2 + 1 gravity with a positive cosmological constant in LQG: a proposal for the physical state

Abstract

In this paper, I investigate the possible quantization, in the context of loop quantum gravity, of three-dimensional gravity in the case of positive cosmological constant Λ and try to make contact with alternative quantization approaches already existing in the literature. Due to the appearance of an anomaly in the constraints algebra, previously studied as a first step of the analysis, alternative techniques developed for the quantization of systems with constraints algebras not associated with a structure Lie group need to be adopted. Therefore, I introduce an ansatz for a physical state, which gives some transition amplitudes in agreement with what one would expect from the Turaev–Viro model. Moreover, in order to check that this state implements the right dynamics, I show that it annihilates the master constraint for the theory up to the first order in Λ.