nLab Supergravity and Superstrings - A Geometric Perspective

This entry contains material related to the textbook

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré,

  Supergravity and Superstrings - A Geometric Perspective

  World Scientific, 1991

  doi:10.1142/0224

  epdf: vol 1, epdf: vol 2, epdf: vol 3

on supergravity and string theory with an emphasis on the D'Auria-Fré formulation of supergravity, based on

• Riccardo D'Auria, Pietro Fré Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140

The book is out of print, unfortunately, but check $gen.lib$. The newer book

• Pietro Fré, Gravity, a Geometrical Course: Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer 2012

is much more brief in comparison, but has more history and exposition, and treats more recent developments, such as aspects related to AdS-CFT.

Description

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.

Further references

The original article that introduced the D’Auria-Fré-formalism is

• geSuGra R. D’Auria, P. Fré Geometric supergravity in $D = 11$ and its hidden supergroup Geometric Supergravity in D=11 and its hidden supergroup

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 the 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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$-algebra connections (web)

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

• Pietro Fré, Pietro Antonio Grassi, Free differential algebras, rheonomy and pure spinors (arXiv:0801.3076)

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

• Urs Schreiber, differential cohomology in a cohesive topos

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, Pietro Antonio Grassi, Free differential algebras, rheonomy, and pure spinors (arXiv:0801.3076)

category: reference