The superspace formulation of eleven-dimensional supergravity with a seven-form

The superspace formulation of eleven-dimensional supergravity with a seven-form field strength is given. CERN-TH/96-159 March 22, 2008 Permanent address: Dept. of Mathematics, King’s College, London The superspace formulation of eleven-dimensional supergravity [1] was given in [2]. The superspace constraints can be understood as being required by kappa-symmetry of the supermembrane action [3] and can also be given an interpretat ion in terms of the geometry of a membrane version of loop superspace [4]. In [4] it was shown that the dimension zero constraints, which arise as integrability conditions in membrane superspace, put the theory on-shell. In this note it is shown that the on-shell constraints are compatible with a superspace seven-form H7 obeying the Bianchi identity dH7 = 1 2 H 4 (1) where H4 is the usual closed four-form field strength. The superspace seven-form should be of interest in the context of the five-brane soliton solution of eleven-dimensional supergravity [5], particularly with a view to understanding the world-vo lume description of the brane [6]. The standard constraints of eleven-dimensional supergravity in superspace are that all components ofH4 which have dimension less than zero vanish together with the dimension zero equations Tαβ c = −i(γ)αβ , (2) Hαβcd = −i(γcd)αβ . (3) Spinor (vector) tangent space indices are denoted by greek (latin) indices from the beginning of the alphabet, T is the torsion tensor and the structure group is the spin group in eleven dimensions acting in the obvious way on spinor and vector indices. These equations imply that the geometry describes the eleven-dimensional supergravity multiplet and furthermore that it is on-shell. All the non-vanishing components of the torsion, curvature and field strength can be expressed in terms of the totally a ntisymmetric superfield Habcd which is the dimension one component of the four-form H4. At dimension one, the torsion is given by Taβ γ = − 1 36 ((γ)β Hbcde + 1 8 (γbcdef)β H). (4) and the connection may be chosen such that Tab c = 0. The dimension three-halves torsion, whose leading term in the θ-expansion is the spacetime field strength tensor of the gravitino is Tab γ = − i 42 (γ)DδHabcd (5) and is γ-traceless. The components of the curvature tensor can be obtained easily using the Bianchi identities. They are expressible in terms of Habcd and its derivatives. It should be emphasised that this on-shell structure is the unique solution of the Bianchi identities, modulo field redefinitions, given the constraints described above. The seven-form field strength H7 can be constructed straightforwardly. It should obey equation (1). Normalising forms such that, for example, H4 = 1 4! EEEEHABCD (6)