R--R Scalars, U--Duality and Solvable Lie Algebras

TL;DR

This paper explores the group-theoretical structure of R--R scalars in string theory's supergravity limit, linking them to solvable Lie algebras within U--duality, and examines their symmetries and algebraic properties.

Contribution

It characterizes the solvable Lie subalgebras associated with R--R scalars and relates them to U--duality and Peccei-Quinn symmetries in supergravity theories.

Findings

Description of maximal rank solvable Lie algebras in supergravities.

Identification of Peccei-Quinn symmetries with abelian ideals.

Explicit example of a rank one solvable algebra from quaternionic spaces.

Abstract

We consider the group theoretical properties of R--R scalars of string theories in the low-energy supergravity limit and relate them to the solvable Lie subalgebra $\IG_s\subset U$ of the U--duality algebra that generates the scalar manifold of the theory: $\exp[\IG_s]= U/H$. Peccei-Quinn symmetries are naturally related with the maximal abelian ideal ${\cal A} \subset \IG_s $ of the solvable Lie algebra. The solvable algebras of maximal rank occurring in maximal supergravities in diverse dimensions are described in some detail. A particular example of a solvable Lie algebra is a rank one, $2(h_{2,1}+2)$--dimensional algebra displayed by the classical quaternionic spaces that are obtained via c-map from the special K\"ahlerian moduli spaces of Calabi-Yau threefolds.