Space, Matter and Interactions in a Quantum Early Universe. Part II : Superalgebras and Vertex Algebras

TL;DR

This paper develops a mathematical framework for quantum gravity by extending Lie algebras into superalgebras and vertex algebras, incorporating scattering mechanisms and Poincaré symmetry.

Contribution

It introduces an infinite-dimensional Lie superalgebra extending e9, with a novel scattering mechanism and integration into a vertex algebra for quantum gravity modeling.

Findings

Defined a new Borcherds algebra-based Lie superalgebra.

Established a Poincaré group action on the superalgebra.

Developed a scattering mechanism including resonant decays.

Abstract

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra $\textbf{${\mathfrak g}_{\mathsf u}$}$ that extends $\mathbf{e_9}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn $\textbf{${\mathfrak g}_{\mathsf u}$}$ into a Lie superalgebra $\textbf{$\mathfrak {sg}_{\mathsf u}$}$ with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincar\'e group on $\textbf{$\mathfrak {sg}_{\mathsf u}$}$, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular {\it resonant scattering}. Finally, we complete the model by merging the local $\textbf{$\mathfrak {sg}_{\mathsf u}$}$ into a vertex-type algebra.