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The $L_\infty$ structure of Free Differential Algebras and $d=11$ Supergravity

L. Castellani, R. D'Auria

Abstract

We discuss free differential algebras (FDA's), a generalization of the Cartan-Maurer equations for the group manifold vielbein, appropriate for theories containing $p$-forms ($p >1$). Their dual formulation is an extension of Lie algebras, called $L_\infty$ algebras, and we illustrate this duality in a simple example. Finally we review the FDA structure and the dual $L_\infty$ structure of $d=11$ supergravity.

The $L_\infty$ structure of Free Differential Algebras and $d=11$ Supergravity

Abstract

We discuss free differential algebras (FDA's), a generalization of the Cartan-Maurer equations for the group manifold vielbein, appropriate for theories containing -forms (). Their dual formulation is an extension of Lie algebras, called algebras, and we illustrate this duality in a simple example. Finally we review the FDA structure and the dual structure of supergravity.

Paper Structure

This paper contains 12 sections, 61 equations.

Table of Contents

  1. Introduction
  2. The supergravity origin of FDA's
  3. FDA structure
  4. $L_\infty$ structure
  5. The example of FDA1
  6. The general case
  7. The duality between FDA and $L_\infty$ Algebra
  8. The construction of the FDA
  9. The FDA associated to the super-Poincaré algebra in d=11
  10. Group-geometrical construction of $d=11$ supergravity
  11. Conclusions
  12. Super-Poincaré Lie Algebra in $d=4$