# Supergravity action as a total integral, over 4 spacetime and 4 Grassmann coordinates

Wess and Bagger, in their Supersymmetry and Supergravity, give the action for a global SUSY, ${\cal N}=1$, $D=4$, Yang-Mills gauge model as an integral over the 4 spacetime coordinates and 4 Grassmann coordinates, and also give an alternative formulation over 4 spacetime and just 2 Grassmann coordinates. I am okay with this. When they locally gauge the global SUSY transform to get supergravity they only give the formulation over 4 spacetime plus 2 Grassmann coordinates. What is the supergravity action in the 4+4-integral form? The rules they give, for going from SUSY to SUGra seem ambiguous, especially with regards to the chiral density. Any links to the explicit construction would be appreciated.