PROC. 19th INT. CONF. HIGH ENERGY PHYSICS TOKYO, 1978 C6 The Superspace Method for Supergravity B. ZUMINO CERN, Geneva Superspace1"3 was introduced by Salam and Minkowski space. Just as Einstein's theory is Strathdee as a technique for the study of obtained when one generalizes from flat representations of the supersymmetry algebra Minkowski space to urved space-time, one in terms of supermultiplets of fields. The expects supergravity (the supersymmetric theory points of superspace are parametrized by coord­ of gravitation) to arise from the differential inates zA = (xa, 6a) where xa are the commuting geometry of curved superspace. Indeed this space-time coordinates while 6a are anticom- superfield approach to supergravity preceded muting variables (whcih are Majorana spinors). that45 based on ordinary fields. Supersymmetry transformations can be realized The first work on curved superspace was as rigid motions in superspace that of Arnowitt and Nath,6 who developed the differential geometry of a super-Riemannian superspace. However, the flat superspace of rigid supersymmetry does not fit naturally as where £a are the infinitesimal anticommuting a special case of a super-Riemannian geometry. spinorial (Majorana) parameters. The com­ For this reason a different geometry of super- mutator of two transformations (1) of param­ space was introduced by Wess and the eters Ci and C 7-13 2 author and by Akulov, Volkov and Soro- ka.14 In this geometry the tangent space (2) group at each point of superspace is not taken to be the graded Lorentz group (orthosimplec- is a translation of parameter —li^yj^^ as tic (4, 4)), but just the ordinary Lorentz group. required by the supersymmetry algebra. A Furthermore the superspace is allowed to have superfield V(x, 6) is a function in superspace. a torsion. More recently this geometry has Since the four components 6 anticommute, in been adopted by other authors.15 In this lec­ particular the square of each one is zero, a a ture we shall follow our own work, as deve­ power series in 0 is at most a polynomial of loped in refs.7"*13. Ours is a second order fourth degree. Therefore a superfield is equi­ formulation, in that we impose on the super- valent to a finite collection of ordinary fields torsion certain conditions ("constraints"), (a supermultiplet), carrying various spinorial which are sufficient to express the superconnec- indices and satisfying corresponding statistics. tion in terms of the supervielbein.* These When the superfield transforms like a scalar conditions also restrict to a certain extent the a a a (da=d/dx ) dV{^{didd )~xr oda)v (3) supervielbein itself. The dynamics is given in the coefficients of its expanision in 6a trans­ terms of a superspace action which must be form into each other. The "covariant deri­ varied keeping the constraints on the torsion vatives" satisfied. Our formulation is exactly equival­ ent to the recently developed formalism for supergravity with a minimal set of auxiliary commute with the infinitesimal operator in (3) and can be used to impose supersymmetric * It would be preferable to have a first order action constraints on superfields. They satisfy in superspace, from which both constraints and equa­ tions of motion follow. A possible choice for a first order action was suggested in ref. 8 and is also dis­ showing that superspace has torsion, even cussed in the lecture by J. H. Schwarz in these Proceed­ 19 though its curvature vanishes. This "flat" ings. However, dimensional arguments indicate that 6 integration of that superspace action does not superspace has been extensively used in the reproduce the correct supergravity action in coordinate study of supersymmetric field theories in space. 556 B. ZUMINO fields.16-18 Indeed this auxiliary field formali­ denote vectorial (bosonic), Greek letters spi­ sm can be derived from our geometrical norial (fermionic) indices. The supervielbein 13 A formulation. matrix EM (z), where A=(a, a), and its inverse M A non-geometric description of supergravity EA (z), can be used to transform world in superspace is favored by Ogievestsky and tensors into tangent space tensors and vice Sokatchev20 and by Warren Siegel.21 The versa (early alphabet letters denote tangent supergravity fields are contained in a superfield space indices). Covariant derivatives involve m B U (x, 6) endowed with a vector index m. The the superconnection 0M A , as in theory, based directly on this superfield, has a certain appeal because it is free of constraints. However, the lack of a complete geometrical Covariant derivatives with tangent space in­ M structure in such a theory makes the con­ dices, &'A=EA £/-x9 satisfy the graded com­ struction of invariants a difficult task. On mutation relations the other hand, from our point of view, the superfield Um emerges in solving the constraints C on the torsion. The supervielbein and the where TAB 9 are the supertorsion coefficients superconnection are expressed in terms of it and RAB is the Lie algebra valued super- and