# First-order formulation of teleparallel gravity and dual loop gravity

## TL;DR

This paper explores the relationship between loop gravity and teleparallel gravity, showing they correspond to different discretizations of the Einstein--Cartan action, with dual loop gravity naturally representing teleparallel gravity.

## Contribution

It establishes a connection between loop gravity and teleparallel gravity through discretizations of the Einstein--Cartan action, highlighting dual loop gravity as a natural discretization of teleparallel gravity.

## Key findings

- Dual loop gravity corresponds to teleparallel gravity discretization.
- Loop gravity relates to the standard metric formulation of GR.
- The Einstein--Cartan action links both formulations via boundary terms.

## Abstract

There are at least two ways to encode gravity into geometry: Einstein's general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein--Cartan action, which were studied recently in the literature. The first discretization leads to the \emph{{loop gravity}} kinematical phase space where the zero torsion condition is enforced {first} and the other is the \emph{{dual loop gravity}} kinematical phase space where curvature is imposed to vanish {first}. Our argument is based on the observation that the GR first-order Einstein--Cartan action can also be seen as a first-order action for teleparallel gravity up to a boundary term. The results of our paper suggest that the \emph{dual loop gravity} framework is a natural discretization of teleparallel gravity, whereas \emph{loop gravity} is naturally related to the standard GR metric description.