# Discretization of 3d gravity in different polarizations

## TL;DR

This paper explores various polarization choices in discretizing 3d gravity within the loop quantum gravity framework, introducing new schemes based on triad variables and linking them to Chern-Simons theory and classical symmetries.

## Contribution

It presents a novel discretization approach using triad variables, connects different polarization choices, and relates the discrete Chern-Simons formulation to classical symmetries and loop gravity.

## Key findings

- New discretization scheme based on triad variables.
- Identification of the Drinfeld double as a classical symmetry.
- Link between discrete Chern-Simons and Goldman bracket.

## Abstract

We study the discretization of 3d gravity with $\Lambda=0$ following the loop quantum gravity framework. In the process, we realize that different choices of polarization are possible. This allows to introduce a new discretization based on the triad as opposed to the connection as in the standard loop quantum gravity framework. We also identify the classical non-trivial symmetries of discrete gravity, namely the Drinfeld double, given in terms of momentum maps. Another choice of polarization is given by the Chern-Simons formulation of gravity. Our framework also provides a new discretization scheme of Chern-Simons, which keeps track of the link between the continuum variables and the discrete ones. We show how the Poisson bracket we recover between the Chern-Simons holonomies allows to recover the Goldman bracket. There is also a transparent link between the discrete Chern-Simons formulation and the discretization of gravity based on the connection (loop gravity) or triad variables (dual loop gravity).