(Super)-Gravities Beyond 4 Dimensions

TL;DR

This collection of lectures frames gravity as a gauge theory framed by fiber bundles, advocating a first-order formalism with independent vielbein and spin connection to extend gravity beyond four dimensions. It surveys Lanczos–Lovelock gravity as the most general torsion-free, second-order theory in $D$ dimensions, then demonstrates how, in odd dimensions, gravity can be reformulated as a Chern–Simons theory for extended (A)dS or Poincaré groups, with torsion playing a natural, sometimes essential, role. The text then extends these ideas to supersymmetry, constructing CS supergravity actions via invariant supertraces of extended AdS superalgebras, and discusses the distinctive dynamical content and Hamiltonian structure of CS theories, including degeneracies and constraint counting. It emphasizes that, when augmented by supersymmetry, CS gravities fix many coupling constants, yield rich topological and black-hole solutions, and offer a promising, gauge-theoretic path toward quantum gravity in higher dimensions, though their full quantum behavior remains an active area of research. Overall, the work argues that CS and LL extensions provide a coherent, geometrically transparent framework linking gravity, topology, and supersymmetry in a way that could illuminate quantum gravitational dynamics in dimensions beyond four.

Abstract

These lectures are 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 action -in the sense of fiber bundles- in more than three dimensions, which could provide a firm ground for constructing a quantum theory of the gravitational field. The case of Chern-Simons gravity and its supersymmetric extension for all odd D is presented. No analogous construction is available in even dimensions.

Figures (5)

• Figure 1: Pythagoras theorem: ${\sf c}^2={\sf (a-b)}^2+4 {\sf [a b]}/2$
• Figure 2: Affine property: L$\parallel$L$^{\prime } \Leftrightarrow \alpha ={\alpha}^{\prime}=\delta = {\delta}^{\prime} \;,\; \beta ={\beta }^{\prime } =\gamma ={\gamma }^{\prime }$
• Figure 3: Constructing parallels using a right-angled triangle ( W) and a straightedge ( R)
• Figure 4: Constructing parallels using an arbitrary angle-preserving wedge ( W) and a straightedge ( R) .
• Figure 5: Constructing parallels using any angle-preserving wedge ( W) and an arbitrary ruler ( R). Any ruler is as good as another.