Supergravity in the group-geometric framework: a primer

TL;DR

This paper reviews the group-geometric approach to supergravity, unifying symmetries as superdiffeomorphisms on supergroup manifolds, and discusses applications to various dimensions including off-shell and Chern-Simons supergravities.

Contribution

It provides a comprehensive overview of the group-geometric framework for supergravity, including recent developments, applications, and technical tools like supermanifold integration and covariant Hamiltonian formalism.

Findings

Unified description of symmetries as superdiffeomorphisms.

Explicit constructions for $d=3,4$ off-shell supergravities.

Discussion of $d=5$ Chern-Simons supergravity and $d=10+2$ supergravity.

Abstract

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.