Quantum Corner Symmetry: Representations and Gluing

TL;DR

This paper explores the quantum properties of corner symmetry groups in gravity, using a 2D gravity toy model to analyze representations, central extensions, and the quantum implementation of gravitational constraints.

Contribution

It introduces a detailed study of the quantum corner symmetry group in 2D gravity, including its representations, central extensions, and a method for gluing corners at the quantum level.

Findings

Description of central extensions and Casimirs of the quantum corner symmetry group

Implementation of gravitational constraints through corner gluing in quantum theory

Analysis of a specific representation for corner gluing in 2D gravity

Abstract

The corner symmetry algebra organises the physical charges induced by gravity on codimension-$2$ corners of a manifold. In this letter, we initiate a study of the quantum properties of this group using as a toy model the corner symmetry group of two-dimensional gravity $\mathrm{SL}\left(2,\mathbb{R}\right)\ltimes \mathbb{R}^2$. We first describe the central extensions and how the quantum corner symmetry group arises and give the Casimirs. We then make use of one particular representation to discuss the gluing of corners, achieved by identifying the maximal commuting sub-algebra. This is a concrete implementation of the gravitational constraints at the quantum level.