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Kac-Moody Spectrum of (Half-)Maximal Supergravities

Eric Bergshoeff, Joaquim Gomis, Teake Nutma, Diederik Roest

TL;DR

The paper develops a precise link between gaugings and massive deformations of half-maximal supergravity with vector multiplets and the generators of the associated very extended Kac–Moody algebras. It introduces the $p$-form algebra as a truncation that captures propagating bosonic fields plus deformation and top-form potentials, and shows how deformation potentials and top-forms map to embedding-tensor data and its quadratic constraints. The half-maximal case matches known gaugings and constraints and yields predictions for higher dimensions, while deformations generically require extensions beyond the $p$-form sector, involving mixed-symmetry generators. Overall, the results place maximal and half-maximal supergravities under a unified KM framework via $SO(8,8+n)^{+++}$, highlighting both the power and limits of the KM approach for capturing deformations and their consistency conditions.

Abstract

We establish the correspondence between, on one side, the possible gaugings and massive deformations of half-maximal supergravity coupled to vector multiplets and, on the other side, certain generators of the associated very extended Kac-Moody algebras. The difference between generators associated to gaugings and to massive deformations is pointed out. Furthermore, we argue that another set of generators are related to the so-called quadratic constraints of the embedding tensor. Special emphasis is placed on a truncation of the Kac-Moody algebra that is related to the bosonic gauge transformations of supergravity. We give a separate discussion of this truncation when non-zero deformations are present. The new insights are also illustrated in the context of maximal supergravity.

Kac-Moody Spectrum of (Half-)Maximal Supergravities

TL;DR

The paper develops a precise link between gaugings and massive deformations of half-maximal supergravity with vector multiplets and the generators of the associated very extended Kac–Moody algebras. It introduces the -form algebra as a truncation that captures propagating bosonic fields plus deformation and top-form potentials, and shows how deformation potentials and top-forms map to embedding-tensor data and its quadratic constraints. The half-maximal case matches known gaugings and constraints and yields predictions for higher dimensions, while deformations generically require extensions beyond the -form sector, involving mixed-symmetry generators. Overall, the results place maximal and half-maximal supergravities under a unified KM framework via , highlighting both the power and limits of the KM approach for capturing deformations and their consistency conditions.

Abstract

We establish the correspondence between, on one side, the possible gaugings and massive deformations of half-maximal supergravity coupled to vector multiplets and, on the other side, certain generators of the associated very extended Kac-Moody algebras. The difference between generators associated to gaugings and to massive deformations is pointed out. Furthermore, we argue that another set of generators are related to the so-called quadratic constraints of the embedding tensor. Special emphasis is placed on a truncation of the Kac-Moody algebra that is related to the bosonic gauge transformations of supergravity. We give a separate discussion of this truncation when non-zero deformations are present. The new insights are also illustrated in the context of maximal supergravity.

Paper Structure

This paper contains 18 sections, 53 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The Kac--Moody approach to supergravity
  3. The $p$--form algebra
  4. Truncation to $p$--forms
  5. Deformation potentials
  6. Top--form potentials
  7. Matter coupled half--maximal supergravity
  8. Kac--Moody and $p$--form algebras
  9. Deformation potentials
  10. Top--form potentials
  11. Deformations
  12. Massive IIA supergravity
  13. Half-maximal supergravity
  14. Conclusions
  15. Terminology and notation
  16. ...and 3 more sections

Figures (10)

  • Figure 1: The Dynkin diagrams of $E_8$\ref{['sub@E8']}, the very extended $E_8^{+++}$\ref{['sub@veE8']}, and its decompositions corresponding to 11D \ref{['sub@E11-11D']}, IIA \ref{['sub@E11-IIA']} and IIB \ref{['sub@E11-IIB']} supergravity. In these decompositions the black nodes are disabled, the white nodes correspond to the gravity line $SL(D,\mathbb{R})$ and the gray node in the last diagram corresponds to the duality group $A_1$.
  • Figure 2: $D_{8}^{+++}$ decomposed as $A_{9}^{}$
  • Figure 3: $D_{8}^{+++}$ decomposed as $A_{8}^{}$
  • Figure 4: $D_{8}^{+++}$ decomposed as $A_{1}^{} \otimes A_{1}^{} \otimes A_{7}^{}$
  • Figure 5: $D_{8}^{+++}$ decomposed as $A_{3}^{} \otimes A_{6}^{}$
  • ...and 5 more figures