Edge modes of gravity -- II: Corner metric and Lorentz charges

TL;DR

This work develops a corner-centered program for quantum gravity by unpacking the corner symmetry algebra and its representations across BF theory and Einstein–Cartan–Holst gravity. By performing detailed 3+1 and horizontal/vertical decompositions, the authors derive a bulk+corner split of the BF and ECH symplectic potentials, revealing a nontrivial corner algebra that extends diffeomorphisms with an SL(2,C) factor and a parallel SL(2,R) sector tied to the corner metric. A key result is that corner area undergoes a Lorentz-covariant discretization via the SL(2,R)_parallel Casimir, resolving a long-standing tension between area quantization and Lorentz invariance. The paper also introduces edge modes to restore corner gauge invariance while preserving nonzero corner charges, establishing a framework where corner degrees of freedom drive the quantum geometry and suggesting a path toward a finite-dimensional corner Hilbert space. Collectively, these advances place corner symmetries and edge modes at the center of a Lorentz-covariant approach to quantum gravity with potential implications for holography and LQG.

Abstract

In this second paper of the series we continue to spell out a new program for quantum gravity, grounded in the notion of corner symmetry algebra and its representations. Here we focus on tetrad gravity and its corner symplectic potential. We start by performing a detailed decomposition of the various geometrical quantities appearing in BF theory and tetrad gravity. This provides a new decomposition of the symplectic potential of BF theory and the simplicity constraints. We then show that the dynamical variables of the tetrad gravity corner phase space are the internal normal to the spacetime foliation, which is conjugated to the boost generator, and the corner coframe field. This allows us to derive several key results. First, we construct the corner Lorentz charges. In addition to sphere diffeomorphisms, common to all formulations of gravity, these charges add a local $\mathfrak{sl}(2,\mathbb{C})$ component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local $\mathfrak{sl}(2,\mathbb{R})$ algebra, whose Casimir corresponds to the corner area element. Due to the space-like nature of the corner metric, this Casimir belongs to the unitary discrete series, and its spectrum is therefore quantized. This result, which reconciles discreteness of the area spectrum with Lorentz invariance, is proven in the continuum and without resorting to a bulk connection. Third, we show that the corner phase space explains why the simplicity constraints become non-commutative on the corner. This fact requires a reconciliation between the bulk and corner symplectic structures, already in the classical continuum theory. Understanding this leads inevitably to the introduction of edge modes.