Higher-dimensional Origin of D=3 Coset Symmetries

TL;DR

The paper develops a comprehensive classification of oxidation endpoints for all three-dimensional scalar cosets G/H with G maximally non-compact, showing that toroidal reductions of higher-dimensional gravities yield these cosets and that the highest dimension from which they originate, D_max, follows regular patterns (often D_max = rank(G) + 2 or +3). It systematically analyzes simply-laced sequences (A_n, D_n, E_n) and non-simply-laced families (B_n, C_n, G_2, F_4), establishing explicit Lagrangians, root structures, and Dynkin-diagram-driven oxidation pathways, including embeddings of non-simply-laced groups into simply-laced ones. A striking feature is the emergence of a magic symmetry table that links oxidation steps across dimensions, revealing deep regularities in U-duality-like structures and the role of hermitian/quaternionic spaces as special cases. The results illuminate how higher-dimensional dualities organize the space of 3D coset theories and provide a framework for understanding hidden symmetries in reduced supergravity theories, with potential implications for non-supersymmetric reductions and extended duality algebras.

Abstract

It is well known that the toroidal dimensional reduction of supergravities gives rise in three dimensions to theories whose bosonic sectors are described purely in terms of scalar degrees of freedom, which parameterise sigma-model coset spaces. For example, the reduction of eleven-dimensional supergravity gives rise to an E_8/SO(16) coset Lagrangian. In this paper, we dispense with the restrictions of supersymmetry, and study all the three-dimensional scalar sigma models G/H where G is a maximally-non-compact simple group, with H its maximal compact subgroup, and find the highest dimensions from which they can be obtained by Kaluza-Klein reduction. A magic triangle emerges with a duality between rank and dimension. Interesting also are the cases of Hermitean symmetric spaces and quaternionic spaces.