Supergravity

Supergravity attempts to use the tools of supersymmetry in some form of gravity, usually general relativity.

Derivations

Derivation 1. If we take an independent supersymmetric transformation at each spacetime point, we arrive at a corresponding gauge symmetry. The (global) supersymmetry generators have the anti-commutation relations $\{Q^{i}_{\alpha},\bar{Q}^{j}_{\beta}\}=2\delta^{ij}(\gamma^{n})_{\alpha\beta}P_{n}$ where $Q$ and $\bar{Q}$ are the spinorial charges (generators of the SUSY algebra), $\gamma$ the Dirac gamma matrix, and $P$ is the four-momentum operator. Observe when we turn this into local symmetries, the four-momentum operator include local spacetime translations. Hence the resulting gauge theory should include gravity.

Derivation 2. Take some superstring theory. The limit as the string length goes to zero results in a supersymmetric gauge theory which includes gravity. It turns outs, these are the more exotic supergravity theories.

Problems

When we consider the simplest model (i.e., $\mathcal{N}=1$) for canonical quantum super-gravity, we find it has no bosonic solutions. This means we cannot interpret “vanilla general relativity” as a “limit” of canonical quantum supergravity.