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

TL;DR

This work surveys a gauge-theoretic route to gravity via Chern–Simons (CS) structures, arguing that in odd spacetime dimensions gravity can be formulated as a CS theory for (A)dS or Poincaré gauge groups with the vielbein and spin connection packaged into a single connection. It first develops the standard, first-order gravity framework, then extends to Lovelock gravity in higher dimensions, and finally elaborates CS gravity and CS supergravity, including AdS algebras, transgression regularization, and boundary charges. A central theme is that odd dimensions admit fully gauge-invariant actions with fixed coupling content, while even dimensions resist a CS construction for gravity, requiring Born–Infeld or other forms. Supersymmetric extensions are shown to close off-shell in CS formulations, yielding AdS CS supergravities with rich fermionic/bosonic content and distinctive dynamical properties, such as degenerate phase spaces and nonstandard DOF distributions. The work thus highlights a unifying geometrical and topological perspective on gravity and its supersymmetric generalizations, with potential implications for quantum gravity, holography, and higher-dimensional black hole physics.

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.