Quantum Corner Symmetry: Representations and Gluing

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.