Origin of the quantum group symmetry in 3D quantum gravity

It is well-known that quantum groups are relevant to describe the quantum regime of 3D gravity. They encode a deformation of the gauge symmetries parametrized by the value of the cosmological constant. They appear as a form of regularization either through the quantization of the Chern-Simons formulation or the state sum approach of Turaev-Viro. Such deformations are perplexing from a continuum and classical picture since the action is defined in terms of undeformed gauge invariance. We present here a novel way to derive from first principles and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deforms the gauge invariance and the boundary symmetries, and makes them depend on the cosmological constant. Second we implement a discretization procedure relying on a truncation of the degrees of freedom from the continuum.