Dual loop quantizations of 3d gravity

Abstract

The loop quantization of 3d gravity consists in defining the Hilbert space of states satisfying the Gau{\ss} constraint and the flatness constraint. The Gau{\ss} constraint is enforced at the kinematical level by introducing spin networks which form a basis for the Hilbert space of gauge invariant functionals. The flatness constraint is implemented at the dynamical level via the Ponzano-Regge state-sum model. We propose in this work a dual loop quantization scheme where the role of the constraints is exchanged. The flatness constraint is imposed first via the introduction of a new basis labeled by group variables, while the Gau{\ss} constraint is implemented dynamically using a projector which is related to the Dijkgraaf-Witten model. We discuss how this alternative quantization program is related to 3d teleparallel gravity.