[Paper Review] From Asymptotic Symmetries to the Corner Proposal

Luca Ciambelli|arXiv (Cornell University)|Dec 27, 2022

Black Holes and Theoretical Physics6 citations

TL;DR

This paper proposes a geometric unification of asymptotic symmetries and corner charges in gauge theories, showing that universal corner symmetry (UCS) emerges from extended phase space formulations. By applying the coadjoint orbit method and Atiyah Lie algebroids, it identifies corner charges as physical observables in quantum gravity, resolving integrability issues via embedding-dependent phase space extensions and providing a framework for unitary representations of gravity's symmetry algebra.

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.

Motivation & Objective

• To unify asymptotic symmetry analysis and corner charge formalism in gauge theories.
• To resolve integrability issues in surface charges by extending the phase space through corner embeddings.
• To provide a geometric foundation for quantum gravity observables using universal corner symmetry (UCS).
• To apply the coadjoint orbit method and algebroid structures to classify physical states in gravity.
• To lay the groundwork for unitary, irreducible representations of UCS in quantum gravity.

Proposed method

• Uses the covariant phase space formalism to derive surface charges from Noether's theorems in gauge theories.
• Introduces corner embeddings to extend the phase space, enabling integrability of charges in dissipative systems.
• Applies the coadjoint orbit method to realize UCS representations as orbits in the dual of the universal corner symmetry algebra.
• Models UCS as automorphisms of an Atiyah Lie algebroid over codimension-2 corners.
• Identifies an affine associated bundle over the corner whose fibers correspond to normal-to-corner coordinates in emergent spacetime.
• Uses moment maps to relate field configurations to points in the coadjoint space, linking symplectic structure to the KKS form.

Experimental results

Research questions

• RQ1 How can asymptotic symmetries be consistently extended to finite-distance corners in gravity?
• RQ2 What is the role of corner embeddings in resolving integrability and finiteness of surface charges?
• RQ3 Can the universal corner symmetry algebra be realized as a coadjoint orbit, and what does this imply for quantum gravity?
• RQ4 How do Atiyah Lie algebroids and associated bundles provide a geometric structure for corner observables?
• RQ5 What is the physical interpretation of the KKS symplectic form on coadjoint orbits in the context of corner charges?

Key findings

• The surface charge algebra projects the asymptotic symmetry algebra, resolving issues of conservation and finiteness in gravity.
• Corner embeddings extend the phase space, enabling integrability of charges even in dissipative systems.
• The universal corner symmetry (UCS) algebra arises naturally as the symmetry algebra of codimension-2 corners in spacetime.
• The coadjoint orbit method provides a geometric classification of physical states, with the KKS form realizing the symplectic structure on field space.
• An affine associated bundle over the UCS algebroid emerges, whose fibers encode normal-to-corner coordinates in an emergent spacetime.
• Moment maps identify field configurations with points in the coadjoint space, embedding classical gravity into abstract orbit theory.