Gauging Maximal Supergravities

TL;DR

Gauged maximal supergravities arise from embedding the gauge group into the duality group ${\rm G}$, with the embedding tensor $${\Theta_M{}^\alpha}$$ encoding the gauging and the ${\rm T}$$-tensor organizing fermion shifts and the scalar potential. The paper surveys the embedding tensor formalism across dimensions, detailing quadratic (closure) and linear (representation) constraints that select admissible gaugings and determine the allowed representations for the $${\rm T}$$-tensor components $${A_1,A_2,A_3}$$. It provides concrete d=5 examples yielding ${\rm CSO}(p,q,r)$ gaugings and irreducible ${\bf 144}_{+1}$ or ${\bf 45}_{+4}$ cases from higher-dimensional origins, and addresses subtleties in d=4 where only electric gaugings can be realized in a given Lagrangian, along with an ${\rm E}_{7(7)}$-invariant quadratic constraint for mixed electric/magnetic charges. The IIB flux route demonstrates a rich set of gaugings in four dimensions by decomposing ${\rm E}_{7(7)}$ and parameterizing the embedding tensor with $\theta_{\Lambda\Sigma\Gamma}{}^\tau$ and $\theta^\Lambda$, enabling the construction of the ${\rm T}$-tensor and potential without a full Lagrangian and revealing domain-wall solutions liftable to ten dimensions. Overall, the work presents a unified, representation-theoretic framework to classify and realize gaugings in maximal supergravity and connect them to higher-dimensional flux compactifications.

Abstract

We review recent progress in gauging maximal supergravity theories