On the formulation of DÂ =Â 11 supergravity and the composite nature of its three-form gauge field

The underlying gauge group structure of the D = 11 Cremmer-Julia-Scherk supergravity becomes manifest when its three-form field A3 is expressed through a set of one-form gauge fields, B1a1a2, B1a1⋯a5, η1α, and Ea, ψα. These are associated with the generators of the elements of a family of enlarged supersymmetry algebras E˜(528|32+32)(s) parametrized by a real number s. We study in detail the composite structure of A3 extending previous results by D'Auria and Fré, stress the equivalence of the above problem to the trivialization of a standard supersymmetry algebra E(11|32) cohomology four-cocycle on the enlarged E˜(528|32+32)(s) superalgebras, and discuss its possible dynamical consequences. To this aim we consider the properties of the first order supergravity action with a composite A3 field and find the set of extra gauge symmetries that guarantee that the field theoretical degrees of freedom of the theory remain the same as with a fundamental A3. The extra gauge symmetries are also present in the so-called rheonomic treatment of the first order D = 11 supergravity action when A3 is composite. Our considerations on the composite structure of A3 provide one more application of the idea that there exists an extended superspace coordinates/fields correspondence. They also suggest that there is a possible embedding of D = 11 supergravity into a theory defined on the enlarged superspace Σ˜(528|32+32)(s).