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On the vacua of N = 8 gauged supergravity in 4 dimensions

G. Dall'Agata, G. Inverso

TL;DR

This work introduces an algebraic approach to finding vacua of $N=8$ gauged supergravity in four dimensions by varying the embedding tensor $\Theta_M{}^\alpha$ and exploiting the coset structure of the scalar manifold $\mathrm{E}_{7(7)}/\mathrm{SU}(8)$. By mapping any critical point to the base point $\phi=0$, the problem reduces to solving linear and quadratic constraints on $\Theta$, enabling analytic reconstruction of known vacua and the discovery of a dozen new analytical solutions, with complete scalar spectra. The analysis clarifies how the residual gauge symmetry fixes the vacuum energy and how vacuum moduli influence the mass spectrum, including moduli-dependent Minkowski vacua related to Scherk–Schwarz-type reductions. The approach provides a systematic, largely analytic path toward classifying gaugings and vacua in maximal 4D supergravity and suggests connections to higher-dimensional flux compactifications and duality frames.

Abstract

We discuss a simple procedure for finding vacua of gauged supergravity models, based on the variation of the embedding tensor rather than on a direct minimization of the scalar potential. We apply this procedure to N=8 gauged supergravity in 4 dimensions. We easily recover many of the previously known vacua, also completing their scalar mass spectrum, and we apply our procedure to find a dozen of new analytical vacuum solutions. The analysis shows an interesting structure on the moduli spaces of these vacua and provides new criteria to determine the expected value of the cosmological constant by a simple inspection of the group properties of the embedding tensor.

On the vacua of N = 8 gauged supergravity in 4 dimensions

TL;DR

This work introduces an algebraic approach to finding vacua of gauged supergravity in four dimensions by varying the embedding tensor and exploiting the coset structure of the scalar manifold . By mapping any critical point to the base point , the problem reduces to solving linear and quadratic constraints on , enabling analytic reconstruction of known vacua and the discovery of a dozen new analytical solutions, with complete scalar spectra. The analysis clarifies how the residual gauge symmetry fixes the vacuum energy and how vacuum moduli influence the mass spectrum, including moduli-dependent Minkowski vacua related to Scherk–Schwarz-type reductions. The approach provides a systematic, largely analytic path toward classifying gaugings and vacua in maximal 4D supergravity and suggests connections to higher-dimensional flux compactifications and duality frames.

Abstract

We discuss a simple procedure for finding vacua of gauged supergravity models, based on the variation of the embedding tensor rather than on a direct minimization of the scalar potential. We apply this procedure to N=8 gauged supergravity in 4 dimensions. We easily recover many of the previously known vacua, also completing their scalar mass spectrum, and we apply our procedure to find a dozen of new analytical vacuum solutions. The analysis shows an interesting structure on the moduli spaces of these vacua and provides new criteria to determine the expected value of the cosmological constant by a simple inspection of the group properties of the embedding tensor.

Paper Structure

This paper contains 10 sections, 76 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. $N=8$ supergravity
  3. Supersymmetry breaking patterns
  4. Finding extrema of the scalar potential
  5. Symplectic frames
  6. Vacua of the gauged theory
  7. $\theta \neq 0$
  8. $\theta \neq 0$ and $\xi \neq 0$
  9. On the spectra of the SO(6,2) and CSO(2,0,6) models
  10. Comments

Figures (2)

  • Figure 1: Stationary points of the scalar potential found for $\phi = \phi'$ can be translated to the fixed point of the scalar manifold $\phi = 0$ by a non-compact isometry transformation $U$ and by a redefinition of the embedding tensor $\Theta' = U \Theta$.
  • Figure 2: Values of the masses of 3 of the scalar fields of the SO(6,2) spectrum as a function of two of the moduli fields. The first and the last plot are referred to massless scalars at the maximally symmetric critical point. It can be seen that in some directions the masses become negative and the potential unstable.