Origin of the quantum group symmetry in 3d quantum gravity

TL;DR

This paper derives the quantum group symmetry in 3d quantum gravity directly from the classical action by canonical transformation and discretization, clarifying its origin and relation to gauge invariance.

Contribution

It introduces a first-principles derivation of quantum group deformation in 3d gravity from classical actions, linking continuum and discrete formulations.

Findings

Quantum group symmetry arises from classical gauge invariance modifications.

Canonical transformation deforms gauge and boundary symmetries based on the cosmological constant.

Discretization with degree truncation reproduces quantum group structures.

Abstract

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 principle and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deformed 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.