Extended Symmetries in Supergravity: the Semi-simple Case

TL;DR

This paper extends the mathematical framework of supergravity theories by exploring Kac-Moody extensions for semi-simple Lie algebras, particularly emphasizing the role of an su(2) factor in N=2 supergravity.

Contribution

It generalizes the Kac-Moody extension theory from simple to semi-simple Lie algebras, including the addition of su(2) factors for N=2 supergravity.

Findings

Extended Kac-Moody framework to semi-simple algebras

Identified the necessity of su(2) augmentation in N=2 supergravity

Provided mathematical tools for analyzing supergravity symmetries

Abstract

The bosonic sector of various supergravity theories reduces to a homogeneous space G/H in three dimensions. The corresponding algebras g are simple for (half-)maximal supergravity, but can be semi-simple for other theories. We extend the existing literature on the Kac-Moody extensions of simple Lie algebras to the semi-simple case. Furthermore, we argue that for N=2 supergravity the simple algebras have to be augmented with an su(2) factor.