Representations of G+++ and the role of space-time

TL;DR

This paper explores the algebraic structure of very extended Kac-Moody algebras G+++ and their relation to space-time, revealing how certain representations relate to space-time translations and brane charges in non-linear realizations.

Contribution

It analyzes the decomposition of G+++ algebras with respect to A type subalgebras, highlighting the correspondence between fundamental and adjoint representations and their implications for space-time emergence.

Findings

Most A type representations in fundamental representations also appear in the adjoint of G+++.

In some cases like An+++, the adjoint contains no space-time translation generators.

A correspondence exists between A representations and brane charge representations.

Abstract

We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras G+++ with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of G+++. In particular, for E8+++, this applies to all its fundamental representations. However, there are some important examples, such as An+++, where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on G+++. Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of G+++ which is consistent with the interpretation of the former as charges associated with brane solutions.