From Asymptotic Symmetries to the Corner Proposal

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.

Anisotropic Scaling Non-Relativistic Holography: A Symmetry Perspective

“…However, in our construction, one can realize the enhanced local symmetry of the boundary field theory as asymptotic symmetry of the dual gravity system in a double expansion in large r and large y. Correspondingly, the dual field theory is supposed to be defined on the corner of the 4D boundary; see, e.g., in [26] for a comprehensive introduction of the corner proposal and references therein.…”

General parallel cosmology

“…Physical observables are described as conserved charges, given by fluxes of field excitations on a closed surface (e.g. [18,[63][64][65][66][67]). This is the operational definition of an observable.…”

Covariant canonical formulations of classical field theories

“…Boundary/bulk correspondence: While this idea of a holographic description already finds a basic expression in Noether's theorems (which imply a relation between gauge symmetries and conserved charges given by surface integrals), different dualities have ben pointed out like the so-called AdS/CFT correspondence or the so-called fluid/gravity duality as well as various generalizations like celestial holography. For an overview of these ideas (including several hundred references to the literature), we refer to the guiding introduction of the recent notes of L. Ciambelli [19] dedicated to asymptotic symmetries and the so-called corner proposal.…”

“…L ξ θ = 0 where ξ denotes the symmetry generating vector field.) The symmetry of Maxwell's theory described by (19) leads to a conserved current density ( j µ ) which is presently described by the following 3-form on P:…”