Plebanski action

General relativity and supergravity in all dimensions meet each other at a common assumption:

Any configuration space can be coordinatized by gauge fields ${\displaystyle A_{a}^{i}}$, where the index ${\displaystyle i}$ is a Lie algebra index and ${\displaystyle a}$ is a spatial manifold index.

Using these assumptions one can construct an effective field theory in low energies for both. In this form the action of general relativity can be written in the form of the Plebanski action which can be constructed using the Palatini action to derive Einstein's field equations of general relativity.

The form of the action introduced by Plebanski is:

${\displaystyle S_{Plebanski}=\int _{\Sigma \times R}\epsilon _{ijkl}B^{ij}\wedge F^{kl}(A_{a}^{i})+\phi _{ijkl}B^{ij}\wedge B^{kl}}$

where

${\displaystyle i,j,l,k}$

are internal indices,${\displaystyle F}$ is a curvature on the orthogonal group ${\displaystyle SO(3,1)}$ and the connection variables (the gauge fields) are denoted by ${\displaystyle A_{a}^{i}}$. The symbol ${\displaystyle \phi _{ijkl}}$ is the Lagrangian multiplier and ${\displaystyle \epsilon _{ijkl}}$ is the antisymmetric symbol valued over ${\displaystyle SO(3,1)}$.

The specific definition

${\displaystyle B^{ij}=e^{i}\wedge e^{j}}$

formally satisfies the Einstein's field equation of general relativity.

Application is to the Barrett–Crane model.[1][2]