Universal Corner Symmetry and the Orbit Method for Gravity

TL;DR

The paper identifies a universal corner symmetry ($UCS$) that organizes gravitational phase space at codimension-2 corners and shows that its representations can be fruitfully studied via the coadjoint orbit method, yielding a unified treatment of finite-distance and asymptotic corner physics. It then provides a geometric reinterpretation of UCS as the symmetry of an Atiyah Lie algebroid over the corner, with moment maps linking the classical field space to the dual algebroid and revealing an affine rank-2 bundle as a concrete representation that encodes bulk metric data near corners. The ECS and ACS subalgebras emerge naturally as orbit directions within this framework, enabling a corner-centric description that bypasses reliance on a fixed spacetime geometry and pointing toward a potential quantum gravitational theory where spacetime configurations arise as semi-classical reconstructions. Overall, the work lays a foundation for a corner-based quantum gravity program, clarifying how corner symmetries encode both finite-distance and asymptotic gravitational data and suggesting new avenues for edge modes, holography, and BRST-enabled quantization.

Abstract

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