The Quantum Gravity Disk: Discrete Current Algebra

We study the quantization of the corner symmetry algebra of 3d gravity, that is the algebra of observables associated with 1d spatial boundaries. In the continuum field theory, at the classical level, this symmetry algebra is given by the central extension of the Poincar\'e loop algebra. At the quantum level, we construct a discrete current algebra based on a quantum symmetry group given by the Drinfeld double $\mathcal{D}\mathrm{SU}(2)$. Those discrete currents depend on an integer $N$, a discreteness parameter, understood as the number of quanta of geometry on the 1d boundary: low $N$ is the deep quantum regime, while large $N$ should lead back to a continuum picture. We show that this algebra satisfies two fundamental properties. First, it is compatible with the quantum space-time picture given by the Ponzano-Regge state-sum model, which provides discrete path integral amplitudes for 3d quantum gravity. The integer $N$ then counts the flux lines attached to the boundary. Second, we analyse the refinement, coarse-graining and fusion processes as $N$ changes, and we show that the $N\rightarrow\infty$ limit is a classical limit where we recover the Poincar\'e current algebra. Identifying such a discrete current algebra on quantum boundaries is an important step towards understanding how conformal field theories arise on spatial boundaries in quantized space-times such as in loop quantum gravity.