we can use the powerful techniques of curvature operator. The bracket in eq. (7) is differential geometry for the construction of an anticommutator if both indices A and B are invariants. spinorial, a commutator otherwise. Torsion The geometric superspace approach gives a and curvature satisfy the Bianchi identities complete and satisfactory method for the study of simple supergravity and its couplings to supersymmetric matter11,12 (both for the and vector-spinor and for the spinor-scalar-pseu- doscalar multiplets). All possible invariants of given dimension are easily constructed and 16 their knowledge can be used to discuss the The cyclic sums are taken with appropriate divergences which arise in perturbation theory. signs: a permutation of two vectorial or one The situation is quige different in the case of vectorial and one spinorial index gives rise to a extended supergravity. Clearly one must use change in sign, while for two spinorial indices here a larger superspace in which the fermionic there is no change in sign. coordinates carry, in addition to the spinorial index, an internal symmetry index, and the Specification of the geometry (kinematics) tangent space group should be the product of We now restrict the tangent space group the Lorentz group with the internal symmetry from the general graded linear group to the 0(AO. The difficulty consists in finding the (x and 0 dependent) Lorentz group. Since the correct separation of the equations into curvature operator belongs to the algebra of kinematical constraints (which are part of the the tangent space group, this implies restric­ specification of the geometry) and equations tions on the curvature coefficients (2a& = of motion (which should follow from the variation of an action, subject to the con­ straints). The results obtained so far22 for Ar=3 extended supergravity in superspace do not make this separation. These difficulties in the superspace method correspond to our (similarly for the superconnection). We also present ignorance as to the correct auxiliary impose the constraints on the supertorsion fields for extended supergravity. Affine superspace r 7 The points of superspace are parametrized but leave Tab, and Tab, undetermined. Using by coordinates zM = (xm, 6ft). Latin letters the Bianchi identities (8) and (9) one can show Super symmetry (Including Super gravity) 557 that (10) and (11) imply that all components chiral infinitesimal parameter function 2 of the supercurvature and supertorsion are expressible in terms of the three superfields and are (Howe and Tucker23) Ga$, WapT, R and their conjugates. Here we have used two-component spinor notation, G„p is hermitean, Wafi7 totally symmetric. For instance For the determinant of the supervielbein, (17) imolv The remaining content of the Bianchi identities while curvature and torsion behave as given by is expressed by the differential relations The superfield Ga& has a simple physical mean­ ing, which can be exhibited16 by considering Since the constraints are invariant under (17) the coefficients of its expansion in 6. For and (18), we can say that the geometry (kine­ instance, at the 66 level one finds a tensor which matics) is invariant under conformal trans­ contains the Einstein tensor, at the 666 level a formations. The dynamics may or may not spinor which is the Rarita-Schwinger ope­ be, depending on the choice of Lagrangian. rator. Similarly, R contains the contracted For instance, the action of supergravity (21), given below, is only invariant under general (scalar) curvature tensor, at the 66 level. Wan contains the Weyl conformal spinor, at the 6 coordinate transformations in superspace and level; for 6=0, it contains the Rarita-Schwinger under local Lorentz transformations, but not field strength. Observe that some components under conformal transformation. The action of conformal supergravity (28), instead, is of R are obtained from those of Gad by tracing over certain indices: in superspace this is ex­ also conformally invariant. pressed by the differential identities (13) in 6. 11 Dynamics Finally we observe that the constraints (11) The supergravity equations of motion are on the torsion imply derived from the action for any vector vA, where Varying (21) subject to the constraints (11) gives11 the equations is the graded determinant of the supervielbein as defined in the first of ref. 6. This formula Because of the identities (\3) thev eive permits integration by parts with covariant derivatives carrying tangent space indices. and this, in turn, implies Conformal transformations The correct conformal transformations in superspace must preserve the constraints (11). On the supergravity mass shell all components These transformation involve a covariantly of the supertorsion and of the supercurvature 558 B. ZUMINO are expressed in terms of the symmetric, chiral spinor superfield Wa^r which now also satisfies It satisfies identically (23), and of its derivatives (and, of course, of the complex conjugate quantities).