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

Jose A. de Azcarraga, Jose M. Izquierdo, Moises Picon, Oscar Varela

TL;DR

The paper introduces the expansion method, a procedure to generate new (super)algebras $\mathcal{G}(N_0,\ldots,N_n)$ from a given algebra $\mathcal{G}$ by rescaling group parameters and expanding Maurer–Cartan forms in powers of $\lambda$. This approach extends contractions and deformations, and, under Weimar–Woods conditions, recovers generalized Inönü–Wigner contractions while also producing higher-dimensional algebras. It is applied to derive the M-theory superalgebra from $\mathrm{osp}(1|32)$ (with Lorentz part included) as $\mathrm{osp}(1|32)(2,1,2)$, and is extended to gauge free differential algebras and Chern–Simons theories, with explicit CS constructions in $D=3$ illustrating the link to CS supergravity. The overall framework provides a versatile tool to relate and build extended symmetry structures in supergravity and string/M-theory contexts, including enlarged superspace algebras and FDA formulations. The method broadens the landscape of symmetry algebras accessible from a given starting point and offers new avenues for constructing actions based on expanded gauge algebras.

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 $λ$ which rescales the coordinates of the Lie (super)group $G$, $g^{i_p} \to λ^p g^{i_p}$, 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 İnönü-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.

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

TL;DR

The paper introduces the expansion method, a procedure to generate new (super)algebras from a given algebra by rescaling group parameters and expanding Maurer–Cartan forms in powers of . This approach extends contractions and deformations, and, under Weimar–Woods conditions, recovers generalized Inönü–Wigner contractions while also producing higher-dimensional algebras. It is applied to derive the M-theory superalgebra from (with Lorentz part included) as , and is extended to gauge free differential algebras and Chern–Simons theories, with explicit CS constructions in illustrating the link to CS supergravity. The overall framework provides a versatile tool to relate and build extended symmetry structures in supergravity and string/M-theory contexts, including enlarged superspace algebras and FDA formulations. The method broadens the landscape of symmetry algebras accessible from a given starting point and offers new avenues for constructing actions based on expanded gauge algebras.

Abstract

We study how to generate new Lie algebras from a given one . The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter which rescales the coordinates of the Lie (super)group , , in a way subordinated to the splitting of as a sum of vector subspaces. We also show that, under certain conditions, one of the obtained algebras may correspond to a generalized İnönü-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 . It is also extended to include gauge free differential (super)algebras and Chern-Simons theories, and then applied to D=3 CS supergravity.

Paper Structure

This paper contains 15 sections, 8 theorems, 113 equations.

Table of Contents

  1. Introduction and motivation: four methods to derive new Lie algebras from given ones
  2. Rescaling of the group parameters and the expansion method
  3. The Lie algebras $\mathcal{G}(N)$ generated from $\mathcal{G}=V_0\oplus V_1$
  4. Structure of the Lie algebras $\mathcal{G}(N)$
  5. The limiting cases $V_0=0,V_1=V$ and $V_0=V,V_1=0$
  6. The case in which $V_0$ is a subalgebra $\mathcal{L}_0\subset \mathcal{G}$
  7. The case in which $V_1$ is a symmetric coset
  8. Structure of ${\mathcal{G}}(N_0,N_1)$ in the symmetric coset case
  9. Different powers rescaling subordinated to a general splitting of ${\mathcal{G}}$
  10. The expansion method for superalgebras
  11. Application: the M-theory superalgebra
  12. Extension to gauge free differential (super)algebras and Chern-Simons theories
  13. Application to Chern-Simons gauge theory of supergravity
  14. Conclusions and outlook
  15. Appendix: expansion of $d\omega^{k_s,\alpha_s}$

Key Result

Theorem 1

Let $\mathcal{G}$ be a Lie algebra, and ${\mathcal{G}} = V_0 \oplus V_1$ (no subalgebra condition is assumed neither for $V_0$ or $V_1$). Let $\{\omega^i \}$, $\{ \omega^{i_0} \}$, $\{ \omega^{i_1} \}$ ($i=1,\ldots, \textrm{dim} \,{\mathcal{G}}$, $i_0=1,\ldots,\textrm{dim}\,V_0$, $i_1=1,\ldots, \tex together with the MC eqs. (eq:cnts) for the structure constants determines a Lie algebra $\mathcal

Theorems & Definitions (8)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • Proposition 3
  • Theorem 3
  • Theorem 4
  • Theorem 5