Universal corner symmetry and the orbit method for gravity

A universal symmetry algebra organizing the gravitational phase space has been recently found. It corresponds to subset of diffeomorphisms that become physical at corners -- codimension-$2$ surfaces supporting Noether charges. applies both finite distance and asymptotic corners. In this paper, we study its representations, via coadjoint orbit method. We show generic orbits split into sub-orbits spanned by corner symmetries, such full gives rise a unified treatment in manifold. then identify geometric structure captures these algebraic properties on corners, which is Atiyah Lie algebroid associated principal $GL(2,\mathbb{R})\ltimes \mathbb{R}^2$-bundle. This suggestive existence novel quantum theory would unitarily glue structures, with spacetime geometries appearing as semi-classical configurations.