# Gravitational edge modes: From Kac-Moody charges to Poincar\'e networks

## TL;DR

This paper explores boundary surface charges in general relativity, revealing a structure of edge modes and networks that extend loop quantum gravity to include Poincaré symmetries and stringy boundary vibrations.

## Contribution

It introduces a boundary charge framework with Kac-Moody edge modes, leading to an extension of spin networks to tube networks with Virasoro and Poincaré representations.

## Key findings

- Identification of boundary electric, magnetic, and momentum charges as conservation laws.
- Development of a boundary surface charge framework with Kac-Moody edge modes.
- Introduction of Poincaré charge networks as a generalization of loop quantum gravity.

## Abstract

We revisit the canonical framework for general relativity in its connection-vierbein formulation, recasting the Gauss law, the Bianchi identity and the space diffeomorphism bulk constraints as conservation laws for boundary surface charges, respectively electric, magnetic and momentum charges. Partitioning the space manifold into 3D regions glued together through their interfaces, we focus on a single domain and its punctured 2D boundary. The punctures carry a ladder of Kac-Moody edge modes, whose 0-modes represent the electric and momentum charges while the higher modes describe the stringy vibration modes of the 1D-boundary around each puncture. In particular, this allows to identify missing observables in the discretization scheme used in loop quantum gravity and leads to an enhanced theory upgrading spin networks to tube networks carrying Virasoro representations. In the limit where the tubes are contracted to 1D links and the string modes neglected, we do not just recover loop quantum gravity but obtain a more general structure: Poincar\'e charge networks, which carry a representation of the 3D diffeomorphism boundary charges on top of the $\mathrm{SU}(2)$ fluxes and gauge transformations.