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On Lagrangians and Gaugings of Maximal Supergravities

Bernard de Wit, Henning Samtleben, Mario Trigiante

TL;DR

This work establishes a unified, group-theoretical framework for classifying consistent gaugings of maximal supergravity across dimensions 3 to 7 by constraining the $T$-tensor to lie in the $912$ representation of the duality group. It systematically analyzes four duality bases in four dimensions, derives exact SU(8) decompositions and T-tensor relations that guarantee supersymmetry, and presents a versatile set of Scherk–Schwarz reductions as explicit examples. The authors demonstrate how known $d=4$ and $d=5$ gaugings emerge from embedding matrices in these bases, including non-semisimple and contracted groups such as CSO variants, and lay out projector machinery for computer-aided classification. The approach enables efficient identification of viable gaugings and clarifies the connection between higher-dimensional reductions and gauged theories, with potential to reveal new consistent gaugings and their scalar potentials.

Abstract

A consistent gauging of maximal supergravity requires that the T-tensor transforms according to a specific representation of the duality group. The analysis of viable gaugings is thus amenable to group-theoretical analysis, which we explain and exploit for a large variety of gaugings. We discuss the subtleties in four spacetime dimensions, where the ungauged Lagrangians are not unique and encoded in an E_7(7)\Sp(56,R)/GL(28) matrix. Here we define the T-tensor and derive all relevant identities in full generality. We present a large number of examples in d=4,5 spacetime dimensions which include non-semisimple gaugings of the type arising in (multiple) Scherk-Schwarz reductions. We also present some general background material on the latter as well as some group-theoretical results which are necessary for using computer algebra.

On Lagrangians and Gaugings of Maximal Supergravities

TL;DR

This work establishes a unified, group-theoretical framework for classifying consistent gaugings of maximal supergravity across dimensions 3 to 7 by constraining the -tensor to lie in the representation of the duality group. It systematically analyzes four duality bases in four dimensions, derives exact SU(8) decompositions and T-tensor relations that guarantee supersymmetry, and presents a versatile set of Scherk–Schwarz reductions as explicit examples. The authors demonstrate how known and gaugings emerge from embedding matrices in these bases, including non-semisimple and contracted groups such as CSO variants, and lay out projector machinery for computer-aided classification. The approach enables efficient identification of viable gaugings and clarifies the connection between higher-dimensional reductions and gauged theories, with potential to reveal new consistent gaugings and their scalar potentials.

Abstract

A consistent gauging of maximal supergravity requires that the T-tensor transforms according to a specific representation of the duality group. The analysis of viable gaugings is thus amenable to group-theoretical analysis, which we explain and exploit for a large variety of gaugings. We discuss the subtleties in four spacetime dimensions, where the ungauged Lagrangians are not unique and encoded in an E_7(7)\Sp(56,R)/GL(28) matrix. Here we define the T-tensor and derive all relevant identities in full generality. We present a large number of examples in d=4,5 spacetime dimensions which include non-semisimple gaugings of the type arising in (multiple) Scherk-Schwarz reductions. We also present some general background material on the latter as well as some group-theoretical results which are necessary for using computer algebra.

Paper Structure

This paper contains 15 sections, 92 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Coset-space geometry and duality
  3. The ${\rm SL}(8,\mathbb{R})$-basis: ${\sf E}={\sf E}_{{\rm SL}(8,\mathbb{R})}={\rm Id}$
  4. The ${\rm E}_{6(6)}$-basis: ${\sf E}={\sf E}_{{\rm E}_{6(6)}}$
  5. The ${\rm SL}(2,\mathbb{R})\times {\rm SO}(1,1)\times {\rm SL}(6,\mathbb{R})$-basis: ${\sf E}={\sf E}_{{\rm SL}(2,\mathbb{R})\times {\rm SO}(1,1)\times {\rm SL}(6,\mathbb{R})}$
  6. The ${\rm SU^\star (8)}$-basis: ${\sf E}={\sf E}_{{\rm SU^\star (8)}}$
  7. The $T$-tensor
  8. Group-theoretic analysis
  9. Scherk-Schwarz reductions
  10. The known gaugings in $d=4$ dimensions
  11. Gauge groups embedded in ${\rm SL}(8,\mathbb{R})$
  12. Gauge groups embedded in the ${\rm E}_{6(6)}$ basis
  13. Gauge groups embedded into ${\rm SU}^\star (8)$
  14. Some gaugings in $d=5$ dimensions
  15. Projectors on the embedding matrix: a general discussion