Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity

TL;DR

This paper presents a method to generate new Lie (super)algebras from existing ones by expanding Maurer-Cartan forms, with applications to supergravity and M-theory, including derivation of the M-theory superalgebra from osp(1|32).

Contribution

The paper introduces a systematic expansion method for Lie (super)algebras that can produce generalized contractions and new algebraic structures relevant to supergravity and M-theory.

Findings

Derived M-theory superalgebra from osp(1|32)

Extended method to gauge free differential superalgebras

Applied to D=3 Chern-Simons supergravity

Abstract

We study how to generate new Lie algebras $\mathcal{G}(N_0,..., N_p,...,N_n)$ from a given one $\mathcal{G}$. The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter $\lambda$ which rescales the coordinates of the Lie (super)group $G$, $g^{i_p} \to \lambda^p g^{i_p}$, in a way subordinated to the splitting of $\mathcal{G}$ as a sum $V_0 \oplus ... \oplus V_p \oplus ... \oplus V_n$ of vector subspaces. We also show that, under certain conditions, one of the obtained algebras may correspond to a generalized \.In\"on\"u-Wigner contraction in the sense of Weimar-Woods, but not in general. The method is used to derive the M-theory superalgebra, including its Lorentz part, from $osp(1|32)$. It is also extended to include gauge free differential (super)algebras and Chern-Simons theories, and then applied to D=3 CS supergravity.