)A 1 ISg~n—g~~.n+(—, ~ .AgIn+gAn,~ . (3)Note that left-up to right-below contractions are the usual contractions and are free from carets, butthat the (unusual) other contractions carry carets on the lower index. In fact, changing an index A to Aintroduces a factor (—)‘~as we shall discuss.* The following is an excellent exercise in tensor analysis. Dynamics starts at eq. (33).
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SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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