# Dual 2+1D Loop Quantum Gravity on the Edge

## TL;DR

This paper explores a new discretization scheme for 2+1D gravity in loop quantum gravity, revealing a family of phase space polarizations, including a dual polarization related to teleparallel gravity.

## Contribution

It introduces a family of phase space polarizations in 2+1D loop quantum gravity, including a novel dual polarization linked to teleparallel gravity.

## Key findings

- Identified a dual polarization of the discrete phase space.
- Analyzed properties and symmetries of the dual polarization.
- Suggested a relation between dual polarization and teleparallel gravity.

## Abstract

In a recent paper, we introduced a new discretization scheme for gravity in 2+1 dimensions. Starting from the continuum theory, this new scheme allowed us to rigorously obtain the discrete phase space of loop gravity, coupled to particle-like "edge mode" degrees of freedom. In this work, we expand on that result by considering the most general choice of integration during the discretization process. We obtain a family of polarizations of the discrete phase space. In particular, one member of this family corresponds to the usual loop gravity phase space, while another corresponds to a new polarization, dual to the usual one in several ways. We study its properties, including the relevant constraints and the symmetries they generate. Furthermore, we motivate a relation between the dual polarization and teleparallel gravity.