# Gravity as an SU(1,1) gauge theory in four dimensions

## TL;DR

This paper reformulates four-dimensional gravity as an SU(1,1) gauge theory, revealing discrete space-like and continuous time-like area spectra, and explores implications for black hole holography within Loop Quantum Gravity.

## Contribution

It introduces a partial gauge-fixing from SL(2,C) to SU(1,1) in gravity, enabling a new phase space parametrization and quantization approach in Loop Quantum Gravity.

## Key findings

- Space-like areas have discrete spectra.
- Time-like areas have continuous spectra.
- Potential for holographic black hole descriptions.

## Abstract

We start with the Hamiltonian formulation of the first order action of pure gravity with a full $\mathfrak{sl}(2,\mathbb C)$ internal gauge symmetry. We make a partial gauge-fixing which reduces $\mathfrak{sl}(2,\mathbb C)$ to its sub-algebra $\mathfrak{su}(1,1)$. This case corresponds to a splitting of the space-time ${\cal M}=\Sigma \times \mathbb R$ where $\Sigma$ inherits an arbitrary Lorentzian metric of signature $(-,+,+)$. Then, we find a parametrization of the phase space in terms of an $\mathfrak{su}(1,1)$ commutative connection and its associated conjugate electric field. Following the techniques of Loop Quantum Gravity, we start the quantization of the theory and we consider the kinematical Hilbert space on a given fixed graph $\Gamma$ whose edges are colored with unitary representations of $\mathfrak{su}(1,1)$. We compute the spectrum of area operators acting of the kinematical Hilbert space: we show that space-like areas have discrete spectra, in agreement with usual $\mathfrak{su}(2)$ Loop Quantum Gravity, whereas time-like areas have continuous spectra. We conclude on the possibility to make use of this formulation of gravity to construct a holographic description of black holes in the framework of Loop Quantum Gravity.