Diffeomorphisms Versus Non Abelian Gauge Transformations: An Example of 1+1 Dimensional Gravity

TL;DR

This paper explores the phase space of 1+1 dimensional gravity modeled as a non-Abelian gauge theory, highlighting the importance of using the universal cover of SL(2,R) for correct quantization and solution classification.

Contribution

It demonstrates that choosing the universal cover of SL(2,R) instead of PSL(2,R) significantly impacts the quantization and spectrum of the model.

Findings

Using the universal cover alters the Dirac observable spectrum.

The standard Hamiltonian formulation can distinguish inequivalent gravitational solutions.

The approach provides insights into gauge group choices in gravity models.

Abstract

We investigate the phase space of a typical model of 1+1 dimensional gravity (Jackiw-Teitelboim model with cylindrical topology) using its reformulation as a non abelian gauge theory based on the sl(2,R) algebra. Modifying the conventional approach we argue that one should take the universal covering of SL(2,R) rather than PSL(2,R) as the gauge group of the theory. We discuss the consequences for the quantization of the model and find that the spectrum of the Dirac observables is sensible to this modification. Our analysis further provides an example for a gravity theory where the standard Hamiltonian formulation identifies gravitationally inequivalent solutions.