# Corner symmetry and quantum geometry

## TL;DR

This paper explores how corner symmetries in general relativity relate to quantum geometry, revealing new insights into quantum gravity, including a covariant area operator with discrete spectrum and connections to loop quantum gravity.

## Contribution

It demonstrates that corner symmetry encodes quantum entanglement and leads to a covariant, discretized area operator, bridging gauge principles with quantum geometry in gravity.

## Key findings

- Corner symmetry encodes quantum entanglement.
- Quantum representation yields a covariant discrete area spectrum.
- Connections established between continuum results and loop quantum gravity.

## 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.