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

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra \textbf{{\mathfrak g}{\mathsf u}} that extends ${\mathbf{e}}_{\mathbf{9}}$. 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.