Coset symmetries and coadjoint orbits
Ismaël Ahlouche Lahlali, Josh A. O'Connor
TL;DR
The notes establish a unified view where particle actions can be derived either from non-linear coset realisations or from coadjoint orbits of symmetry groups, with both routes yielding identical dynamics at the level of worldline actions. They proceed from empty-space constructions through concrete particle models (massive, massless, and in backgrounds like EM fields) and extend to extended objects, conformal and affine symmetries, and gravity as a non-linear realisation, highlighting how MC data, inverse Higgs constraints, and invariant measures generate familiar Lagrangians such as the Nambu–Goto and Einstein–Hilbert actions. A thorough introduction to symplectic geometry motivates coadjoint orbits as natural classical phase spaces, with Kostant–Kirillov–Souriau forms and symplectic reduction providing geometric actions whose extrema reproduce particle trajectories, including massive and spinning cases in AdS and Minkowski spaces. The final discussion on semidirect-product groups (notably the Poincaré group) shows how coadjoint orbits decompose into momentum orbits and little-group dynamics, linking classical phase-space pictures to Wigner’s quantum classifications and suggesting paths toward geometric quantization of these structures. Overall, the work offers a cohesive framework connecting coset methods, symplectic geometry, and representation theory to construct and understand particle dynamics in a broad symmetry context, with explicit constructions and exemplars throughout.
Abstract
In these lectures we review two approaches to constructing particle actions from coset spaces of symmetry groups: non-linear realisations and coadjoint orbits. At the level of particle actions, we observe that they coincide. We also provide an introduction to symplectic geometry and we sketch the theory of coadjoint orbits for the Poincaré group.