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This entry contains material related to the textbook
Leonardo Castellani, Riccardo D'Auria, Pietro Fré,
Supergravity and Superstrings - A Geometric Perspective
World Scientific, 1991
on supergravity and string theory with an emphasis on the D'Auria-Fré formulation of supergravity, based on
The book is out of print, unfortunately, but check $gen.lib$. The newer book
is much more brief in compariso, but has more history and exposition, and treats more recent developments, such as aspects related to AdS-CFT.
This book focuses on the discussion of supergravity-aspects of string theory from the point of view of the D'Auria-Fré formulation of supergravity, which is implicitly a formulation via supergeometry higher Cartan geometry. Therefore, while far, far from being written in the style of a mathematical treatise, this book stands out as making a consistent proposal for what the central ingredients of a mathematical formalization might be: as explained at the above link, secretly this book is all about describing supergravity in terms of infinity-connections with values in super L-infinity algebras such as the supergravity Lie 3-algebra.
See also higher category theory and physics.
The original article that introduced th D’Auria-Fré-formalism is
The geometric perspective discussed there is both the emphasis of working over base supermanifolds and combined with that the the approach that here we call tthe D’Auria-Fré-formalism .
The interpretation of the D’Auria-Fré-formalism in terms of ∞-Lie algebra valued forms together with a discussion of the supergravity Lie 3-algebra in the context of String Lie n-algebras was given in
Apart from that the first vague mention of the observation that the “FDA”-formalism for supergravity is about higher categorical Lie algebras (as far as I am aware, would be grateful for further references) is page 2 of
An attempt at a comprehensive discussion of the formalism in the context of cohesive (∞,1)-topos-theory for smooth super ∞-groupoids is in the last section of
Here are some more references:
Pietro Fré, M-theory FDA, twisted tori and Chevalley cohomology (arXiv)
Pietro Fré, Pietro Antonio Grassi, Pure spinors, free differential algebras, and the supermembrane (arXiv:hep-th/0606171)
Pietro Fré and Pietro Antonio Grassi, Free differential algebras, rheonomy, and pure spinors (arXiv:0801.3076)