Dualities in the classical supergravity limits

TL;DR

This work analyzes duality symmetries in classical supergravity, focusing on maximally dualised formulations and their organization into U-duality groups $G$ with coset scalar spaces $G/K$ and twisted self-duality relations. It develops three guiding principles (silver rules) linking scalar cosets, local $K$-gauge invariance, and twisted self-duality, and then interrogates how (un)dualisations reshape the duality algebras, often contracting them via abelian ideals in Borel subalgebras. Through Dynkin-diagram analysis, it shows how higher-form content is read from internal symmetries, clarifying dualities among $k$-forms and suggesting larger unifying structures like $F_{11}$ that encode Type II/I and heterotic relations. The paper proposes a universal twisted self-duality framework (TS) and its extension to F-duality, aiming to unify spacetime, internal, and gauge symmetries and guide a coherent picture of classical limits and quantum unification in M-theory contexts.

Abstract

Duality symmetries of supergravity theories are powerful tools to restrict the number of possible actions, to link different dimensions and number of supersymmetries and might help to control quantisation. (Hodge-Dirac-)Dualisation of gauge potentials exchanges Noether and topological charges, equations of motion and Bianchi identities, internal rigid symmetries and gauge symmetries, local transformations with nonlocal ones and most exciting particles and waves. We compare the actions of maximally dualised supergravities (ie with gauge potential forms of lowest possible degree) to the non-dualised actions coming from 11 (or 10) dimensions by plain dimensional reduction as well as to other theories with partial dualisations. The effect on the rigid duality group is a kind of contraction resulting from the elimination of the unfaithful generators associated to the (inversely) dualised scalar fields. New gauge symmetries are introduced by these (un)dualisations and it is clear that a complete picture of duality (F(ull)-duality) should include all gauge symmetries at the same time as the rigid symmetries and the spacetime symmetries. We may read off some properties of F-duality on the internal rigid Dynkin diagram: field content, possible dualisations, increase of the rank according to the decrease of space dimension... Some recent results are included to suggest the way towards unification via a universal twisted self-duality (TS) structure. The analysis of this structure had revealed several profound differences according to the parity mod 4 of the dimension of spacetime (to be contrasted with the (Bott) period 8 of spinor properties).