Supergravity in the Group-Geometric Framework: A Primer

We review the group-geometric approach to supergravity theories, in the perspective of recent developments and applications. Usual diffeomorphisms, gauge symmetries and supersymmetries are unified as superdiffeomorphisms in a supergroup manifold. Integration on supermanifolds is briefly revisited, and used as a tool to provide a bridge between component and superspace actions. As an illustration of the constructive techniques, the cases of d=3,4 off-shell supergravities and d=5 Chern-Simons supergravity are discussed in detail. A cursory account of d=10+2 supergravity is also included. We recall a covariant canonical formalism, well adapted to theories described by Lagrangians d-forms, that allows to define a form hamiltonian and to recast constrained hamiltonian systems in a covariant form language. Finally, group geometry and properties of spinors and gamma matrices in d=s+t dimensions are summarized in Appendices.