From Asymptotic Symmetries to the Corner Proposal

TL;DR

The work proposes a corner-centric reformulation of gauge gravity, linking Noether charges at codimension-2 corners to a universal symmetry UCS and an extended phase space that makes all corner charges integrable. It surveys the covariant phase space formalism, Noether theorems, and the treatment of asymptotic and near-horizon symmetries, then develops the geometric machinery of embeddings (corners), edge modes, and algebroids to realize representations via coadjoint orbits. Central to the program is the universal corner symmetry and its finite-distance subalgebras (ECS) as organizing principles for observables, with explicit demonstrations in AdS3/Brown–Henneaux and finite-distance corner settings. By recasting embeddings as dynamical fields and employing Atiyah algebroids and moment maps, the paper outlines a mathematically coherent path toward quantum gravity where corner charges and their representations encode the physics, potentially surviving beyond classical spacetime. The approach bridges asymptotic symmetries, holography, and the corner program, highlighting open questions on unitary representations and concrete realizations of quantum gravity from corner data.

Abstract

These notes are a transcript of lectures given by the author in the XVIII Modave summer school in mathematical physics. The introduction is devoted to a detailed review of the literature on asymptotic symmetries, flat holography, and the corner proposal. It covers much more material than needed, for it is meant as a lamppost to help the reader in navigating the vast existing literature. The notes then consist of three main parts. The first is devoted to Noether's theorems and their underlying framework, the covariant phase space formalism, with special focus on gauge theories. The surface-charges algebra is shown to projectively represent the asymptotic symmetry algebra. Issues arising in the gravitational case, such as conservation, finiteness, and integrability, are addressed. In the second part, we introduce the geometric concept of corners, and show the existence of a universal asymptotic symmetry group at corners. A careful treatment of corner embeddings provides a resolution to the issue of integrability, by extending the phase space. In the last part we bridge asymptotic symmetries and corners by formulating the corner proposal. In essence, the latter focuses on the central question of extracting from classical gravity universal results that are expected to hold in the quantum realm. After reviewing the coadjoint orbit method and Atiyah Lie algebroids, we apply these concepts to the corner proposal. Exercises are solved in the notes, to elucidate the arguments exposed.