# $SU(1,1)$ Barbero-like variables derived from Holst action

## TL;DR

This paper derives an $SU(1,1)$ gauge theory formulation of gravity from the Holst action with a cosmological constant, providing a Lorentz-covariant solution to second-class constraints and connecting to Ashtekar-Barbero variables.

## Contribution

It introduces a Lorentz-covariant solution to second-class constraints and formulates gravity as an $SU(1,1)$ gauge theory, extending the Ashtekar-Barbero approach.

## Key findings

- Lorentz-covariant solution to second-class constraints
- $SU(1,1)$ gauge theory formulation of gravity
- Relation to Ashtekar-Barbero variables

## Abstract

We work on a spacetime manifold foliated by timelike leaves. In this setting, we explore the solution of the second-class constraints arising during the canonical analysis of the Holst action with a cosmological constant. The solution is given in a manifestly Lorentz-covariant fashion, and the resulting canonical formulation is expressed using several sets of real variables that are related to one another by canonical transformations. By applying a gauge fixing to this formulation, we obtain a description of gravity as an $SU(1,1)$ gauge theory that resembles the Ashtekar-Barbero formulation.