On the exceptional generalised Lie derivative for $d\ge 7$

Abstract

In this work we revisit the $E_8\times {\mathbb{R}}^{+}$ generalised Lie derivative encoding the algebra of diffeomorphisms and gauge transformations of compactifications of M-theory on eight-dimensional manifolds, by extending certain features of the $E_7\times {\mathbb{R}}^{+}$ one. Compared to its $E_d\times {\mathbb{R}}^{+},d\le 7$ counterparts, a new term is needed for consistency. However, we find that no compensating parameters need to be introduced, but rather that the new term can be written in terms of the ordinary generalised gauge parameters by means of a connection. This implies that no further degrees of freedom, beyond those of the field content of the $E_8$ group, are needed to have a well defined theory. We discuss the implications of the structure of the $E_8\times {\mathbb{R}}^{+}$ generalised transformation on the construction of the $d=8$ generalised geometry. Finally, we suggest how to lift the generalised Lie derivative to eleven dimensions.

Version accepted to JHEP

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• The exceptional story of massive IIA supergravity
• The Geometry, Branes and Applications of Exceptional Field Theory
• Generalised Scherk-Schwarz reductions from gauged supergravity
• The Classical Double Copy for M-theory from a Kerr-Schild Ansatz for Exceptional Field Theory
• Effective Action for Non-Geometric Fluxes from Duality Covariant Actions
• Exceptional geometry and Borcherds superalgebras
• $L_{\infty }$ algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity
• Finite Transformations in Doubled and Exceptional Space
• Finite Gauge Transformations and Geometry in Extended Field Theory
• Extended Drinfel'd algebras and non-Abelian duality
• Strings, Branes and the Self-dual Solutions of Exceptional Field Theory
• Reductions of Exceptional Field Theories
• The landscape of G-structures in eight-manifold compactifications of M-theory
• Duality Covariant Solutions in Extended Field Theories
• Exotic Aspects of Extended Field Theories