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

Question

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.

This post imported from StackExchange Physics at 2018-01-20 17:49 (UTC), posted by SE-user Michael C Price

Answer 1

Detailed discussion of construction of supergravity theories in "superspace" is in the excellent textbook  Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991). Specifically 4d SuGra is discussed in chapter III.3.5 and III.4 and V.4

Their approach is both elegant and transparent as well as fully detailed.

Answer 2

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.