Plenbanski action

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

Any configuration space can be coordinatized by gauge fields <math>A^i_a<math>, where the index <math>i<math> is lie algebra indes and <math>a<math> is 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 Plebanski action which can be constructed using the Palatini action to derive Einstein's field equations of general relativity.

The form of Plebanski action is: <math>S_{Plebanski} = \int_{\Sigma \times R} \epsilon_{ijkl} B^{ij} \wedge F^{kl} (A^i_a) + \phi_{ijkl} B^{ij} \wedge B^{kl} <math>

where <math>i, j, l, k<math> are internal indices, <math>F<math> is a curvature on <math>SO(3, 1)<math> Connection (mathematics) variables (the gauge fields <math>A^i_a<math>. The symbol <math>\phi_{ijkl}<math> is the Lagrangian multiplier and <math>\epsilon_{ijkl}<math> is the antisymmetric symbol valued over <math>SO(3, 1)<math>.