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

TL;DR

The paper develops a group-theoretic framework for RR versus NS scalars in string-derived supergravity by isolating the solvable subalgebra $G_s$ that generates the scalar manifold $U/H$. It shows how decomposing $G_s$ with respect to ST-duality clarifies which scalars are RR vs NS and analyzes maximal rank algebras across maximal supergravities, including explicit NS/RR counts and the structure of the electric subgroup. The c-map examples illustrate a rank-one quaternionic fiber with a PQ-like abelian ideal, and the results connect to partial SUSY breaking via gauging of nilpotent generators of the abelian ideal. The framework provides a unified algebraic lens on dualities, Peccei-Quinn symmetries, and potential nonperturbative facets of string theory.

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 Kählerian moduli spaces of Calabi-Yau threefolds.