The maximal D=4 supergravities

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.

• The paper introduces an embedding tensor that encodes the invariant E7(7) subgroup to manage both electric and magnetic charges.
• It employs a novel framework using tensor gauge fields to enforce supersymmetry and gauge invariance without extra degrees of freedom.
• The findings refine duality properties and offer a standardized method to explore varied gaugings in maximal D=4 supergravity models.

Insights into the Maximal D=4 Supergravities

This paper provides a comprehensive analysis of maximal supergravity theories in four-dimensional spacetime, with a focus on their embedding within extended symmetry frameworks. Specifically, the work explores the formulation of gauged supergravities, characterized by their interaction with gauge fields, and examines the constraints necessary to preserve supersymmetry.

The authors propose a novel approach to characterizing the gaugings of these theories through the use of an embedding tensor. This tensor acts as a crucial component, encoding the subgroup of the E7(7) duality group that remains invariant under local transformations. Known for its capacity to account for electric and magnetic charges, the embedding tensor allows a systematic exploration of possible gaugings. The definition of this tensor comes with two primary constraints: a linear constraint, ensuring that it belongs to the representation of E7(7), and a quadratic constraint, which guarantees closure of the gauge group and enforces consistency of the Lagrangian.

A key methodological advancement presented in the paper is the ability to handle both electric and magnetic charges via the embedding tensor formalism. This is achieved without introducing additional degrees of freedom, by employing a new framework that incorporates tensor gauge fields within the adjoint representation of E7(7). Consequently, all derived supergravity actions become invariant under both supersymmetry and gauge transformations. Notably, the formulation's robustness remains irrespective of the chosen electric/magnetic duality basis, underlining its versatility.

The implications of the findings are twofold. Practically, the framework offers a standardized path to explore diverse gaugings of N = 8 supergravity, accommodating configurations from disparate origins, such as those emerging from compactifications of higher-dimensional theories. Theoretically, it refines the understanding of the duality properties inherent in supergravity, paving the way for potential exploration of dualities within string theory and M-theory contexts.

The numerical results corroborate earlier findings, such as the viability of specific gauged supergravity models in relation to flux compactifications and Scherk-Schwarz reductions. Moreover, the universal structure of the scalar potential, invariant under electric/magnetic duality transformations, emphasizes the theoretical coherence of these supergravity models, providing a promising avenue for further study.

In conclusion, the paper delineates a rigorous theoretical framework for the maximal supergravities in four dimensions by extending their gauge interactions through the embedding tensor. The work not only resolves several complexities associated with electric/magnetic charges but also opens new directions for research into the intricate tapestry of higher-dimensional supergravity theories and string dualities. Future inquiries might well revolve around the functional integration of these theories with gauged R-symmetries, stability analyses of resulting scalar potentials in cosmological models, or a deeper exploration of their role in non-perturbative effects in string/M-theory landscapes.