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Solvable Lie Algebras in Type IIA, Type IIB and M Theories

L. Andrianopoli, R. D'Auria, S. Ferrara, P. Fré, R. Minasian, M. Trigiante

TL;DR

Solvable Lie algebras provide a unifying algebraic framework for the scalar sectors of maximal supergravity arising in type IIA, type IIB, and M-theory compactifications. By identifying the U-duality cosets $U/H$ with group manifolds of solvable algebras, the authors map NS and RR scalars to distinct root spaces and classify translational symmetries through maximal abelian nilpotent ideals, linking to brane wrapping. They develop embedding chains of $E_{r+1(r+1)}$ that encode IIA, IIB, and M-theory reductions, relate abelian ideals to brane wrapping, and provide a concrete gauging recipe including both compact and translational isometries, aided by a canonical coset parametrization via boosted structure constants. The work sheds light on dualities, gauging, and potential partial supersymmetry breaking within string/M-theory, offering a group-theoretical lens to identify vacua and nonperturbative structures across dimensions.

Abstract

We study some applications of solvable Lie algebras in type IIA, type IIB and M theories. RR and NS generators find a natural geometric interpretation in this framework. Special emphasis is given to the counting of the abelian nilpotent ideals (translational symmetries of the scalar manifolds) in arbitrary D dimensions. These are seen to be related, using Dynkin diagram techniques, to one-form counting in D+1 dimensions. A recipy for gauging isometries in this framework is also presented. In particular, we list the gauge groups both for compact and translational isometries. The former agree with some results already existing in gauged supergravity. The latter should be possibly related to the study of partial supersymmetry breaking, as suggested by a similar role played by solvable Lie algebras in N=2 gauged supergravity.

Solvable Lie Algebras in Type IIA, Type IIB and M Theories

TL;DR

Solvable Lie algebras provide a unifying algebraic framework for the scalar sectors of maximal supergravity arising in type IIA, type IIB, and M-theory compactifications. By identifying the U-duality cosets with group manifolds of solvable algebras, the authors map NS and RR scalars to distinct root spaces and classify translational symmetries through maximal abelian nilpotent ideals, linking to brane wrapping. They develop embedding chains of that encode IIA, IIB, and M-theory reductions, relate abelian ideals to brane wrapping, and provide a concrete gauging recipe including both compact and translational isometries, aided by a canonical coset parametrization via boosted structure constants. The work sheds light on dualities, gauging, and potential partial supersymmetry breaking within string/M-theory, offering a group-theoretical lens to identify vacua and nonperturbative structures across dimensions.

Abstract

We study some applications of solvable Lie algebras in type IIA, type IIB and M theories. RR and NS generators find a natural geometric interpretation in this framework. Special emphasis is given to the counting of the abelian nilpotent ideals (translational symmetries of the scalar manifolds) in arbitrary D dimensions. These are seen to be related, using Dynkin diagram techniques, to one-form counting in D+1 dimensions. A recipy for gauging isometries in this framework is also presented. In particular, we list the gauge groups both for compact and translational isometries. The former agree with some results already existing in gauged supergravity. The latter should be possibly related to the study of partial supersymmetry breaking, as suggested by a similar role played by solvable Lie algebras in N=2 gauged supergravity.

Paper Structure

This paper contains 14 sections, 82 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The solvable Lie algebra structure: NS and RR scalar fields
  3. Counting of massless modes in sequential toroidal compactifications of $D=10$ type IIA superstring
  4. $E_{r+1}$ subalgebra chains and their string interpretation
  5. Structure of the $E_{r+1(r+1)}$ root spaces and of the associated solvable algebras
  6. Simple roots and Dynkin diagrams
  7. String theory interpretation of the sequential embeddings: Type $IIA$, type $IIB$ and $M$ theory chains
  8. The maximal abelian ideals ${\cal A}_{r+1} \subset Solv_{r+1}$ of the solvable Lie algebra
  9. The maximal abelian ideal from an algebraic viewpoint
  10. The maximal abelian ideal from a physical perspective: the vector fields in one dimension above and translational symmetries
  11. Maximal abelian ideal and brane wrapping
  12. Gauging
  13. Filtration of the $E_{r+1}$ root space, canonical parametrization of the coset representatives and boosted structure constants
  14. Gauging of compact and translational isometries

Figures (4)

  • Figure 1:
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  • Figure 3:
  • Figure 4: