Corner symmetry and quantum geometry

TL;DR

The work develops a continuum, boundary-centered framework where corner (codimension-2) symmetries, arising from gauge redundancies in gravity, organize both entanglement and dynamics across subregions. By formulating generalized Noether charges and their algebras, it shows how quantum geometry emerges from representations of the corner symmetry group, connecting seamlessly with Loop Quantum Gravity’s boundary data and twisted geometries. A key result is that the Barbero–Immirzi parameter β induces a noncommutative boundary geometry and a Lorentz-covariant, discrete area spectrum via the SL(2,R)^S (or SL(2,C)^S) corner symmetry, with representations labeled by punctures carrying (j,Δ,s) quantum numbers. The analysis of null boundaries reveals Carrollian fluid dynamics on the boundary, with a boundary Lagrangian encoding radiation and a boundary symplectic structure matching the bulk–boundary corner-symmetry picture, thereby unifying black-hole entropy, holography, and quantum geometry in a local, continuum setting.

Abstract

By virtue of the Noether theorems, the vast gauge redundancy of general relativity provides us with a rich algebra of boundary charges that generate physical symmetries. These charges are located at codimension-2 entangling surfaces called corners. The presence of non-trivial corner symmetries associated with any entangling cut provides stringent constraints on the theory's mathematical structure and a guide through quantization. This report reviews new and recent results for non-perturbative quantum gravity, which are natural consequences of this structure. First, we establish that the corner symmetry derived from the gauge principle encodes quantum entanglement across internal boundaries. We also explain how the quantum representation of the corner symmetry algebra provides us with a notion of quantum geometry. We then focus our discussion on the first-order formulation of gravity and show how many results obtained in the continuum connect naturally with previous results in loop quantum gravity. In particular, we show that it is possible to get, purely from quantization and without discretization, an area operator with discrete spectrum, which is covariant under local Lorentz symmetry. We emphasize that while loop gravity correctly captures some of the gravitational quantum numbers, it does not capture all of them, which points towards important directions for future developments. Finally, we discuss the understanding of the gravitational dynamics along null surfaces as a conservation of symmetry charges associated with a Carrollian fluid.