N=8 gaugings revisited: an exhaustive classification

TL;DR

This work solves the long-standing problem of all possible electric gaugings in $N=8$ supergravity by recasting the consistency conditions as a single algebraic constraint on the embedding of the gauge group into $SL(8,\mathbb{R})\subset E_{7(7)}$. The authors derive a 36-parameter solution for the embedding matrix ${\cal E}$ and show that, after factoring out $SL(8,\mathbb{R})$ conjugations, the physically distinct theories are organized into a discrete set of 36 orbits, each corresponding to a real form of $SO(8)$ (and its contractions). They prove that Hull’s gaugings constitute the complete classification of $N=8$ models and identify a unique 7-dimensional abelian gauging CSO$(1,7)$, which intriguingly lies outside the maximal abelian ideal, implying subtle structure for flat directions and supersymmetry breaking. The paper provides an explicit construction of the embedding, using the $56$-dimensional fundamental of $E_{7(7)}$ and the $SL(8,\mathbb{R})$ subalgebra data, and discusses the implications for $p$-brane physics and AdS holography. Overall, the work delivers a complete, constructive taxonomy of $N=8$ gaugings with a clear link to Hull’s earlier results, enhancing our understanding of maximal supergravity gaugings and their geometric origin.

Abstract

In this paper we reconsider, for N=8 supergravity, the problem of gauging the most general electric subgroup. We show that admissible theories are fully characterized by a single algebraic equation to be satisfied by the embedding of the gauge group G within the electric subalgebra SL(8,\IR) of E_{7(7)}. The complete set of solutions to this equation contains 36 parameters. Modding by the action of SL(8,\IR) conjugations that yield equivalent theories all continuous parameters are eliminated except for an overall coupling constant and we obtain a discrete set of orbits. This set is in one--to--one correspondence with 36 Lie subalgebras of SL(8,\IR), corresponding to all possible real forms of the SO(8) Lie algebra plus a set of contractions thereof. By means of our analysis we establish the theorem that the N=8 gaugings constructed by Hull in the middle eighties constitute the exhaustive set of models. As a corollary we show that there exists a unique 7--dimensional abelian gauging. The corresponding abelian algebra is not contained in the maximal abelian ideal of the solvable Lie algebra generating the scalar manifold E_{7(7)}/SU(8).

Figures (3)

• Figure 1: Decomposition in irreducible representations of a tensor of type $T^A{}_{BCD}$
• Figure 2: Decomposition in irreducible representations of a tensor of type $Z^{CD}_{ABEF}$
• Figure 3: $E_7$ Dynkin diagram and root labeling

Theorems & Definitions (5)

• Theorem 3.1
• Proof 3.1
• Proposition 5.1
• Proof 5.1
• Proposition 5.2