Edge modes of gravity -- I: Corner potentials and charges
Laurent Freidel, Marc Geiller, Daniele Pranzetti
TL;DR
The paper develops a framework for gravity where the bulk theories are equivalent but different Lagrangian choices induce distinct corner symmetries via corner terms in the symplectic potential. It shows, in metric gravity, that EH introduces a nontrivial sl(2,R)^S_⊥ boost sector absent in GR, while GH relates to a relative boost angle conjugate to area; in tetrad gravity, the corner symmetry is diff(S) ⋉ sl(2,C)^S, with edge modes encoding boundary degrees of freedom that enable gauge restoration and potential bulk reconstruction. A detailed bulk–boundary decomposition clarifies how Holst (Barbero–Immirzi) parameter and internal normals modify boundary charges and induce additional edge data, guiding the quantization program through edge modes and representation theory of corner algebras. The results advocate a maximally extended corner symmetry as a guiding principle for quantum gravity and lay the groundwork for subsequent Edge-Mode papers focusing on the explicit edge content and its quantization. Overall, the work provides a systematic method to relate Lagrangian choices, corner symmetries, and edge-mode degrees of freedom to the quest for quantum gravity.
Abstract
This is the first paper in a series devoted to understanding the classical and quantum nature of edge modes and symmetries in gravitational systems. The goal of this analysis is to: i) achieve a clear understanding of how different formulations of gravity provide non-trivial representations of different sectors of the corner symmetry algebra, and ii) set the foundations of a new proposal for states of quantum geometry as representation states of this corner symmetry algebra. In this first paper we explain how different formulations of gravity, in both metric and tetrad variables, share the same bulk symplectic structure but differ at the corner, and in turn lead to inequivalent representations of the corner symmetry algebra. This provides an organizing criterion for formulations of gravity depending on how big the physical symmetry group that is non-trivially represented at the corner is. This principle can be used as a "treasure map" revealing new clues and routes in the quest for quantum gravity. Building up on these results, we perform a detailed analysis of the corner symplectic potential and symmetries of Einstein-Cartan-Holst gravity in [1], use this to provide a new look at the simplicity constraints in [2], and tackle the quantization in [3].