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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.

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There are a lot of ways to get to SUGRA, in the memoirs https://arxiv.org/abs/1702.00743v2 they are described in a historical way how they have appeared and the complete Lagrangian. A discusion

In particular, superconformal formalism is an approach that employs many of the good-benefits of superspace formalism for rigid (global) supersymmetry such as the structure of multpletes. In this formalism the integral you are looking for is in equation 2 of https://arxiv.org/abs/1104.2598, (something quite complicated in itself). I know that you will find in Equation 15 the familiar structure of the rigid case, with the extra of a superfield called compensator, typical of the Superconformal formalism. But do not be fooled! these terms F and D in the Lagrangian are not calculated in such a simple way to the rigid case, equations 13 and 14 show you how they are calculated so that everything is superconforming invariant.

If you want a more pedagogical introduction look the book of Daniel Z. Freedman and Antoine Van Proeyen "Supergravity (2012)" isbn 9780521194013.

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