The maximal D=4 supergravities

TL;DR

The paper presents a complete, duality-covariant framework for all gaugings of four-dimensional maximal ${N=8}$ supergravity, encoded by the embedding tensor $oldsymbol{ heta}$ in ${f 56} imes{f 133}$ of ${ m E}_{7(7}}$. Consistency requires a linear ${f 912}$ constraint and a quadratic constraint to ensure closure and supersymmetry; magnetic charges are accommodated via a tensor hierarchy with ${f 133}$-valued fields, yielding a universal Lagrangian and transformation rules expressed through the ${ m T}$-tensor. The authors derive explicit universal formulas for the gauged Lagrangian, fermionic mass terms, and the scalar potential, and discuss how various gaugings arise from higher-dimensional fluxes, twists, and Scherk–Schwarz reductions. This formalism unifies electric/magnetic frames and clarifies the role of the embedding tensor in connecting four-dimensional gaugings to string/M-theory compactifications. The work provides a practical blueprint for constructing and analyzing all consistent maximal gaugings in four dimensions, including flux-induced and higher-dimensional-origin scenarios, with explicit group-theoretical constraints guiding the allowed deformations.

Abstract

All maximal supergravities in four space-time dimensions are presented. The ungauged Lagrangians can be encoded in an E_7(7)\Sp(56,R)/GL(28) matrix associated with the freedom of performing electric/magnetic duality transformations. The gauging is defined in terms of an embedding tensor which encodes the subgroup of E_7(7) that is realized as a local invariance. This embedding tensor may imply the presence of magnetic charges which require corresponding dual gauge fields. The latter can be incorporated by using a recently proposed formulation that involves tensor gauge fields in the adjoint representation of E_7(7). In this formulation the results take a universal form irrespective of the electric/magnetic duality basis. We present the general class of supersymmetric and gauge invariant Lagrangians and discuss a number of applications.