Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008)

TL;DR

This paper provides a comprehensive introduction to Chern-Simons gravity and supergravity, emphasizing their gauge invariance in higher odd dimensions and their potential role in quantum gravity theories.

Contribution

It introduces a gauge-invariant formulation of gravity in odd dimensions using Chern-Simons actions, including supersymmetric extensions with off-shell closure.

Findings

Chern-Simons actions generalize Einstein gravity in higher odd dimensions.

Theories exhibit gauge invariance under (anti-)de Sitter or Poincare groups.

Supersymmetric extensions exist in all odd dimensions with off-shell closure.

Abstract

This is intended as a broad introduction to Chern-Simons gravity and supergravity. The motivation for these theories lies in the desire to have a gauge invariant system --with a fiber bundle formulation-- in more than three dimensions, which could provide a firm ground for constructing a quantum theory of the gravitational field. The starting point is a gravitational action which generalizes the Einstein theory for dimensions D>4 --Lovelock gravity. It is then shown that in odd dimensions there is a particular choice of the arbitrary parameters of the action that makes the theory gauge invariant under the (anti-)de Sitter or the Poincare groups. The resulting lagrangian is a Chern-Simons form for a connection of the corresponding gauge groups and the vielbein and the spin connection are parts of this connection field. These theories also admit a natural supersymmetric extension for all odd D where the local supersymmetry algebra closes off-shell and without a need for auxiliary fields. No analogous construction is available in even dimensions. A cursory discussion of the unexpected dynamical features of these theories and a number of open problems are also presented.