The quantum gravity disk: Discrete current algebra

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